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high-energy cosmic neutrinos +Damiano F. G. Fiorillo +∗ and Mauricio Bustamante +† +Niels Bohr International Academy, Niels Bohr Institute, +University of Copenhagen, DK-2100 Copenhagen, Denmark +(Dated: January 3, 2023) +The origin of the bulk of the high-energy astrophysical neutrinos seen by IceCube, with TeV– +PeV energies, is unknown. If they are made in photohadronic, i.e., proton-photon, interactions in +astrophysical sources, this may manifest as a bump-like feature in their diffuse flux, centered around a +characteristic energy. We search for evidence of this feature, allowing for variety in its shape and size, +in 7.5 years of High-Energy Starting Events (HESE) collected by the IceCube neutrino telescope, +and make forecasts using larger data samples from upcoming neutrino telescopes. +Present-day +data reveals no evidence of bump-like features, which allows us to constrain candidate populations +of photohadronic neutrino sources. +Near-future forecasts show promising potential for stringent +constraints or decisive discovery of bump-like features. Our results provide new insight into the +origins of high-energy astrophysical neutrinos, complementing those from point-source searches. +I. +INTRODUCTION +What is the origin of the bulk of the high-energy as- +trophysical neutrinos discovered by IceCube [1–7], with +TeV–PeV energies? They are likely predominantly pro- +duced by one or more populations of extragalactic sources +capable of accelerating cosmic rays to EeV-scale energies. +Yet, so far, less than a handful of sources have been iden- +tified [8–12]—though more conceivably will be [13, 14]. +Unquestionably, looking for individual sources is chal- +lenging [15], due to the need to detect coincident electro- +magnetic emission from them, incomplete catalogs, large +trial factors, and low detection rates. +To overcome these limitations, here we adopt a dif- +ferent strategy: rather than resolving individual sources, +we look, in a single swathe, for the population, or pop- +ulations, of sources responsible for the bulk of the high- +energy neutrinos. We inspect the diffuse neutrino energy +spectrum, made up of the aggregated neutrino emission +from all sources, for evidence of distinct features that may +reveal contributions to it from tributary populations. +We consider two broad classes of candidate high-energy +neutrino sources: those where neutrinos are made pri- +marily in cosmic-ray interactions with ambient matter— +i.e., proton-proton (pp) sources [16–18]—and those where +neutrinos are made primarily in cosmic-ray interac- +tions with ambient radiation—i.e., photohadronic (pγ) +sources [17, 19, 20]. +In both, neutrinos come from +the decay of the short-lived particles—pions and muons, +mostly—born from these interactions. +However, they +emit neutrinos with different energy spectra. Based on +this, we use their spectra as proxies of their contributions +to the diffuse neutrino flux. +Neutrinos from pp sources have a power-law spectrum, +inherited from their parent cosmic rays. Candidate pp +sources include starburst galaxies [21–28], galaxy clus- +ters [29–32], and low-luminosity active galactic nuclei (LL +AGN) [33, 34]. Neutrinos from pγ sources have instead a +“bump-like” spectrum, centered around an energy de- +termined by the properties of the interacting photons +and cosmic rays [35–39]. Candidate pγ sources include +gamma-ray bursts (GRBs) [36, 40–45], LL AGN [33, 34], +radio-quiet AGN (RQ AGN) [46], radio-loud AGN (RL +AGN) [47–49], BL Lacertae AGN (BL Lacs) [50, 51], flat- +spectrum radio quasars (FSRQs) [50, 52–54], and tidal +disruption events (TDEs) [55–63]. +The above classification is admittedly approximate: in +reality, most candidate source classes may produce neu- +trinos via both pp and pγ interactions, though not nec- +essarily in equal measure. However, we do not test pre- +dictions of specific source models, but the presence of +generic spectral features due to pp and pγ production in +the neutrino data. Still, we do interpret our results, with +caveats, in terms of population properties (Section V B). +Present-day IceCube data are described well by a dif- +fuse pure-power-law spectrum, i.e., Φν ∝ E−γ, with +γ ≈ 2.37 [65] or 2.87 [7], depending on the data used. +Naively, this would suggest that the diffuse flux is due to +a single population of pp sources. However, it is widely +understood that that conclusion would be premature: +the large present-day uncertainty in the measured en- +ergy spectrum might be hiding deviations from a pure +power law. +To wit, while there is no marked prefer- +ence for alternatives to a pure power law today, they +are not strongly disfavored [7]. More complex possibili- +ties are not excluded either, e.g., a two-component model +with pp sources dominating up to PeV energies and pγ +sources dominating above [28, 66], or pγ sources opaque +to gamma rays [67, 68] dominating below 60 TeV [69]. +Motivated by these works, we perform a systematic +search for the presence of power-law and bump-like dif- +fuse flux components in present-day IceCube data, and +make near-future forecasts using the combined exposure +of upcoming neutrino telescopes, up to the year 2040. We +model the power-law spectrum from the general class of +pp sources and the bump-like spectrum from the general +class of pγ sources with flexible parametrizations that +capture the rich interplay of their relative contributions. +For our present-day results, we use the recent 7.5-year +public sample of IceCube High-Energy Starting Events +(HESE) [7], which have high astrophysical purity and en- +ergy resolution, and the associated Monte-Carlo sample +arXiv:2301.00024v1 [astro-ph.HE] 30 Dec 2022 + +2 +7.5 +10 +20 +30 +40 +80 120 +Equivalent IceCube exposure [years] +2020 +2025 +2030 +2035 +Year +1 +101 +102 +103 +104 +105 +Strength of evidence for two components in ν flux, B +B +C +Barely worth mentioning +Substantial +Strong +Very strong +Decisive +IceCube (IC) +IC–Gen2 (8 IC) +Baikal-GVD (1.5 IC) +KM3NeT (2.8 IC) +P-ONE (3.2 IC) +TAMBO (0.5 IC) +TRIDENT (7.5 IC) +Using HESE only +All detectors: IceCube HESE-detection efficiency +All detectors +IceCube+Gen2 only +IceCube only +A +D +10−10 +10−9 +10−8 +10−7 +10−6 +A +7.5 years exposure +IceCube HESE +(7.5 years) +10−10 +10−8 +10−8 +10−7 +All-flavor diffuse ν flux, E2 +ν Φν [GeV cm−2 s−1 sr−1] +B 2025 (proj.) +Two-component fit +One-component fit +10−10 +10−8 +10−8 +10−7 +C 2030 (proj.) +105 +106 +107 +Neutrino energy, Eν [GeV] +10−10 +10−8 +10−8 +10−7 +D 2035 (proj.) +FIG. 1. Evidence for the existence of a PeV bump in the diffuse flux of high-energy astrophysical neutrinos. The +discovery potential is quantified via a Bayes factor (Section III) that compares the evidence for a two-component flux fit—a +power law plus bump—vs. a one-component flux fit—a power law—after marginalizing over all flux parameters (Section II D). +Present-day results (snapshot A) are obtained using the 7.5-year public IceCube HESE sample [7, 64]; the best-fit parameter +values are in Table I. Projections (snapshots B, C, D) are obtained using scaled-up event rates, adopting the present-day best-fit +two-component flux as the true flux. We assume that upcoming neutrino telescopes will have the same HESE-detection efficiency +as IceCube. Left: Evolution of discovery potential with time, using combined detector exposure. Right: Best-fit and 68% allowed +ranges of the one- and two-component flux fits for snapshots A–D. A prominent PeV bump may be discovered decisively already +by 2027, by combining IceCube, Baikal-GVD, and KM3NeT. (In contrast, constraining or discovering subdominant bumps will +require adding more detectors; see Section V.) See Section IV for details. +of simulated HESE events [64], which includes detector +details and backgrounds, and which we reweigh to test +our flux predictions. For our forecasts, given the absence +of details on upcoming detectors, we assume for them +HESE-detection capabilities with IceCube-like efficiency. +Our goal below is double. First, we show that, thanks +to larger statistics, we may soon distinguish decisively +between a single-component (pp only or pγ only) and +a multi-component (pp and pγ) description of the diffuse +neutrino flux. Second, we show that, even if future obser- +vations were to favor a dominant power-law diffuse flux +from pp sources, sub-dominant bump-like contributions +from pγ sources could still be discovered or constrained. +The latter case would in turn constrain the properties of +the pγ source population, independently of constraints +from point-source searches [15]. +Figure 1 shows how our results address the first of these +two goals (we defer details to Section IV). It shows how +the evidence in favor of a particular two-component dif- +fuse flux—a power law and a PeV bump, hinted at by +present-day data—may grow with time, assuming this +is indeed the true flux. +Already by 2027, the com- +bined exposure of IceCube, Baikal-GVD [70, 71], and +KM3NeT [72–74] could decisively favor this explanation. +Figure 1 illustrates a larger point: growing statistics will +allow us to probe progressively more inconspicuous fea- +tures of the diffuse flux, offering powerful discrimination +between competing source models. +Our methods (Section III) are not dissimilar from those +used in collider physics to search for particle resonances: +like them, we hunt for statistically significant bumps in +an otherwise smooth landscape—in our case, in a power- +law neutrino spectrum. Discovering a bump would signal +the existence of a population of pγ sources. Not finding +any would constrain their contribution. In both cases, +the power of the method grows with statistics. +In Section II, we review the current state of the diffuse +flux of high-energy neutrinos and introduce parametriza- + +3 +tions for the power-law and bump-like flux components. +In Section III, we describe the present-day IceCube data +and the methods we use to compare them to our flux +predictions. In Section IV, we focus on the case of a PeV +bump. In Section V, we focus on subdominant bumps +in the TeV–PeV range. In Section VI, we list possible +future directions. In Section VII, we summarize. +II. +THE DIFFUSE FLUX OF +HIGH-ENERGY ASTROPHYSICAL NEUTRINOS +The sources responsible for the diffuse flux of TeV–PeV +astrophysical neutrinos seen by IceCube are unknown. +Yet, different neutrino production mechanisms, promi- +nent in different candidate source classes, are expected to +make neutrinos with different energy spectra. Thus, we +use the diffuse neutrino spectrum as proxy of the identity +of the population, or populations, of neutrino sources. +A. +Overview: one or more source populations? +At present, because the origin of the bulk of high- +energy astrophysical neutrinos is unknown, models of +their diffuse flux are many and varied. +Viable models +must be able to explain the diffuse flux seen by Ice- +Cube [7, 65], or a fraction of it. But, beyond that con- +straint, there is significant room for variety in the predic- +tions of the flux shape and size from various candidate +astrophysical sources; see, e.g., Ref. [14] for an overview. +Further, the diffuse neutrino flux could conceivably be +the superposition of contributions from multiple source +populations, each contributing a flux component with +a differently shaped energy spectrum and size. (Refer- +ences [75–77] estimated the size of these contributions, +though based on searches for point and stacked sources, +rather than on the diffuse flux.) Identifying these com- +ponents in the diffuse flux—and, hence, identifying the +contribution of multiple source classes—requires distin- +guishing between their different spectral shapes. Later, +in Sections IV and V, we show that the main challenge +to do that is the paucity in high-energy neutrino data. +Fortunately, this will be surmounted in the near future. +Below, we consider two broad classes of candidate +sources that roughly map to two different neutrino pro- +duction mechanisms: sources that make neutrinos via +proton-proton interactions and sources that make neu- +trinos via proton-photon interactions. Later, we look for +their imprints in the diffuse flux of high-energy neutrinos. +B. +Neutrinos from pp vs. pγ sources +Because the diffuse flux of TeV–PeV astrophysical +neutrinos seen by IceCube is seemingly isotropic, the +astrophysical sources responsible for it are likely pre- +dominantly extragalactic. +Their identity is presently +unknown; except for a few notable exceptions [8–12]. +They are purportedly high-energy non-thermal astro- +physical sources able to accelerate cosmic-ray protons +and charged nuclei to energies of at least 100 PeV; see, +e.g., Refs. [78, 79] for an overview. Thus, in many mod- +els, they are also sources of ultra-high-energy cosmic rays +(UHECRs) and high-energy gamma rays. For simplicity, +we frame our discussion below in terms of UHECR pro- +tons; however, the mass composition of UHECRs is key +to understanding the production of UHECRs and of the +associated high-energy neutrinos [80–82]. +In the sources, diffusive shock acceleration may gener- +ate UHECRs with a power-law spectrum ∝ E−γp +p +, where +Ep is the proton energy and γp ≳ 2. +UHECRs inter- +act with ambient matter, in proton-proton (pp) interac- +tions, or ambient radiation, in photohadronic (pγ) inter- +actions. Both pp and pγ interactions make high-energy +pions that, upon decaying, make the high-energy neutri- +nos that IceCube detects, i.e., π+ → µ++νµ, followed by +µ+ → e++νe+¯νµ, and their charge-conjugated processes. +Each neutrino carries, on average, 5% of the energy of the +parent proton, i.e., Eν ≈ Ep/20. However, while both pp +and pγ interactions can produce high-energy neutrinos, +they may yield markedly different neutrino spectra. +In pp interactions, deep inelastic scattering produces +multiple π+ and π−, in roughly equal proportions. Be- +cause UHECR protons collide with ambient protons that +are comparatively at rest, the resulting neutrino spec- +trum is entirely determined by the spectrum of the high- +energy protons. Thus, the neutrino spectrum emitted by +a pp source is a power law ∝ E−γ +ν , where γ ≈ γp, up to +a maximum neutrino energy Eν,cut = Ep,max/20, where +Ep,max is the maximum energy to which the source can +accelerate UHECR protons. The latter depends on the +properties of the source that drive particle acceleration, +e.g., the size of the acceleration region, the bulk Lorentz +factor in it, the intensity of the magnetic field, and the +fraction of available of energy that is imparted to non- +thermal protons; see, e.g., Refs. [39, 83, 84]. +In pγ interactions, UHECR protons interact with am- +bient photons whose spectrum is concentrated around a +characteristic photon energy E⋆ +γ. +The value of E⋆ +γ de- +pends on the origin of the ambient photon field, e.g., +synchrotron or synchrotron self-Compton emission by ac- +celerated electrons or protons. Pion production via the +∆(1232) resonance dominates around center-of-mass en- +ergy of 1.232 GeV, i.e., p + γ → ∆+ → n + π+, and, +at higher energies, deep inelastic scattering yields multi- +ple π+ and π−, in roughly equal proportions. Thus, the +neutrino spectrum emitted by a pγ source stems from +the interplay of the interacting protons and photons: +to produce a ∆+ resonance, their energies must satisfy +EpEγ ≈ 0.2 GeV2; see, e.g., Refs. [20, 39]. Hence, most +∆+-producing pγ interactions occur between photons of +energy E⋆ +γ and protons of energy E⋆ +p ≈ 0.2 GeV2/E⋆ +γ. +As a result, the neutrino spectrum is bump-like, con- +centrated around the characteristic neutrino energy of +Eν,bump ≈ E⋆ +p/20 ≈ 0.01 GeV2/E⋆ +γ. (The high-energy + +4 +neutrino spectrum from certain classes of pp sources, like +starburst galaxies [85], might be a power law with a spec- +tral kink. In those cases, it may also be approximated by +a bump, centered at the spectral kink; see Fig. 2.) +In our analysis, we do not model the intrinsic proper- +ties of pp or pγ sources, particle acceleration, radiation +processes, or specific shapes of the ambient photon field +that are integral to building complete source models of +neutrino production. Instead, for pp sources, we model +directly the neutrino spectra that they emit as a power +law ∝ E−γ +ν +augmented with an exponential suppression +around Eν,cut. For pγ sources, we model directly the neu- +trino spectra that they emit as a bump-like flux, centered +at Eν,bump. This strategy allows us to describe many dif- +ferent candidate pp and pγ source populations under a +common, albeit simplified, framework (see Section VI for +proposed refinements). We give details in Section II D. +The shapes of the neutrino spectra above are for in- +dividual pp or pγ sources. However, the diffuse neutrino +flux from a population of pp sources or pγ sources is ex- +pected to approximately retain the shape of the energy +spectra emitted by the individual sources that make up +the population—a power-law flux or a bump-like flux, re- +spectively. In the diffuse flux, the spectral features of in- +dividual sources are averaged by the spread in the source +properties that affect UHECR acceleration and neutrino +production—luminosity, density, magnetic field intensi- +ties, etc.—and are softened by the effect of cosmological +expansion on the neutrino energies, and by the distri- +bution of sources with redshift. Nevertheless, the fun- +damental difference between the diffuse neutrino energy +spectra from a population of pp and pγ sources remains +and is what motivates us to model them as two differ- +ently shaped flux components, a power law and a bump. +By varying the values of their shape parameters in fits +to data (more on this later), we capture the interplay +between them and, indirectly, the effects of spectral av- +eraging and softening on them. +C. +Overview of source candidates +Presently, it is unknown whether the diffuse flux of +high-energy astrophysical neutrinos seen by IceCube is +due to a single population of pp sources, a single popula- +tion of pγ sources, or a superposition of pp and pγ source +populations. For comparison, the unresolved diffuse flux +of GeV–TeV gamma rays is likely due to various popu- +lation of sources, see, e.g., Refs. [86–88], including unre- +solved blazars, star-forming galaxies, and radio galaxies, +which, incidentally, may also be neutrino sources. In con- +trast, identifying the contributions from multiple source +populations in the diffuse flux of high-energy neutrinos +is hampered by low neutrino event rates. Nevertheless, +weak hints in present-day observations of the diffuse neu- +trino flux suggest that different source populations may +contribute at different energies. We sketch them below. +In the 10–100 TeV range, the flux of astrophysical neu- +trinos seen by IceCube suggests an origin in pγ sources +that are opaque to gamma rays [69]. +These sources +must be opaque, i.e., must attenuate gamma rays via +electron-positron pair production, in order for the flux +of gamma rays co-produced with neutrinos not to ex- +ceed the isotropic diffuse gamma-ray background seen +by Fermi-LAT [67–69]. +Various candidate pγ sources +with potential high opacity have been proposed, includ- +ing low-luminosity and choked GRBs [43, 89–91] and su- +pernovae [92–94], and hidden cores of AGN [95]. Notably, +in AGN corona models [95] neutrino production via pγ +and pp might be comparable. Nevertheless, because our +analysis uses detected events with energies above 60 TeV +(Section III A)—to reduce the contamination of atmo- +spheric backgrounds—it is largely insensitive to bumps +that peak below this energy. +Around 100 TeV, the flux of astrophysical neutrinos +seen by IceCube may originate in pp sources. +Exam- +ples include cosmic reservoirs, like star-forming galax- +ies [21, 24, 27, 28, 32, 96, 97] and galaxy clusters [30– +32, 98, 99]. Cosmic reservoirs are believed to be cosmic- +ray calorimeters: they confine cosmic rays for a long time, +boosting their chances of interacting with interstellar ma- +terial and making neutrinos. They can explain the co- +incidence observed between the energy generation rate +of UHECRs and of high-energy neutrinos. However, for +some of these sources, e.g., star-forming galaxies, it is +challenging to model the acceleration of UHECRs and, +therefore, the production of PeV-scale neutrinos [24, 27]. +In the PeV range, pγ sources like blazars [50–54, 100], +GRBs [36, 40, 42, 44, 101], and TDEs [55–63], may domi- +nate neutrino production. This is expected because these +sources are all candidate UHECR accelerators, and they +are all known to contain eV–MeV photon fields that can +act as targets for photohadronic interactions. (There are +also models of PeV-scale neutrino production via pp in- +teractions of UHECRs on nuclei from the host galaxy; +see, e.g., Ref. [85].) +Thus, the picture that tentatively emerges is that a +low-energy population of pγ sources may dominate neu- +trino production below 100 TeV—though our analysis +is largely insensitive to it—pp sources may dominate it +up to a few hundred TeV, and a different population of +pγ sources may dominate it at higher energies, up to a +few PeV. References [28, 66, 102, 103] proposed multi- +component flux models based on this tentative picture. +In short, above 60 TeV, where our analysis is sensitive, +the diffuse neutrino flux may be a power law up to a few +hundred TeV, followed by a bump centered at PeV en- +ergies. Indeed, in Sections IV and V, we find marginal +evidence for this in present-day IceCube data. Still, as +part of our analysis, we explore many alternative super- +positions of a power law and bump flux components. + +5 +D. +Power-law and bump flux components +Following the tenet of our work, laid out in Sec- +tion II B, we forego modeling in detail the neutrino emis- +sion from individual pp and pγ sources and computing the +diffuse neutrino flux from the aggregated contributions of +their populations. Instead, we directly model the diffuse +neutrino flux without recourse to any particular source +model. This strategy allows us to describe a vast number +of possible superpositions of pp and pγ neutrino source +populations within the same framework. +In practice, our goal is to assess the evidence in fa- +vor of the existence of a bump in an otherwise feature- +less power-law diffuse flux of high-energy astrophysical +neutrinos. Discovering a bump would be evidence of a +pγ source population (or of a pp source population with +a spectral kink; see Section II B). Hence, we model the +diffuse flux as the superposition of two components: a +power-law flux, representative of neutrino production in +pp sources, and a log-parabola bump-like flux, represen- +tative of neutrino production in pγ sources (or pp sources +with a spectral kink). By construction, the parametriza- +tions that we adopt have the flexibility to capture the +variety in the interplay between power laws and bumps +of various shapes and relative sizes. Below, we describe +them. Later, when computing the evidence for a bump in +Sections III–V, we vary the values of the flux parameters. +The diffuse power-law flux component is +E2 +ν +dΦPL +dEνdAdtdΩ = Φ0,PL +� +Eν +100 TeV +�2−γ +e +− +Eν +Eν,cut , +(1) +where Φ0,PL is a normalization parameter, γ is the spec- +tral index, and Eν,cut is the neutrino cut-off energy. +Equation (1) describes the diffuse flux of neutrinos pro- +duced in pp interactions of UHECRs that have a rel- +atively soft spectrum ∝ E−γp +p +with γp ≳ 2, as ex- +pected from diffusive shock acceleration [104, 105]; see- +Section II B. Below, instead of modeling specific flux pre- +dictions, we vary the values of Φ0,PL, γ, and Eν,cut in fits +to present-day and projected samples of detected events. +The diffuse bump-like flux component is +E2 +ν +dΦbump +dEνdAdtdΩ = +� +E2 +ν,bumpΦ0,bump +� +× exp +� +−αbump log2 +� +Eν +Eν,bump +�� +,(2) +i.e., a log-parabola, where E2 +ν,bumpΦ0,bump is a normal- +ization parameter, Eν,bump is the energy at which the +bump is centered, and αbump defines the width of the +bump, which is approximately Eν,bump/α1/2 +bump. Most of +the neutrinos are concentrated around energy Eν,bump. +The value of αbump controls whether the spectrum is +wide around this energy—if αbump is small—or narrow— +if αbump is large. Equation (2) represents the diffuse flux +of neutrinos produced in pγ interactions (or in pp inter- +actions with a spectral kink); see Section II B. Below, +instead of modeling specific flux predictions, we vary the +values of E2 +ν,bumpΦ0,bump, αbump, and Eν,bump in fits to +present-day and projected samples of detected events. +Figure 2 compares our log-parabola bump-like flux, +Eq. (2), with detailed models of the diffuse high-energy +neutrino emission from various classes of sources, taken +from the literature, both pγ—blazars [50], low-luminosity +GRBs [106], and TDEs [107]—and pp sources—starburst +galaxies [85]. +These models illustrate that, in reality, +bumps may be asymmetric around Eν,bump and may fea- +ture a plateau rather than a peak. For the case of TDEs, +for example, the flux at energies below the peak flattens +out due to a contribution from pγ interactions on a sec- +ond target of X-ray photons, which is not captured by our +parametrization, Eq. (2). We leave searches for these fea- +tures to future dedicated studies (Section VI). Figure 2 +shows that, in all cases, the log-parabola bump-like flux, +Eq. (2), is a reasonable fit to the flux models, especially +close to the peak of the bump, where the flux component +contributes the most to the rate of detected events, and +especially for more symmetric model predictions. This +validates the use of Eq. (2) in our analysis. +An alternative origin of a bump in the diffuse flux, +from beyond the Standard Model, is from the decay of +heavy dark matter particles between 100 TeV and 10 PeV +into high-energy neutrinos [108–120]. Dark-matter decay +would yield a neutrino spectrum that peaks at an energy +determined by the mass of the dark matter particle, with +a width determined by its decay width and by the dis- +tribution of its dark-matter density with redshift. While +our analysis below focuses on bumps as coming from the +neutrino production mechanism, it can be repurposed to +perform searches for neutrino bumps from dark-matter +decay. +Previous studies, e.g., Refs. [117, 118], have +shown that including a bump-like high-energy neutrino +flux component from the decay of PeV-scale dark matter +can marginally improve fits to IceCube data. Below, we +find a similar result, though motivated differently. +(A +separate issue is that, for most dark matter decay chan- +nels, gamma rays co-produced with neutrinos may be in +tension with observations; see, e.g., Refs. [121, 122].) +Thus, our diffuse flux model is two-component, the +superposition of the power-law and bump components, +Eqs. (1) and (2), i.e., +E2 +νΦν(Eν) ≡ E2 +ν +�dΦPL(Eν; Φ0,PL, γ, Eν,cut) +dEνdAdtdΩ ++ +dΦbump(Eν; E2 +ν,bumpΦ0,bump, αbump, Eν,bump) +dEνdAdtdΩ +� +.(3) +The physical parameters of our model are Φ0,PL, γ, +Eν,cut, E2 +ν,bumpΦ0,bump, αbump, and Eν,bump. Later, in +our statistical analysis in Section III, we introduce addi- +tional nuisance parameters, related to atmospheric neu- +trino and muon backgrounds. Table I summarizes the +free parameters of our analysis. +We assume that the diffuse flux is made up of νe, νµ, ντ, +¯νe, ¯νµ, and ¯ντ in equal proportions. This is the canonical + +6 +105 +106 +107 +Neutrino energy, Eν [GeV] +10−10 +10−9 +10−8 +10−7 +10−6 +All-flavor diffuse ν flux, E2 +νΦν [GeV cm−2 s−1 sr−1] +IceCube 7.5-yr HESE (PRD 2021) +Blazars (Palladino et al., 2018) +GRBs (Tamborra et al., 2015) +TDEs (Winter et al., 2022) +SBGs (Condorelli et al., 2022) +Flux from source model +Log-parabola bump-like approximation +FIG. 2. +Diffuse neutrino fluxes from representative +source models of neutrino production via pγ and pp in- +teractions vs. approximations using the bump-like flux +from our work, Eq. (2). For blazars (mainly pγ), the flux +is scenario 1 from Ref. [50], with constant baryon loading +for all sources. For gamma-ray bursts (GRBs, mainly pγ), +the flux is from low-luminosity bursts, from Ref. [106]. For +tidal disruption events (TDEs, mainly pγ), the flux is from +Ref. [107], including interactions with optical and ultraviolet +photons. For starburst galaxies (SBGs, mainly pp), the flux +is from Ref. [85], from pp interactions of UHECRs. For com- +parison, we show the 68% allowed flux band from the 7.5-year +IceCube HESE analysis, assuming a pure power-law [7]. We +do not test the source flux models shown in this figure, nor +any specific source flux models; we show them here merely as +representative examples to validate Eq. (2). Our log-parabola +bump-like flux approximates the source flux models reasonably +well, especially where they are highest, and especially if they +are symmetric around their peak energy. +expectation for high-energy neutrinos produced in pion +decays (Section II B), after flavor oscillations have acted +on them en route to Earth [123, 124], and is compati- +ble with IceCube measurements of the flavor composi- +tion [125]. Present uncertainties in the values of the neu- +trino mixing parameters lead to uncertainties in the pre- +dicted flavor composition of the neutrino flux, but they +should be rendered negligible in the next decade by up- +coming oscillation experiments [124], so we ignore them. +Figure 3 illustrates the role of the different flux param- +eters on the shape of the neutrino diffuse flux, and singles +out the impact that varying the width, αbump, has on the +bump component. Also, the dip in the flux in-between +the cut-off of the power-law component and the rise of +the bump component is a feature that could reflect the +transition from a pp to a pγ source population. +105 +106 +107 +Neutrino energy, Eν [GeV] +10−10 +10−9 +10−8 +10−7 +10−6 +All-flavor diffuse ν flux, E2 +νΦν [GeV cm−2 s−1 sr−1] +Power law +Bump +αbump = 10 +αbump = 0.5 +Φ0,PL +γ +Eν,cut +E2 +ν,bumpΦ0,bump +α−1/2 +bump +Eν,bump +FIG. 3. +Illustration of the power-law and bump flux +components used in our analysis, Eqs. (1)-(3), and ef- +fect of their free parameters. +For this figure only, the +values of the flux parameters are fixed to their best-fit values +obtained in a two-component flux fit to the public 7.5-year +IceCube HESE data [7, 64]; see Table I. In our analysis, we +allow the values of the parameters to vary in fits to data, ei- +ther present-day or projected. See Table I for a summary of +the parameters and Section II D for details. +The IceCube Collaboration itself has explored various +possible shapes of the diffuse neutrino spectrum when +fitting to detected data, including their default pure +power law, i.e., one without a high-energy cut-off, a dou- +ble power law, a pure log-parabola, a segmented power +law, and fluxes from different astrophysical models; see +Refs. [3–5, 7, 65, 126–130]; see also Refs. [28, 66, 102, 103] +for independent analyses. Present-day statistics are in- +sufficient to yield a conclusive preference for any of these +models. Below, we reach the same conclusion when com- +paring the present-day preference for a one-component +flux model vs. a two-component flux model. +III. +HUNTING FOR BUMPS +We look for bump-like features in the diffuse flux +of high-energy neutrinos by using IceCube High-Energy +Starting Events (HESE), with high astrophysical purity. +We account for detector effects and the irreducible con- +tamination from atmospheric neutrinos and muons by +using the public IceCube Monte Carlo HESE sample to +compute event rates. We scan wide ranges of possible +values of the flux model parameters (Table I) and, when +computing evidence, account for the appearance of spu- + +7 +rious bump-like features (the “look-elsewhere effect”). +A. +IceCube High-Energy Starting Events (HESE) +IceCube is the largest high-energy neutrino telescope +in operation: roughly 1 km3 of underground Antarctic +ice instrumented with photomultipliers. It is an optical +Cherenkov detector: it collects the light made by radi- +ating secondary particles in showers born from neutri- +nos interacting in the ice. The main interaction chan- +nel is neutrino-nucleon deep inelastic scattering (DIS). +In it, a high-energy neutrino scatters off of a constituent +parton of the nucleon—a quark or a gluon—and breaks +up the nucleon in the process. +The high-energy final- +state particles—electrons, muons, tauons, and hadrons— +initiate showers whose charged particles emit Cherenkov +radiation that propagates through the ice and is recorded +by the photomultipliers. From the amount of light de- +posited, and from its spatial and temporal distribution, +IceCube reconstructs the neutrino energy, direction, and +flavor, with varying degrees of precision [131]. +In our analysis, we focus on IceCube High-Energy +Starting Events (HESE). These are events where the neu- +trino interaction occurs inside the instrumented volume. +They undergo a self-veto that reduces the contamina- +tion from atmospheric muons, which would otherwise be +dominant. By design, HESE samples are the most astro- +physically pure out of all of the event samples selected +by IceCube. See Refs. [2, 3, 132, 133] for details. This, +coupled to the fact that their energy resolution is high +(more on this below), makes them the most suitable kind +of events to look for features in the neutrino energy spec- +trum. +Later (Section VI), we comment on the use of +the other main event sample, of through-going muons. +When making projections that involve other detectors, +we assume that they will also collect HESE samples, a +capability that they will arguably likely have, and that +their HESE-detection efficiency will be equal to that of +IceCube, which is admittedly a necessary simplification, +born from the absence of details on future detectors. +In the TeV–PeV range, there are two main light-profile +topologies of HESE events: cascades and tracks. Cas- +cades are made mainly by charged-current DIS of νe or +ντ (i.e., νl + N → l + X, where l = e, τ, N is a nucleon, +and X are final-state hadrons), and also by neutral- +current DIS of all flavors (i.e., νl + N → νl + X, where +l = e, µ, τ). Tracks are made by charged-current DIS of +νµ (i.e., νµ + N → µ + X), where the final-state muon +leaves an track of light in its wake, km-scale in length. +In a DIS interaction, +on average, +the final-state +hadrons receive about 25% of the initial neutrino energy, +and the final-state lepton receives 75% [134–136]. Thus, +in cascades, essentially all of the neutrino energy is de- +posited in the ensuing shower, which grants them good +energy resolution. In tracks, because the track escapes +the instrumented detector volume, energy resolution is +somewhat poorer (but the muon energy can be approxi- +mated by how much energy the track deposits inside the +detector [131]). The energy resolution of HESE events +is about 10% in the logarithm of the event energy. Con- +versely, because cascades have a roughly spherical light +profile centered on the neutrino interaction vertex, their +angular resolution may be as poor as tens of degrees, +while tracks, because they are elongated, have sub-degree +angular resolution. For details, see Ref. [131]. +At a few PeV, in addition, charged-current DIS of ντ +may produce “double bangs” or “double cascades” [137]. +In them, the neutrino-nucleon DIS produces a first cas- +cade; the final-state tauons propagate away from the in- +teraction vertex, decay, and produce a second cascade. +Recently, IceCube identified the first two candidate dou- +ble bangs [138]. However, they are not captured by the +public IceCube HESE Monte Carlo sample on which we +base our analysis [7, 64] (Section III B). +Beside neutrino-nucleon DIS, high-energy neutrinos +are also detected via neutrino-electron scattering. This +interaction channel is negligible except in a narrow +energy +range +around +6.3 +PeV—the +Glashow +reso- +nance [139]—where ¯νe may produce an on-shell W boson, +which enhances the expected event rate massively [140– +146]. Recently, IceCube observed the first Glashow reso- +nance candidate [147]. The public IceCube HESE Monte +Carlo sample that we use (Section III B) does contain +contributions from Glashow resonance, but it does not +contain the dedicated analysis that was needed to dis- +cover that one candidate, which was a partially contained +shower, rather than a fully contained one. +Thus, IceCube HESE events are cascades, tracks, and +double cascades; we keep this classification also when +making forecasts for future detectors. Because we have +assumed equal proportion of νe, νµ, and ντ in the flux +(Section II D), we do not attempt to infer the flavor com- +position from the relative numbers of events of different +classes, like Refs. [123, 124, 128, 129, 138, 148–150] do. +After neutrinos reach the surface of the Earth, they +propagate underground, for a length of up to the diame- +ter of the Earth, until they reach IceCube. Inside Earth, +they undergo DIS on nucleons [134–136], which damp- +ens their flux. The effect is stronger at higher energies, +where the neutrino-nucleon cross section is larger, and +for neutrinos that travel longer paths inside the Earth, +which encounter a larger column depth of nucleons. For +ντ in particular, charged-current DIS produces tauons +which decay back into ντ with lower energy, that partially +counteract the dampening of its flux; this is known as “ντ +regeneration.” While this effect is present in our analy- +sis, it is significant mainly at energies above 100 PeV, +higher than the ones we use. Overall, the propagation of +high-energy neutrinos inside the Earth affects their flux +in an energy-, direction-, and flavor-dependent manner; +see, e.g., Ref. [134–136, 151–154] for explicit examples. +The above effects are built into the public IceCube +HESE Monte Carlo sample that we use in our analysis. +The sample is generated assuming the neutrino-nucleon +cross section from Ref. [155], for the propagation of neu- + +8 +trinos inside Earth and their detection at IceCube, and +the Preliminary Earth Reference Model [156] for the in- +ternal matter density of Earth. For details, see Ref. [64]. +B. +HESE public data and Monte Carlo +Recently, the IceCube Collaboration made public the +7.5-year HESE sample [7] and an accompanying Monte +Carlo (MC) simulation of the performance of the detec- +tor [64]. We build our analysis on them. +The 7.5-year HESE sample contains 102 events in to- +tal. In our analysis, we use only the 60 events with re- +constructed shower deposited energy larger than 60 TeV; +there are 41 cascades, 17 tracks, and 2 double cascades. +Above 60 TeV, the irreducible contamination from at- +mospheric neutrinos and muons that pass the HESE self- +veto (Section III C) is small [132, 133, 157], since their +fluxes decrease faster with energy than the flux of astro- +physical neutrinos. Because of the event selection, most +events are downgoing, i.e., coming from the Southern +Hemisphere. For details, see Ref. [7]. +The HESE MC sample contains 821764 simulated +HESE events, generated using the same detector simu- +lation used in the analysis of the 7.5-year HESE sample +by the IceCube Collaboration. They are initiated by all +neutrino flavors, produce cascades, tracks, and double +cascades, from all directions, and cover the energy range +that is relevant for our analysis. +Events in the MC sample were generated assuming a +reference diffuse high-energy astrophysical neutrino flux; +see Ref. [64] and Section III E. In our analysis, we com- +pute HESE events corresponding to different choices of +the high-energy astrophysical neutrino flux by reweighing +the events in the MC sample; we describe the procedure +in Section III D 1. Thus, our predicted event rates inher- +ently include the detailed IceCube HESE response. +Compared to the 7.5-year analysis by the IceCube +Collaboration [7], we adopt a simplified treatment of +three nuisance detector systematic uncertainties—the ef- +ficiency of digital optical modules, the head-on efficiency, +and the lateral efficiency—in order to reduce the time +needed for our computations. Whereas the IceCube anal- +ysis allows the values of these parameters to float in fits +to observed data, with narrow prior distributions, we +keep their values fixed to their nominal expectations, i.e., +where their priors are maximum. (For the same reason, +we also keep the shapes of the atmospheric background +distributions fixed; see Section III C.) In Section III E, we +verify that the impact of fixing their values is limited, by +approximating closely the IceCube fit from Ref. [7]. +For +each +simulated +event +in +the +MC +sample, +we use its primary neutrino quantities—neutrino en- +ergy, flavor, and zenith angle—and its reconstructed +event quantities—reconstructed deposited energy, recon- +structed zenith angle, and event topology. In our statisti- +cal analysis (Section III D), we compare predicted vs. ob- +served event rates using reconstructed quantities, since +these are accessible experimentally, but reweigh events +in the MC sample using primary neutrino quantities. +C. +Irreducible atmospheric backgrounds +The HESE sample contains events initiated not only by +astrophysical neutrinos, but also by the irreducible back- +ground flux of atmospheric neutrinos and muons, created +in cosmic-ray interactions in the atmosphere of the Earth, +that escape the HESE self-veto [132, 133, 157]. There +are three contributions to it—conventional atmospheric +muons, conventional atmospheric neutrinos, and prompt +atmospheric neutrinos—born from the decay of mesons +and muons produced by the cosmic rays. For all of them, +we use the same flux prescriptions as the IceCube 7.5- +year HESE analysis [7], via their implementations in the +HESE MC sample. Below, we sketch them; for details, +see Ref. [7], especially Figs. IV.3, IV.4, and IV.6 therein. +The conventional atmospheric muon flux comes from +the decay of pions and kaons. Compared to the parent +cosmic rays, the atmospheric muon spectrum is softer +due to the energy losses of the pions and kaons prior to +their decaying and of the muons themselves. The baseline +muon flux prescription that we use comes from air-shower +simulations made with CORSIKA [158], using the Hillas- +Gaisser H4a cosmic-ray flux model [159] and the Sibyll +2.1 hadronic interaction model [160]. +The conventional atmospheric neutrino flux comes +from the decay of pions, kaons, and muons. +Like the +conventional atmospheric muon flux, because of energy +losses, its spectrum is softer than that of the parent cos- +mic rays. The baseline conventional neutrino flux pre- +scription that we use is from Ref. [161], obtained using +the modified DPMJET-III generator [162]. +The prompt atmospheric neutrino flux comes from the +decay of charmed mesons. Because they are short-lived, +they experience little to no energy losses before decaying. +As a result, the spectrum of prompt neutrinos that they +produce is harder than that of conventional atmospheric +neutrinos, and closer to that of the parent cosmic rays. +The baseline prompt neutrino flux prescription that we +use is from Ref. [163]. To date, the prompt neutrino flux +remains unobserved; still, we include its possible contri- +bution to the HESE rate. +Accordingly, in all our fits +to HESE data below, we find that the contribution of +prompt atmospheric neutrinos is compatible with zero. +Later, +as +part +of +our +statistical +analysis +(Sec- +tion III D 2), we let the normalization of the conventional +muon flux, conventional neutrino flux, and prompt neu- +trino flux float freely in fits to HESE data, like the Ice- +Cube analysis in Ref. [7], and using the same priors. +Reference [7] included extra parameters that affect the +shape, not just the normalization, of the energy spec- +tra of the atmospheric backgrounds: the spectral index +of the cosmic-ray spectrum, the ratio of kaons to pions +produced, and the ratio of neutrinos to anti-neutrinos +produced. In our analysis, we keep these shape parame- + +9 +ters fixed at their nominal values, given in Table IV.1 of +Ref. [7], to reduce the time needed for our computations. +This is justified because the atmospheric backgrounds are +subdominant in the HESE event rate above 60 TeV, i.e., +in the energy range of our analysis. Like for detector sys- +tematics (Section III B), we verify in Section III E that +the impact of fixing the shape parameters is limited. +D. +Statistical procedure +Our analysis compares expected HESE event rates— +induced by our two-component astrophysical neutrino +flux model (Section II D) and by atmospheric back- +grounds (Section III C)—against the public IceCube 7.5- +year HESE sample [7, 64] (Section III B), and against +projected versions of it with larger statistics. To com- +pute event rates for arbitrary flux choices, we reweigh +the HESE MC sample and, when making projections, +re-scale it by longer detector exposure times. To com- +pare event-rate predictions with observations, we adopt +a Bayesian approach, binned in reconstructed event en- +ergy and direction (Section III B), and allow astrophys- +ical and background flux parameters (Table I) to float +freely. Below, we describe this in detail. +1. +Astrophysical neutrinos +The set of flux parameters introduced in Section II D, +θ ≡ +� +Φ0,PL, γ, Eν,cut, E2 +ν,bumpΦ0,bump, αbump, Eν,bump +� +, +defines a specific realization of our two-component diffuse +flux of high-energy astrophysical neutrinos, Eq. (3). In +the fits to HESE data below, we let the value of each +parameter float independently of each other. +For a given realization of θ, we compute the expected +mean number of HESE events due to the corresponding +astrophysical neutrino flux by reweighing the sample of +MC HESE events; we explain the reweighing procedure +below. After reweighing, the mean number of astrophys- +ical events in the i-th bin of reconstructed shower en- +ergy, Edep, and the j-th bin of reconstructed direction, +cos θrec +z , is µν,ast +ij,t (θ); we introduce our choice of binning +later (Section III D 3). We treat events of each topology +(t) separately, i.e., cascades (c), tracks (tr), and double +cascades (dc). We do the same for atmospheric events. +The flux-reweighing procedure is as follows: from the +public HESE data release [7, 64], we extract the weight +wk +ref,t associated with the k-th MC event of topology t, +generated by a neutrino of energy Eν,k. Events in the MC +sample were originally generated assuming as reference +flux the best-fit pure-power-law flux from the 7.5-year +HESE analysis [7], Φν,ref = Φ0,ref(Eν/100 TeV)−2.87, +with Φ0,ref = 5.68 × 10−18 GeV−1 cm−2 s−1 sr−1, and +exposure time Tref = 2635 days. Given a new flux Φν, +i.e., our two-component model in Eq. (3), and exposure +time T, the mean number of events of topology t is +µν,ast +ij,t (θ) = +� +k +Φν(Eν,k, θ)T +Φν,ref(Eν,k)Tref +wk +ref,t , +(4) +where the sum is restricted to MC events whose recon- +structed deposited energy, Edep,k, falls within the i-th +bin and whose reconstructed deposited direction, cos θrec +z,k, +falls within the j-th bin. +2. +Atmospheric neutrinos and muons +To account for the irreducible atmospheric background +(Section III C), we extract from the IceCube HESE MC +sample [64] the baseline number of conventional atmo- +spheric neutrinos, N ν,c +ij,t, prompt atmospheric neutrinos, +N ν,pr +ij,t , and atmospheric muons, N µ +ij,t. (In practice, we +do this by setting the astrophysical flux to zero in the +MC reweighing, and extracting the resulting event rates, +which are purely atmospheric.) The baseline atmospheric +event rates in the MC sample were produced using the +MC generator of Ref. [164]; see Section III C for details. +We keep the shape of the atmospheric background +event distributions fixed (Section III C), but allow their +normalization constants, N ν,c, N ν,pr, and N µ, to float +independently of each other. For a specific choice of their +values, the number of background events of topology t is +µatm +ij,t (η) = N ν,cN ν,c +ij,t + N ν,prN ν,pr +ij,t + N µN µ +ij,t , +(5) +where η ≡ (N ν,c, N ν,pr, N µ). +3. +Likelihood function +The mean number of HESE events of topology t in +each bin, of astrophysical and atmospheric origin, is +µij,t(θ, η) = µν,ast +ij,t (θ) + µatm +ij,t (η) . +(6) +We use the same binning as in Ref. [7]: NEdep = 21 bins +evenly spaced in log10(Edep/GeV), between 60 TeV and +10 PeV, and Ncθrec +z += 10 bins evenly spaced in cos θrec +z , +between -1 and 1. +To compare our predicted HESE event rate, µij,t, +vs. the observed 7.5-year HESE sample or projected ver- +sions of it, N data +ij,t , we use a binned Poissonian likelihood, +L (θ, η) = +NEdep +� +i=1 +Ncθrec +z +� +j=1 +{c,tr,dc} +� +t +Lij,t(θ, η) , +(7) +where the likelihood in each bin, for event topology t, is +Lij,t(θ, η) = µij,t(θ, η)N data +ij,t +N data +ij,t ! +e−µij,t(θ,η) . +(8) + +10 +TABLE I. Free model parameters, their priors, best-fit values and allowed ranges, from a fit to the IceCube 7.5-year +HESE event sample [7, 64]. Allowed parameter ranges are 68% one-dimensional marginalized credible intervals. Results +are for fits with a pure power law (“Pure PL”), a power law with an exponential cut-off (“PL w/ cut-off”), and a power law +with a cut-off plus a bump (“PL + B”). The former two serve as validation of our method; we find parameter values similar +to Ref. [7]. For the latter, we only show the values of the parameters that maximize the posterior, Eq. (9). (We keep some +nuisance parameters of the original HESE analysis[7] fixed to their nominal values.) See Sections III E and IV A for details. +Parameter +Prior +Fit to 7.5-yr IceCube HESE sample +Symbol +Units +Description +Pure PLa +PL w/ cut-offb +PL + Bc +Physical parameters, θ +Power law +Φ0,PL +GeV−1 cm−2 s−1 sr−1 +Flux norm. at 100 TeV +Log10-uniform ∈ [−20, −15] +5.9+2.1 +−1.1 +5.9+1.7 +−1.3 +1.5 × 10−17 +γ +— +Spectral index +Uniform ∈ [2.0, 3.5] +2.88 ± 0.21 +2.76+0.27 +−0.22 +2.3 +Eν,cut +GeV +Cut-off energy +Log10-uniform ∈ [4, 8] +— +8.9+72.4 +−7.5 × 106 +1.7 × 105 +Bump +E2 +ν,bumpΦ0,bump +GeV cm−2 s−1 sr−1 +Flux norm. at Eν,bump +Log10-uniform ∈ [−10, −5] +— +— +3 × 10−8 +αbump +— +Energy width of bump +Uniform ∈ [0.1, 10] +— +— +3.4 +Eν,bump +GeV +Central energy of bump +Log10-uniform ∈ [4, 7] +— +— +1.4 × 106 +Nuisance parameters, η +N ν,c +— +Flux norm., convent. atm. ν +Gaussian, µ = 1,σ = 0.4 +1.08 ± 0.39 +1.09 ± 0.39 +0.96 +N ν,pr +— +Flux norm., prompt atm. ν +Uniform ∈ [0, 10] +0.94+0.39 +−0.90 +0.90+0.27 +−0.83 +0.17 +N µ +— +Flux norm. atm. µ +Gaussian, µ = 1, σ = 0.5 +1.20 ± 0.46 +1.24+0.38 +−0.50 +1.05 +a Mean value of the one-dimensional marginalized posterior ± 68% C.L. range. The mean value coincides with the best-fit value. +b Mean value of the one-dimensional marginalized posterior ± 68% C.L. range. The mean value coincides with the best-fit value. +c Best-fit, or maximum a posteriori, value of the full posterior. Because of correlations between parameters in the full posterior, Eq. (9), +this value does not coincide with the mean value when using the 7.5-year IceCube HESE sample; see Fig. B1. +The likelihood in Eq. (7) accounts for the contribution of +events in all energy and direction bins, and of all topolo- +gies. The associated posterior probability distribution is +P(θ, η) = L (θ, η) π (θ) π (η) +Z +, +(9) +where π(θ) and π(η) are the prior distributions for the +astrophysical-flux parameters, θ, which are physical, and +of the atmospheric-background parameters, η, which are +nuisance. In Eq. (9), the denominator is the evidence, +i.e., the posterior marginalized over all parameters, +Z = +� +dθ +� +dη L (θ, η) π (θ) π (η) . +(10) +We use UltraNest [165], an efficient importance nested +sampler [166, 167], to maximize the posterior, find the +best-fit and allowed ranges of parameter values, and com- +pute the evidence. +4. +Parameter priors and look-elsewhere effect +Table I summarizes our choice of priors. For the phys- +ical parameters, θ, we adopt uniform priors over wide +ranges to avoid introducing bias in the fit. We use log- +uniform priors for the flux normalization of the power-law +and bump components, Φ0,PL and E2 +ν,bumpΦν,bump, the +energy of the exponential cut-off of the power law, Eν,cut, +and the central energy of the bump, Eν,bump. This allows +them to more easily vary over wide ranges of values in +order to capture a vast array of possibilities for the rela- +tive contributions of the power-law and bump-like com- +ponents. For the power-law spectral index, we restrict +γ ≥ 2, as typically expected for pp sources with diffusive +shock acceleration (Section II). For the energy width of +the bump, αbump, we choose αbump > 0.1, to avoid intro- +ducing bumps so wide as to be mistaken for hard power +laws over the entire energy range of our analysis, and +αbump < 10, since narrower bumps are likely unrealistic; +see Appendix A for details. +For the nuisance parameters, η, we adopt the same +priors used in the IceCube 7.5-year HESE analysis [7], +which are extracted from Ref. [168]. They represent the +uncertainty in the underlying models of cosmic-ray spec- +trum and hadronic interaction. For the prompt neutrino +flux normalization, N ν,pr, we adopt a uniform prior up to +10, rather than a positive unbounded one as in Ref. [7]. +Since our fits below are all compatible with N ν,pr = 0 +(see Table I), our use of a more restrictive prior does not +modify our results significantly compared to Ref. [7]. +In analogy with searches for resonances in collider data, +in searching for bump-like features in the diffuse high- +energy neutrino spectrum we must account for the trials +factor, or “look-elsewhere effect.” This is the decrease in +the statistical significance with which the existence of a +bump can be claimed due to the possibility of there be- +ing spurious bump-like features, mere random statistical +fluctuations of the event rate, anywhere in the energy +range that is relevant to our analysis. +In a Bayesian +approach like ours, integrating the likelihood over wide +prior ranges in order to compute the evidence, Eq. (10), +automatically accounts for the look-elsewhere effect by +penalizing large prior volumes. + +11 +5. +Bump discovery Bayes factor +We evaluate the preference for a two-component, +power-law-plus-bump flux model (PL+B), Eq. (3), vs. a +one-component, power-law flux model (PL), Eq. (1), via +the Bayes factor +B = ZPL+B +ZPL +. +(11) +We compute the evidence ZPL+B using Eq. (9), and the +evidence ZPL using Eq. (9) with E2 +ν,bumpΦ0,bump = 0, +i.e., with only the power-law flux component. The higher +the value of B, the higher the preference of the data for +the two-component flux model. Broadly stated, narrow +bumps are hard to identify—unless they are very tall— +because they only affect the event rate within a narrow +energy window, while wide bumps are hard to identify be- +cause they may resemble a power law. In-between these +extremes, discovery may be more feasible. We adopt Jef- +freys’ criteria to classify the preference qualitatively into +barely worth mentioning, 100 ≤ B < 100.5; substantial, +100.5 ≤ B < 101; strong, 101 ≤ B < 101.5; very strong, +101.5 ≤ B < 102; and decisive, B ≥ 102. +Our likelihood, Eq. (7), is valid but approximate. +Because our predicted astrophysical HESE event rates +are obtained by reweighing the HESE MC sample (Sec- +tion III D 1), Ref. [169] proposed using a more sophisti- +cated, though computationally expensive, likelihood pre- +scription that accounts for random fluctuations intrinsic +to the MC sample itself. However, in our analysis, we +forego this after verifying, in Section III E below, that our +approach reproduces closely the best-fit values and al- +lowed intervals reported in the analysis performed by the +IceCube Collaboration [7] using a one-component power- +law-flux fit to the 7.5-year HESE sample. +E. +Validation: power-law fits to present-day data +As validation of our method, we fit a pure power law +and a power law with exponential cut-off to the 7.5-year +HESE event sample, as in the IceCube analysis in Ref. [7]. +Table I shows the best-fit and 68% confidence inter- +vals for the free parameters in each case (“Pure PL” and +“PL w/ cut-off”). In both cases, our results approximate +those of Ref. [7]. +For the power law with exponential +cut-off, the best-fit value of Eν,cut is at a few PeV, as +in Ref. [7], but has a large uncertainty, so it should be +treated only as a weak suggestion, which we do below. +IV. +A PEV BUMP? +First, we apply our methods above to the present-day, +7.5-year IceCube HESE data sample. We find marginal +preference (B ≈ 0.7) for a one-component flux model—a +power law flux with a cut-off at a few hundred TeV— +vs. a two-component flux model. +Then we adopt the +best-fit two-component flux that we find—a steep power +law with a bump centered at roughly 1 PeV—as template +for a possible real two-component flux scenario. We use +it to forecast what detector exposure would be needed +to discover a PeV bump, which would require combining +contributions of several neutrino telescopes. +A. +Bump-hunting in present-day IceCube data +Applying the statistical procedure introduced in Sec- +tion III D to the present-day, 7.5-year IceCube HESE +sample, we find a value for the Bayes factor, Eq. (11), +of B = 0.7 ± 0.1. Following Jeffreys’ criteria, this rep- +resents mild preference for a one-component power-law +flux, to explain the data vs. a two-component power-law- +plus-bump flux. Table I shows the best-fit values of the +model parameters and their allowed ranges in each case, +i.e., “PL w/cut-off” vs. “PL + B”. Appendix B contains +details on the full posterior for the latter case. +Figure 1 (also Fig. 3) shows the present-day best-fit +two-component flux: a steep power law, with γ = 2.75 +and cut-off at Eν,cut = 170 TeV, followed by a prominent, +relatively wide bump centered at Eν,bump = 1.4 PeV. +This flux explains the dearth of HESE events between +300 TeV and 1 PeV (see snapshot A in Fig. 1) by this +being the energy range where the power-law flux from a +population of pp sources vanishes and before the bump- +like flux from a population of pγ sources becomes ap- +preciable. +A PeV bump could be indicative, e.g., of +blazars [50], low-luminosity GRBs [106], or TDEs [107] +as sources of PeV-scale neutrinos; see Fig. 2 +Although we find evidence against a two-component +flux model to explain the present-day HESE data, in +what follows we entertain the possibility that instead the +present-day best-fit two-component flux is borderline pre- +ferred, for two reasons. First, the present-day preference +against the two-component flux is only marginal. Small +changes to the priors, data, or analysis methods, could +conceivably change the value of B = 0.7±0.1 that we find +into B ≳ 1, which would represent no preference between +the one-component and two-component flux models, or +marginal preference for the latter. Second, our preference +for a two-component flux with a PeV bump is compati- +ble with similar results from previous works [28, 66, 130], +obtained using different methods or event samples (see +also Ref. [118] for an origin in dark matter decay). +Thus, below we adopt our best-fit two-component flux +to forecast the near-future prospects of discovering a PeV +bump, using larger HESE samples. (Later, in Section V, +we consider bumps centered at other energies.) + +12 +10−1 +100 +101 +102 +Astrophysical ν: +Power-law flux +Atm. muons +Atm. ν, conventional +Atm. ν, prompt +Astrophysical ν +Astrophysical ν: +Power-law flux +105 +106 +Deposited energy, Edep [GeV] +10−1 +100 +101 +102 +Astrophysical ν: +Power-law + PeV bump flux +−1.0 +−0.5 +0.0 +0.5 +1.0 +Reconstructed direction, cos θrec +z +Astrophysical ν: +Power-law + PeV bump flux +Expected HESE event rate (2035, all detectors, proj.) +FIG. 4. +Projected HESE event rates by the year 2035. The combined detector exposure is due to all the neutrino telescopes +expected at that time (see Fig. 1): Baikal-GVD [70, 71], IceCube, IceCube-Gen2 [170], KM3NeT [72–74], P-ONE [171], +TAMBO [172], and TRIDENT [173]. Events are distributed in reconstructed deposited energy (left) and reconstructed direction +(right). Top: Assuming a power law with a high-energy cut-off, with flux parameters fixed at their present-day best-fit values +(“PL w/ cut-off” in Table I). Bottom: Assuming a power law with a high-energy cut-off plus a PeV bump, with flux parameters +fixed at their present-day best-fit values (“PL + B” in Table I). The HESE-detection efficiency of upcoming detectors is assumed +to be equal to that of IceCube today [7]; their combined exposure by 2035 is equivalent to 159 years of IceCube exposure. +B. +Modeling near-future neutrino telescopes +1. +Assumptions about future neutrino telescopes +Following Section IV A, we forecast the discovery +prospects of our best-fit two-component flux (Table I) +based on larger HESE event sample made possible by +upcoming TeV–PeV neutrino telescopes, currently in op- +eration, construction, and planning stages [14]. Because +detailed information about their detection capabilities, +or simulations of them, are not publicly available at the +time of writing, and because all of them are in-water or +in-ice optical Cherenkov detectors (with the exception of +TAMBO [172], see below), we model each as a re-scaled +version of IceCube. While this simple procedure admit- +tedly does not capture the differences between detector +designs, photomultiplier efficiency, backgrounds, atten- +uation and scattering length of light in water and ice, +systematic errors, and analysis techniques, it allows us +to produce informed estimates of upcoming event rates. +Figure 1 shows the effective volume of each detector, +relative to IceCube, and their tentative start dates, which +may change. +By 2030, we expect nearly an order-of- +magnitude increase in the combined detector exposure +to high-energy astrophysical neutrinos, thanks to the +continuing operation of IceCube and the completion of +Baikal-GVD [70, 71] and KM3NeT [72, 74]. After 2030, +we expect a faster growth of the event rate thanks to +the construction of new detectors IceCube-Gen2 [170], +P-ONE [171], TAMBO [172], and TRIDENT [173]. +To compute future event samples of a detector, we re- +scale the number of events in the IceCube MC sample +by a factor equal to the size of the detector relative to +IceCube. We only account for the contribution of a detec- +tor after it has reached its full target size; by doing this, +we ignore possible contributions from partially finished +detector configurations, which may be small. +Given the commonalities between detectors, we safely +assume that they will all be capable of detecting HESE or +HESE-like events. (This is less clear for TAMBO, which +is the only detector among the ones that we consider that +is a surface array of water Cherenkov tanks. However, +TAMBO, whose science case is specific to multi-PeV ντ +detection, represents only a small contribution to the to- +tal event rate.) Further, we assume that their efficiency +to detect HESE events will be the same as in IceCube. +This is likely an optimistic assumption, which implies +that the bump discovery prospects that we find later are, + +13 +too. This assumption could be revisited in revised fore- +casts, as details on upcoming detectors become available. +Below, we sketch the relevant features of each detector. +(Combining multiple neutrino telescopes at differ- +ent locations into a global monitoring system, like +PLEνM [174], would also increase the field of view to +high-energy neutrinos and significantly boost the chances +of discovering point sources. See Ref. [174] for details.) +2. +Overview of near-future neutrino telescopes +Baikal-GVD [70, 71], the successor of Baikal NT- +200 [175], is an in-water detector currently under con- +struction in Lake Baikal, Russia. +It has been operat- +ing in partial configuration since 2018; in 2022, its effec- +tive volume was about 0.35 km3. Recently, it reported +the detection [176] of a high-energy astrophysical neu- +trino from the TXS 0506+056 blazar previously observed +by IceCube [8, 9], and of the IceCube diffuse flux of +high-energy astrophysical neutrinos, with a significance +of about 3σ [177]. We assume a start date for the full +Baikal-GVD of 2025, with an effective volume of 1.5 km3. +IceCube-Gen2 [170] is the envisioned upgrade of Ice- +Cube. We consider its in-ice optical array, composed of +120 new detector strings, that will extend the effective +volume of IceCube. Because the new strings will be more +sparsely deployed than in IceCube, the HESE detection +efficiency of IceCube-Gen2 might be lower; this is not +captured by our forecasts. (There is an additional en- +visioned radio-detection component that targets the dis- +covery of ultra-high-energy neutrinos [178].) We assume +a start date for the full IceCube-Gen2 optical array of +2030, with an effective volume of 8 km3. +KM3NeT [72, 74], the successor of ANTARES [179], is +an in-water detector currently under construction in the +Mediterranean Sea. Its high-energy component, ARCA, +targets high-energy astrophysical neutrinos. Of the 230 +detection units planned at ARCA, 19 units are already +deployed and operating in 2022. It is expected that a +building block of 115 units will be able to measure the +diffuse flux detected by IceCube in about a year of ob- +servation. We assume a start date for the full KM3NeT +of 2025, with an effective volume of 2.8 km3. +P-ONE [171], the Pacific Ocean Neutrino Experiment, +is an in-water detector, currently under planning and pro- +totyping, to be deployed in the Cascadia Basin, Canada. +P-ONE will have 70 detector strings with 20 detector +modules each, instrumented over a cylindrical volume +with radius 1 km and height 1 km. The first prototype +string is expected to be deployed in 2023. We assume a +start date for the full P-ONE of 2030, with an effective +volume of 3.2 km3. +TAMBO [172], the Tau Air-Shower Mountain Based +Observatory, +is a proposed surface array of water +Cherenkov tanks to be located in a canyon in Peru. It tar- +gets Earth-skimming ντ with energies of 1–100 PeV that +interact on one side of the canyon and produce a high- +energy tauon whose decay triggers a particle shower that +is detected on the opposite of the canyon. The detection +strategy of TAMBO is different from IceCube, and its en- +ergy range, while overlapping, extends to higher values. +However, because detailed simulations are unavailable at +the time of writing, we model it as a small version of +IceCube. We assume a start date for the full TAMBO of +2030, with a target effective volume of 0.5 km3. +TRIDENT [173], The tRopIcal DEep-sea Neutrino +Telescope, is a proposed in-water detector to be located +in the South China Sea. +TRIDENT is expected to +be able to detect a transient neutrino source like TXS +0506+056 [8, 9] with 10σ significance and the steady- +state neutrino source NGC 1068 [12] within two years of +operation. We assume a start date for the full TRIDENT +of 2030, with a target effective volume of 7.5 km3. +C. +Projected discovery prospects +In the near future, the increased event statistics pro- +vided by the combined exposure of the above detec- +tors will enhance our ability to discriminate between a +one-component and a two-component diffuse flux model. +To quantify this, below we forecast and compare future +HESE event rates for both flux models. For benchmark- +ing, we assume that the true diffuse flux is the present- +day best-fit two-component flux found in Section IV A—a +steep power law followed by a PeV bump. We follow the +same procedure detailed in Section IV A to compute the +projected Bayes factor, Eq. (11), that compares the evi- +dence for the benchmark two-component flux vs. the evi- +dence for the one-component flux. We do this for increas- +ing values of the IceCube-equivalent combined detector +exposure, as delineated in Section IV B, from halfway +through the year 2017—the end of data-taking of the +7.5-year HESE data sample—to the year 2035; see Fig. 1 +To produce our forecasts, we assume that the future +observed event rates coincide with the expected event +rates, which amounts to using an Asimov data sam- +ple [180] to find representative results for the Bayes fac- +tor. In a real future event sample, Poisson fluctuations +would naturally be present, which could bias the value +of the Bayes factor. By using an Asimov data sample, +we obtain the median value of the logarithm of the ev- +idence, Eq. (10), for each flux model, i.e., ZPL+B and +ZPL in Eq. (11). If the distribution of the Bayes factor +is Gaussian, as expected from the central limit theorem, +this median value coincides with the expected value. Fur- +ther, for growing detector exposure, the relative size of +the Poisson fluctuations in the observed event sample +shrinks by a factor of 1/ +√ +N, where N is the total num- +ber of observed events, so that their impact on the Bayes +factor wanes at longer exposures. +Figure 4 shows, as illustration, the event rates ex- +pected in 2035 assuming as true flux the present-day +best-fit one-component flux and best-fit two-component +flux (“PL w/ cut-off” and “PL + B” in Table I, respec- + +14 +tively). The combined detector exposure corresponds to +roughly 159 years of equivalent IceCube HESE exposure +(see Fig. 1) and is due to all of the neutrino telescopes +that we consider (Section IV B). The energy distributions +of the events for the one-component and two-component +cases are noticeably different. As expected, for the latter +there is a visible excess of events in the PeV region due to +its PeV bump. In contrast, the angular distributions of +events are nearly identical, since they are mostly driven +by the isotropy of the high-energy astrophysical neutrino +fluxes and by neutrino absorption inside Earth. Differ- +ences in the energy spectra between the two cases only +affect the angular distributions indirectly, by changing +the intensity of neutrino attenuation inside Earth; these +differences are small in the TeV–PeV range. +Figure 1 shows how the Bayes factor grows with com- +bined detector exposure. +Its rate of growth increases +when new detectors are added to the combined expo- +sure; in Fig. 1, this is seen as a kink in the slope of +the Bayes factor curve. As expected, because of growing +event rates, the longer the exposure, the clearer the sep- +aration between the evidence for the one-component and +two-component flux fits. We illustrate the growing sepa- +ration via four snapshots of the best-fit and 68% allowed +bands of the fluxes, A–D, from present-day to 2035. +We conclude that the combined exposure of IceCube, +Baikal-GVD, and KM3NeT may provide decisive evi- +dence in favor of a two-component flux with a PeV bump +already by 2027. +(This is contingent on future detec- +tors having IceCube-like HESE-detection capabilities; see +Section IV B.) Alternatively, IceCube plus IceCube-Gen2 +may achieve the same by 2031. In any case, a prominent +population of pγ sources of PeV neutrinos could be dis- +coverable in the diffuse flux within only a few years. +V. +HUNTING FOR TEV–PEV BUMPS +Section IV explored the discovery of a prominent PeV +bump in the diffuse high-energy neutrino flux, which is +only marginally disfavored by present-day HESE data. +Next, we use the same statistical methods to explore the +more general case of constraining or discovering a bump +of varying size anywhere in the TeV–PeV range. +A. +Constraining subdominant bumps +If future HESE observations were to still favor a one- +component power-law description of the diffuse flux, we +could place upper limits on the height of a coexistent +bump component, which must be necessarily subdom- +inant so that it does not disrupt the preference for a +power-law description. We compute the limits as follows. +For given values of the position of the bump, Eν,bump, +which we vary in Fig. 5, and of its width, which we keep +fixed at the representative value of αbump = 1 in the main +text, we compute the posterior under the two-component +105 +106 +107 +Central energy of ν bump, Eν,bump +10−10 +10−9 +10−8 +10−7 +10−6 +ν bump height, E2 +ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] +Projected upper limits (95% C.L.): +Present-day best-fit +power law ∝ E−γ +ν e−Eν/Eν,cut +(γ = 2.75, Eν,cut = 4 PeV) +68% C.L. 2035, all detectors (proj.) +All limits: bump width αbump = 1 +IceCube HESE 7.5 yr (95% C.L.) +2025, IceCube only +2035, IceCube only +2035, IceCube+Gen2 only +2035, all detectors +FIG. 5. Upper limits on the height of a bump in the dif- +fuse flux of high-energy astrophysical neutrinos. The +bump flux component, Eq. (2), is centered at energy Eν,bump, +has height E2 +ν,bumpΦν,bump, and width αbump = 1, and is over- +laid on a power-law flux ∝ E−γ +ν +eEν/Eν,cut, with parameter +values given by the best fit to the 7.5-year IceCube HESE +sample [7] (“PL w/ cut-off” in Table I), shown for comparison. +See Section II D and Fig. 3 for the definitions of the flux com- +ponents. Today, IceCube limits the height of a bump centered +at a few hundred TeV to be, at most, comparable to the size of +the dominant power-law component. In the future, the upper +limit may be tightened to tens of percent of the power-law com- +ponent. Figure 6 shows how this translates into constraints on +candidate neutrino source populations. See Section III for the +statistical analysis and Section V A for details on this plot. +flux model, Eq. (9), and marginalize it over all the free +model parameters (see list in Table I), except for the +bump height, E2 +ν,bumpΦ0,bump. We integrate the resulting +one-dimensional marginalized posterior to find the 95% +credible interval on the bump height, for each value of the +bump position. Differently from our previous results, in +drawing constraints on the bump height we adopt a flat +linear prior on it, rather than a logarithmic one. (Oth- +erwise, because the posterior is flat for arbitrarily low +values of the bump height, limits drawn using a logarith- +mic prior would differ depending on our arbitrary choice +of the lower end of the logarithmic prior.) +Figure 5 shows the results. Present-day limits, based +on the 7.5-year IceCube HESE sample, disfavor especially +the presence of relatively wide bumps (αbump = 1) cen- +tered around 200 TeV, where event statistics are higher. +For all values of Eν,bump, the limit lies above the present- +day best-fit power-law flux, meaning that a sizable con- +tribution to the diffuse flux from a population of photo- +hadronic sources cannot presently be excluded. The lim- + +15 +its are weaker for bumps centered at lower energies, where +the atmospheric background is higher, and at higher en- +ergies, where statistics are poorer. The weakening above +500 TeV reflects the fact that a two-component flux with +a bump between hundreds of TeV and a few PeV is only +marginally disfavored in present-day data (Section IV A). +Figure 5 shows limits for αbump = 1, but marginal- +izing over αbump yields comparable results; see Fig. C2 +in Appendix C. If the dominant power-law component is +harder, e.g., ∝ E−2.5 +ν +, the limits weaken at low energies +and strengthen at high energies, but the overall conclu- +sions are unchanged; see Fig. D1 in Appendix D. +The limits are expected to strengthen with more statis- +tics, made available by the continued operation of Ice- +Cube and by upcoming detectors. We forecast limits us- +ing larger combined detector exposure. To do this, we as- +sume that the true diffuse flux coincides with the present- +day best-fit power-law flux, “PL w/ cut-off” in Ta- +ble I. For upcoming detectors, we use the same IceCube- +equivalent exposures as in Section IV B. We choose two +reference years, 2025, using IceCube only, and 2035, us- +ing IceCube only, IceCube plus IceCube-Gen2, and the +combination of all detectors available by then (see Fig. 1). +Figure 5 shows that future HESE data may finally limit +the bump height to be a fraction of the size of the domi- +nant power-law component, especially at energies below +1 PeV. The limits strengthen roughly as the square root +of the ratio of future combined exposure to present-day +exposure. Unlike present-day limits, they do not weaken +above 500 TeV because they are obtained from Asimov +event samples generated assuming a power-law flux and, +therefore, are by design inconsistent with the presence +of a bump. Figure 5 shows that by 2035, IceCube could +limit the height of a bump with αbump = 1 and centered +at 100 TeV to be 86% of the present-day best-fit power- +law component; combined with IceCube-Gen2, 66%; and, +combining all detectors, 47%. +The above findings reveal promising power, accessible +by 2035 and with IceCube alone, to constrain a poten- +tial dominant contribution of photohadronic sources at +around 100 TeV. With the help of future detectors, con- +straints may improve by about a factor of 2 by 2035 +and apply also to bumps at PeV energies, contingent +on having IceCube-like HESE-detection capabilities (Sec- +tion IV B). Below, we discuss what these limits entail for +the properties of candidate source populations. +B. +Constraints on source populations +In Section II, we motivated the existence of a bump-like +component in the diffuse flux as coming from a popula- +tion of sources that make neutrinos via pγ interactions. +Below, we translate the upper limits that we found in +Section V A on the bump height into upper limits on the +local (i.e., redshift z = 0) high-energy neutrino lumi- +nosity density of candidate pγ source populations. The +translation depends on the values of the bump parame- +ters. As benchmark, we pick Eν,bump = 1 PeV for the +central energy of the bump and αbump = 1 for its width. +Appendix A shows how we relate the size of the diffuse +neutrino flux to the local neutrino luminosity density. We +show results for steady-state sources only, though similar +results can be obtained for transient sources. +Figure 6 shows results using present-day, 7.5-year Ice- +Cube HESE data, and the same projections of combined +detector exposure as in Fig. 5. Following Ref. [181], we +consider three different possibilities for the redshift evo- +lution of the source luminosity density, representative of +different candidate source classes: no evolution, evolu- +tion following the star-formation rate (SFR) [182], and +strong, FSRQ-like evolution [183]. Each source class is +assumed to be independently dominant, i.e., to saturate +the local high-energy neutrino luminosity density [181]. +Present-day point-source limits from IceCube [170, +181] already disfavor FSRQs, BL Lacs, and galaxy clus- +ters as the dominant source class. +In contrast, our +present-day limits from bump search are too weak to con- +strain any of the candidate source classes in Fig. 6 as the +dominant pγ emitter of PeV neutrinos. This is consistent +with our finding in Section V A that present-day data al- +low for a bump taller than the power-law component. +By 2035, the situation evolves favorably for our limits +from bump search. There, our limits match the power +of point-source limits drawn from ten years of IceCube- +Gen2. If there is indeed no evidence for a PeV bump, +our limits using the combined exposure of IceCube plus +IceCube-Gen2 could put to test the independent domi- +nance, as PeV pγ sources, of all the source classes in +Fig. 6. +In fact, using IceCube alone already provides +nearly the same power (although, if IceCube-Gen2 is +present, its contribution quickly becomes dominant after +2035). The combined detector exposure of all detectors +by 2035 affords even more stringent limits. +Since Fig. 5 shows that the projected limits on the +bump height strengthen for bumps centered at hundreds +of TeV, we expect the corresponding limits on the lu- +minosity density of pγ sources that emit those bumps +to strengthen, too. +Similarly, the limits on the lumi- +nosity density for wider and narrower bumps, and for +a harder power-law flux component, trace the limits on +bump height shown in Figures C2 and D1. +C. +Discovering bumps +In Section V B, we placed limits on the height of bumps +in the diffuse flux if no evidence for them is found. Now +we answer a related question: if a bump exists, how large +should the detector exposure be to discover it? +Like in Section V B, we take the true flux to be +the present-day best-fit power-law flux (“PL w/ cut-off” +in Table I), but now we add a subdominant bump to +it. +We vary the bump position, Eν,bump, and height, +E2 +ν,bumpΦν,bump, and, like before, we fix the bump width +at a representative value of αbump = 1. For each choice of + +16 +1040 +1042 +1044 +1046 +1034 +1035 +1036 +1037 +1038 +Lum. density, z = 0 [erg s−1 Mpc−3] +Projected upper limits (95% C.L.): +All limits: bump width αbump = 1 +No redshift evolution +2025, IceCube only +2035, IceCube only +2035, IceCube+Gen2 only +2035, all detectors +1040 +1042 +1044 +1046 +Source luminosity [erg s−1] +Point sources +(IceCube, 10 yr) +Point sources +(IC-Gen2, 10 yr) +SFR redshift evolution +IceCube HESE 7.5 yr (95% C.L.) +1040 +1042 +1044 +1046 +Strong redshift evolution +FIG. 6. Upper limits on the local (i.e., redshift z = 0) high-energy neutrino luminosity density of steady-state source +candidates. Our new limits apply to pγ source populations that emit a diffuse neutrino spectrum with a bump centered at +Eν,bump = 1 PeV and with width αbump = 1; they are interpretations of the limits from Fig. 5. Within a population, all sources +are identical; they have the same neutrino luminosity in their rest frame. We show candidate source classes without distinction +between mostly pp and mostly pγ sources. In each panel, the neutrino luminosity density evolves with redshift differently: no +evolution (left), star-formation rate (SFR) evolution (center), and strong (FSRQ-like) evolution (right); see Appendix A. For +each source class, its local luminosity density is chosen to saturate the present-day high-energy neutrino flux [181]. Our limits +put this assumption to test. Limits from searches for point neutrino sources are from Ref. [170]. Our limits show that by 2035 +the combined exposure IceCube plus IceCube-Gen2, or of all available detectors, could constrain the source luminosity density +of pγ to a fraction of what is needed to saturate the diffuse flux at 1 PeV. See Section V B for details. +parameter values, we compute the Bayes factor for bump +discovery, Eq. (11), following the methods in Section III. +Figure 7 shows the results computed at the same snap- +shots of combined detector exposure used in Figs. 5 and +6. For comparison, we include the present-day best-fit +power-law flux and its 68% allowed band. Our results +mirror what we found for the bump constraints in Sec- +tion V B: discovering a subdominant bump component +that is smaller than the dominant power-law component +will require the combined exposure of all the detectors +available by 2035 (see Fig. 1). Further, it will only be +possible if the bump is located in the energy region with +higher statistics, around 100 TeV. Appendix D shows +results obtained using instead a harder power-law flux, +with γ = 2.5, and no cut-off. +Figure 8 illustrates the projected 68% allowed flux +bands obtained from one-component and two-component +fits to a specific realization of the true flux, picked from +Fig. 7: a power law with a subdominant bump centered +at 141 TeV. Broadly stated, the one-component and two- +component explanations can be discriminated between +when their allowed flux bands shrink to a size compa- +rable to the difference between the true power-law flux +and the true power-law-plus-bump flux. Because such a +difference is tiny, this is only possible with the combined +detector exposure expected by 2035. +VI. +FUTURE DIRECTIONS +Using other bump shapes.— We searched for log- +parabola bumps in the diffuse flux as generic proxies of +the different bump shapes expected from different source +classes; see Fig. 2. Future dedicated searches for the im- +prints of specific photohadronic source classes could use +alternative bump shapes predicted by source models. +Varying systematic detector parameters.— In our anal- +ysis, we varied the normalization of the atmospheric neu- +trino and muon backgrounds, but fixed other parameters +associated to their shape and to detector systematics to +their nominal expectations (Section III B), in order to re- +duce the time needed for our large parameter space scans. +Nevertheless, the IceCube HESE MC sample allows for +varying them as well. Doing so would naturally reduce +the sensitivity of our analysis. Yet, the fact that in the +analysis performed by the IceCube Collaboration [7] most +of these parameters affect the fits only weakly might be +indicative of their possibly limited effect on our results. +Using other event types.— So far, our analysis has +used only HESE data. Using other event types would +come at the expense of introducing a larger atmospheric +background and poorer energy reconstruction, but may +be worth it. Including the IceCube 9.5-year sample of +through-going muons [65] would increase the statistics +massively. +Reference [174] shows an example of char- + +17 +10−9 +10−8 +10−7 +10−6 +Present-day best-fit +power law ∝ E−γ +ν e−Eν/Eν,cut +(γ = 2.75, Eν,cut = 4 PeV) +68% C.L. +2025, IceCube only (proj.) +Upper limit (95% C.L.), IceCube HESE 7.5 yr +10−9 +10−8 +10−7 +10−6 +Bump height E2 +ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] +2035, IceCube only (proj.) +10−9 +10−8 +10−7 +10−6 +2035, IceCube+Gen2 only (proj.) +105 +106 +107 +Central energy of ν bump, Eν,bump [GeV] +10−10 +10−9 +10−8 +10−7 +10−6 +All panels: bump width αbump = 1 +2035, all detectors (proj.) +0.0 +0.5 +1.0 +1.5 +2.0 +Bayes factor for bump discovery, log10 B +Substantial +Strong +Very strong +Decisive +FIG. 7. Projected discovery potential of a bump in the +diffuse flux of high-energy neutrinos. +The bump flux +component, Eq. (2), is centered at energy Eν,bump, has height +E2 +ν,bumpΦν,bump, and width αbump = 1, and is overlaid on a +power-law flux ∝ E−γ +ν +eEν/Eν,cut, with parameter values given +by the best fit to the 7.5-year IceCube HESE sample [7] (“PL +w/ cut-off” in Table I), shown for comparison. +The Bayes +factor that quantifies the discovery potential, Eq. (11), is ob- +tained in a two-component flux fit to projected event distribu- +tions, and is marginalized over the power-law flux parameters. +Figure 5 shows a corresponding plot of bump constraints; +from top to bottom, the snapshots here are the same as in +that figure. Decisive discovery of a subdominant bump may be +achieved by 2035, using IceCube-Gen2 or, more prominently, +using all planned upcoming neutrino telescopes available at the +time (see Fig. 1). See Section III for the statistical analysis +and Section V C for details on this plot. The white star (⋆) +marks the bump flux parameters chosen to make Fig. 8. +acterizing the diffuse flux using through-going muons in +10−9 +10−8 +10−7 +10−6 +2025, IceCube only (proj.) +True power law +True bump +10−9 +10−8 +10−7 +10−6 +All-flavor diffuse ν flux, E2 +ν Φν [GeV cm−2 s−1 sr−1] +2035, IceCube only (proj.) +True two-component flux ( ) +Two-component fit (68% C.L.) +One-component fit (68% C.L.) +10−9 +10−8 +10−7 +10−6 +2035, IceCube+Gen2 only (proj.) +105 +106 +107 +Neutrino energy, Eν [GeV] +10−10 +10−9 +10−8 +10−7 +10−6 +2035, all detectors (proj.) +FIG. 8. +Illustration +of +the +separation +between +a +one-component vs. a two-component fit. +We assume +as the true flux, +picked from Fig. 7 (marked with +⋆ +therein), a bump with normalization E2 +ν,bumpΦν,bump = 3.6 × +10−8 GeV cm−2 s−1 sr−1, width αbump = 1, and centered at +energy Eν,bump = 141 TeV. From top to bottom, the snap- +shots and corresponding combined detector exposure here are +the same as in Figs. 5 and 7. See Section III for the statistical +analysis and Section V C for details on this plot. +PLEνM. Including the sample of medium-energy starting +events (MESE) [126] would allow us to look for bumps +below 60 TeV. This is particularly interesting in view +of the suggested photohadronic origin of medium-energy +neutrinos; see, e.g., Refs. [32, 69]. +Using priors informed by point-source and stacked- +source searches.— To avoid introducing bias to our re- +sults above, we adopted flat, uninformed priors for the +flux parameters (Section III D and Table I). Yet, point- + +18 +source and stacked-source searches carried out in parallel +may provide complementary limits, hints, and discover- +ies on individual sources and source classes that could +be interpreted as informed priors on the power-law and +bump parameters of our analysis, strengthening it. +Considering more flux components.— Reference [66] +considered, in addition to pp and pγ neutrino flux com- +ponents of extragalactic origin, similar to ours, a pp neu- +trino component of Galactic origin, subdominant to the +other components and contributing mainly below about +1 PeV. So far, the contribution of Galactic neutrinos to +the diffuse flux is limited to be at most a few tens of per- +cent of the total [15, 184–187], but this may change with +more data. Further, ANTARES recently reported the de- +tection of TeV neutrinos from the Galactic Ridge [188]. +Thus, future versions of our analysis could include a +Galactic component, which may induce a directionally +dependent excess of events towards the Galactic Center +in the low-energy range of the event sample. +VII. +SUMMARY AND OUTLOOK +Despite important experimental advances, the origin +of the bulk of the TeV–PeV astrophysical neutrinos dis- +covered by IceCube remains unknown. Recent success +in discovering point neutrino sources [8–12], while out- +standing, accounts for only a small fraction of the to- +tal number of neutrinos detected. +Thus, we have ex- +plored a parallel strategy to probing their origin: to glean +from the shape of the diffuse energy spectrum of high- +energy neutrinos—made up of the contributions of all +high-energy neutrino sources—insight into the identity +of dominant, co-dominant, and subdominant classes of +neutrino source populations. +Motivated by previous analyses that looked for dif- +ferently shaped diffuse energy spectra [3–5, 7, 65, 126– +130] or contributions of multiple source populations to +it [28, 66], we performed a systematic search in the energy +spectrum of present-day IceCube data and made fore- +casts based on expected future data. We looked for fea- +tures that could be imprinted on the diffuse spectrum by +two broad source classes: sources that make neutrinos via +proton-proton (pp) interactions—like starburst galaxies +and galaxy clusters—and sources that make neutrinos via +photohadronic, i.e., proton-photon (pγ) interactions— +like active galactic nuclei, gamma-ray bursts, and tidal +disruption events. Generally, the former are expected to +yield a power-law flux; the latter, a bump-like flux con- +centrated around a characteristic energy (Section II B). +The strength of our analysis is triple. First, we use the +same observed and mock data as the IceCube Collabora- +tion uses in their own analysis [7, 64], including detailed +detector resolution and geometry, and atmospheric neu- +trino and muon backgrounds. Second, because we adopt +flexible spectral shapes for the power-law and bump-like +fluxes, we probe many different shapes and relative sizes +of them. Third, we extend our analysis to the expected +combined exposure of multiple upcoming neutrino detec- +tors, to deliver on the full potential of our methods. +As observed data, we use the recent IceCube 7.5- +year public HESE (High-Energy Starting Event) sam- +ple [7, 64], because of its high purity in astrophysical +neutrinos (Section III A). To test different shapes of the +diffuse spectrum, we used the public HESE Monte Carlo +event sample provided by the IceCube Collaboration [64] +(Section III B). Our statistical analysis is Bayesian, and +uses wide, unbiased priors for the model parameters to +avoid introducing bias (Section III D). +Overall, we find that hunting for bumps in the diffuse +high-energy neutrino flux may indeed reveal important +insight about a photohadronic origin of the neutrinos. +Below we summarize our findings. +Bump-hunting could test whether PeV neutrinos are +made by the same population of pp sources that make +100-TeV neutrinos, or by a separate population of pho- +tohadronic sources, a scenario that has been proposed +before [28, 66]. +We find that present-day HESE data +are best described by a power-law diffuse flux, though +that description is only marginally preferred over an al- +ternative one containing in addition a PeV bump (Sec- +tion IV A). If this bump is truly present, we find that it +could be decisively discovered already by 2027 using the +combined exposure of IceCube, Baikal-GVD [70, 71], and +KM3NeT [72, 74], or by 2031 using the combined expo- +sure of IceCube and IceCube-Gen2 [170] (Section IV B). +Even if the diffuse neutrino flux were dominated by a +population of pp sources producing a power-law flux, a +second population of photohadronic sources could still +produce a subdominant bump-like flux. +Present-day +HESE data only place weak constraints on the contri- +bution of this second population (Section V A). By 2035, +however, the combined exposure of neutrino telescopes +available at the time may limit the contribution of pho- +tohadronic sources to the diffuse flux at 100 TeV to be +no more than a few tens of percent. This would imply +upper limits on the local high-energy neutrino luminos- +ity density of photohadronic source populations, based +on the spectral shape of their flux alone (Section V B). +In contrast, discovering a subdominant bump in HESE +data, with decisive evidence, will be comparatively more +challenging. Only subdominant bumps centered around +100 TeV are likely to be discovered, and only using the +combined exposure of multiple detectors (Section V C). +Our results demonstrate the power to test the pos- +sible photohadronic origin of high-energy astrophysical +neutrinos by looking for bump-like features in the dif- +fuse flux. Our results are complementary to those from +point-source and stacked-source searches, but obtained +independently of them. In the coming years, they might +reveal not just the existence of a population of photo- +hadronic neutrino sources, but possibly also its identity. + +19 +ACKNOWLEDGEMENTS +We thank Kohta Murase for illuminating discussion. +DF and MB are supported by the Villum Fonden un- +der project no. 29388. This work used resources provided +by the High Performance Computing Center at the Uni- +versity of Copenhagen. +Appendix A: Connection between bump parameters +and astrophysical source parameters +Since the diffuse flux of high-energy neutrinos is the ag- +gregated contribution of all neutrino sources, the bump- +like diffuse flux component, Eq. (2) in the main text, is +the combination of the individual bumps emitted by all +photohadronic sources in the population. Connecting the +bump parameters—i.e., height, Eν,bump, width, αbump, +and position, Eν,bump—to the parameters that describe +the population of sources—i.e., the neutrino luminosity +density and the local number density of sources—allows +us to translate the limits that we have obtained on the +former into limits on the latter. Below we describe our +procedure; Fig. 6 in the main text shows the results. +Our approach is approximate and based on simple +physical considerations, the main one of which is that +all sources emit neutrinos with the same spectrum; the +only difference between them is the redshift at which +they are located. We leave refinements, such as using a +luminosity-dependent redshift evolution, for future work. +The comoving neutrino spectrum emitted by any one +source in the population is +E2 +ν +dNν +dEνdt = Lνω(Eν) , +(A1) +where Lν is the total neutrino luminosity, i.e., the lumi- +nosity integrated over all energies, and ω describes the +shape of the neutrino spectrum, normalized so that +� ∞ +0 +dEν +Eν +ω(Eν) = 1 . +(A2) +In what follows, we focus on a bump-like spectral shape. +The diffuse neutrino energy spectrum at Earth is +E2 +ν +dΦν +dEνdΩ = Lνnν +4π +� ∞ +0 +dz +ρ(z) +H(z)(1 + z)2 ω [Eν(1 + z)] , +(A3) +where H is the Hubble parameter, ρ is the number +density of sources, normalized so that ρ(0) = 1, and +nν is the local source number density. +We assume +a ΛCDM cosmology, with the Hubble constant H0 = +67.4 km s−1 Mpc−1, and adimensional energy density +parameters Ωm = 0.315, ΩΛ = 0.685 [189]. +In Eq. (A3), the product Lνnν is the local (i.e., at +z = 0) high-energy neutrino luminosity density that ap- +pears in Fig. 6. +The bump width, αbump, in the dif- +fuse spectrum at Earth is determined by the two factors: +the intrinsic spread in energy of the bump in ω and the +spread in redshift of the sources in the population, given +by ρ. Connected to the latter, in the right-hand side of +Eq. (A3), ω is evaluated at an energy (1+z) times higher +than at Earth to account for cosmological expansion. +On the one hand, if ω is a very narrow bump peaked +at comoving energy E⋆ +ν, then the width of the bump in +the diffuse spectrum is entirely determined by the spread +in redshift of the sources. In this case, the diffuse flux +can be approximated as +E2 +ν +dΦν +dEνdΩ = Lνnν +4π +E⋆ +ν +Eν +φ +�E0 +E − 1 +� +, +(A4) +where φ(z) ≡ ρ(z)/[H(z)(1 + z)2]. Using for ρ the star- +formation rate from Ref. [182], we have verified that the +function φ has a bump structure that can be fitted by +our log-parabola bump, Eq. (2) in the main text, with +a width αbump ≈ 2. Therefore, we conclude that bumps +much narrower than that one, with αbump ≫ 2, are not +realizable by photohadronic sources, due to the intrinsic +spread in their redshift. Of course, this conclusion de- +pends on the choice of redshift evolution of the source +number density. Accounting for the spread in other pa- +rameters of the sources, e.g., the comoving neutrino lumi- +nosity or the comoving peak energy, would only increase +the width of the bump in the diffuse spectrum. +On the other hand, for wide bumps in the diffuse spec- +trum, with αbump ≪ 2, we can assume that the spread +mostly comes almost completely from the intrinsic width +of ω, since by itself it is larger than the spread induced +by the redshift distribution of sources. +In the main text, we adopt this approximation already +when we produce results for αbump = 1 in Fig. 6. Do- +ing this allows us to connect the diffuse spectrum to the +emitted spectrum by the simpler relation +E2 +ν +dΦν +dEνdΩ = nνLν +4π ω(2Eν) +� ∞ +0 +dz +ρ(z) +H(z)(1 + z)2 , (A5) +where we have assumed SFR evolution, so that contri- +butions mostly come from sources at z ≈ 1. For other +choices of redshift evolution, the relation between the +peak energy of the diffuse spectrum and the peak en- +ergy of the individual source spectrum changes. How- +ever, evaluating the diffuse flux at its peak value, and +assuming for ω the same log-parabola form, Eq. (2), that +we use for the diffuse flux, but normalized according to +the condition Eq. (A2), we can still obtain the connection +between the diffuse bump normalization and the intrinsic +source luminosity, i.e., +E2 +ν,bumpΦν,bump = nνLν +4π +�αbump +π +� ∞ +0 +dz +ρ(z) +H(z)(1 + z)2 . +(A6) +Numerically, this is +E2 +ν,bumpΦν,bump = 1.13 × 10−7 GeV cm−2 s−1 sr−1 +× +nν +10−6 Mpc−3 +Lν +1043 erg s−1 +√αbump ξz , +(A7) + +20 +where ξz is defined as in Eq. (5) of Ref. [190], and is equal +to 2.8 for SFR evolution, 0.6 for no redshift evolution, and +8.4 for strong, FSRQ-like evolution; see also Ref. [181]. +We use Eq. (A7) to produce Fig. 6. +Appendix B: Details of the posterior probability +distribution +In the main text (Section III D), we described our sta- +tistical procedure to fit the present-day, 7.5-year IceCube +HESE sample [7, 64] using a two-component flux model +composed of a power law and a bump, Eq. (3). Here we +provide more details on the fit. +In our fits, we scan over a nine-dimensional parameter +space. Table I shows the free parameters: three phys- +ical parameters describe the power-law flux component +(Φ0,PL, γ, Eν,cut), three physical parameters describe the +bump-like flux component (E2 +ν,bumpΦ0,bump, αbump), and +three nuisance parameters vary the normalization of the +atmospheric backgrounds (N µ, N ν,c, N ν,pr). +Figure B1 shows the resulting 1σ and 2σ contours of +the posterior in the planes of each pair of parameters, +marginalized over all the remaining ones. Results are for +present-day IceCube data and for 2035 forecasts, using +the combined exposure of all future detectors (see Fig. 1). +For the present-day results, in the planes involving the +bump parameters, the posterior is multimodal, since the +data can be explained either by a soft power law with +no bump, by a relatively harder power law and a bump +at hundreds of TeV, or by a power law with a cut-off +at hundreds of TeV and a PeV bump; see Table I. The +latter leads to the combination of parameters that max- +imizes the posterior, i.e., the maximum a posteriori esti- +mator in Fig. B1. Qualitatively, this solution is a multi- +component flux model on par with those proposed, e.g., +in Refs. [28, 66], where the power-law component was +associated with SBGs and the bump component was as- +sociated with photohadronic sources such as blazars or +TDEs (see also Refs. [102, 103]). +However, the region of parameter space corresponding +to this maximum a posteriori solution is tiny, since it re- +quires a tuning between the power-law and the bump pa- +rameters such that the bump takes over from the power +law after its cut-off. For this reason, the marginalized +two-dimensional posterior in Fig. B1 favors instead an +explanation of the data with a single power-law compo- +nent. This is evidence by the mean value of the posterior +corresponding to a low value of log10 E2 +ν,bumpΦbump. +Figure B1 also shows the projected contours in 2035, +taking the true flux as given by the the present-day max- +imum a posteriori solution, i.e., a power law followed by +a PeV bump. The contours shrink significantly, which al- +lows a remarkably precise measurement of the bump pa- +rameters. However, the planes involving the power-law +parameters show evident degeneracy, mainly because our +true flux includes a power law cut-off at a few hundred +TeV, which could just as well be explained by a very soft +power law without a cut-off. +Appendix C: Limits on subdominant bumps of +different widths +In the main text (Section V A), we showed limits on the +height of subdominant bumps, assuming a fixed bump +width of αbump = 1; see Fig. 5. Here we show how the +limits change with the bump width. +Figure C1 shows present-day and projected limits on +the height of subdominant wide (αbump = 0.5) and nar- +row bumps (αbump = 2). +For wide bumps, the limits +weaken for bumps centered in the high-statistics energy +region, i.e., around Eν ≈ Eν,bump ≈ 200 TeV, compared +to the limits obtained for αbump = 1 in Fig. 5. This is +because wide bumps introduce less sharply defined spec- +tral features into the diffuse flux, spread out over a wide +energy range; they are, therefore, harder to spot. For nar- +row bumps, in contrast, the limits strengthen for bumps +centered in the high-statistics region, because they intro- +duce sharper spectral features that are easier to spot us- +ing high statistics. However, for narrow bumps the limits +weaken at the lowest energies, because the HESE sam- +ple that we use contains only events with energy above +60 TeV (Section III B), which makes the analysis sensi- +tive to bumps centered at low energies only if they are +wide enough to affect also higher energies. +Figure C2 shows analogous limits after the posterior +has been marginalized over the bump width, αbump. +Compared to Figs. 5 and C1, the limits are signifi- +cantly weakened at low values of Eν,bump, because narrow +bumps are essentially undetectable if centered below the +60-TeV cut in the HESE sample. On the other hand, the +main conclusion that we had found in the main text for +αbump = 1 in Fig. 5 is fortified: by 2035, the combined +detector exposure may limit the contribution of a popula- +tion of photohadronic sources to a fraction of the diffuse +flux from 100 TeV to 1 PeV, regardless of the bump width. +Appendix D: Limits and discovery of subdominant +bumps assuming a harder power-law spectrum +In Section V and Appendix C, we placed limits on and +computed the discovery potential to subdominant bumps +under the assumption that the true diffuse neutrino spec- +trum is the best-fit one-component flux from present-day +data (“PL w/ cut-off” in Table I), i.e., γ = 2.75 and +Eν,cut ≈ 10 PeV. However, a spectrum this soft may be +difficult to reconcile with theory expectations of realis- +tic cosmic-ray acceleration. Thus, here we show how the +limits would change if the true diffuse spectrum were in- +stead a harder power law ∝ E +−2.5 +ν +, with no cut-off. +To find the normalization of the new power law, we fit +it to the public 7.5-year IceCube HESE sample [7, 64]. +We perform two-component fits to the data, present and +future, using the same methods as before (Section V), to + +21 +0 +1 +l10Φ0,PL +0 +1 +2 +3 +N ν,pr +0 +1 +2 +N ν,c +0 +1 +2 +N µ +0 +5 +10 +αbump +4 +5 +6 +7 +l10Eν,bump +−2 +0 +2 +l10E2 +ν,bumpΦ0,bump +5 +6 +7 +8 +l10Eν,cut +2.0 +2.5 +3.0 +3.5 +γ +2.0 +2.5 +3.0 +3.5 +γ +5 +6 +7 +8 +l10Eν,cut +−2 +0 +2 +l10E2 +ν,bumpΦ0,bump +4 +5 +6 +7 +l10Eν,bump +0 +5 +10 +αbump +0 +1 +2 +N µ +0 +1 +2 +N ν,c +Mean value, 7.5 years exposure +Maximum a posteriori estimator, 7.5 years exposure +68% C.L. +95% C.L. +0 +1 +2 +3 +N ν,pr +IceCube HESE 7.5 yr +2035, all detectors (proj.) +FIG. B1. Posterior probability distribution using present-day HESE data and projected HESE data in the year +2035. Each panel shows the two-dimensional posterior, Eq. (9) in the main text, for a pair of parameters, marginalized over +all the other parameters. The units for the dimensional parameters are as follows: log10 +� +ΦPL/ +� +10−18 GeV−1 cm−2 s−1 sr−1�� +, +log10(Eν,cut/GeV), log10 +� +E2 +ν,bumpΦbump/ +� +GeV cm−2 s−1 sr−1�� +, and log10(Eν,bump/GeV). Projected results assume as true +flux the one described by the present-day maximum a posteriori estimators. +place upper limits on the bump height and to compute +the bump discovery Bayes factor. +Figure D1 shows the resulting limits. +Compared to +Figs. 5 and C2, they are significantly weaker for bumps +centered at high energies, due to the larger number of +events there that can be explained solely by a hard power- +law flux component, which falls more slowly with energy +than in Fig. C2. Regardless, the main conclusion that +we found in the main text, based on Fig. C2, holds: by +2035, the combined exposure of all detectors may limit +the contribution of a population of photohadronic sources +to a fraction of the diffuse flux in the 100–500 TeV range. + +22 +105 +106 +107 +Central energy of ν bump, Eν,bump +10−10 +10−9 +10−8 +10−7 +10−6 +ν bump height, E2 +ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] +Projected upper limits (95% C.L.): +Present-day best-fit +power law ∝ E−γ +ν e−Eν/Eν,cut +(γ = 2.75, Eν,cut = 4 PeV) +68% C.L. 2035, all detectors (proj.) +All limits: bump width αbump = 0.5 +IceCube HESE 7.5 yr (95% C.L.) +2025, IceCube only +2035, IceCube only +2035, IceCube+Gen2 only +2035, all detectors +105 +106 +107 +Central energy of ν bump, Eν,bump +10−10 +10−9 +10−8 +10−7 +10−6 +ν bump height, E2 +ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] +Projected upper limits (95% C.L.): +Present-day best-fit +power law ∝ E−γ +ν e−Eν/Eν,cut +(γ = 2.75, Eν,cut = 4 PeV) +68% C.L. 2035, all detectors (proj.) +All limits: bump width αbump = 2 +IceCube HESE 7.5 yr (95% C.L.) +2025, IceCube only +2035, IceCube only +2035, IceCube+Gen2 only +2035, all detectors +FIG. C1. Upper limits on the height of a bump in the diffuse flux of high-energy neutrinos, for varying bump +width, αbump. Same as Fig. 5, which assumed αbump = 1, but for wide bumps (α = 0.5, left) and narrow bumps (αbump = 2, +right). See Section V A in the main text and Appendix C for details. +Figure D2 shows the resulting discovery potential. +105 +106 +107 +Central energy of ν bump, Eν,bump +10−10 +10−9 +10−8 +10−7 +10−6 +ν bump height, E2 +ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] +Projected upper limits (95% C.L.): +Present-day best-fit +power law ∝ E−γ +ν e−Eν/Eν,cut +(γ = 2.75, Eν,cut = 4 PeV) +68% C.L. 2035, all detectors (proj.) +All limits: marginalized over bump width, αbump +IceCube HESE 7.5 yr (95% C.L.) +2025, IceCube only +2035, IceCube only +2035, IceCube+Gen2 only +2035, all detectors +FIG. C2. +Upper limits on the height of a bump in +the diffuse flux of high-energy neutrinos, marginalized +over the bump width, αbump. +Same as Figs. 5 and C1, +which assumed fixed values of αbump = 0.5, 1, 2, but marginal- +izing the posterior distribution function over αbump. See Sec- +tion V A in the main text and Appendix C for details. +Comparec to Fig. 7, the main change is at low values +of Eν,bump, where the harder power law predicts fewer +events and, therefore, a bump in the spectrum would +stand out more clearly. Here also the main conclusion +that we found in the main text, based on Fig. 7, holds: +by 2035, the combined exposure of all detectors may dis- +cover a bump whose height is comparable to or smaller +than the diffuse flux in the 100–500 TeV range. + +23 +105 +106 +107 +Central energy of ν bump, Eν,bump +10−10 +10−9 +10−8 +10−7 +10−6 +ν bump height, E2 +ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] +Projected upper limits (95% C.L.): +Present-day best-fit +power law ∝ E−γ +ν +(γ = 2.5) +68% C.L. 2035, all detectors (proj.) +All limits: bump width αbump = 1 +IceCube HESE 7.5 yr (95% C.L.) +2025, IceCube only +2035, IceCube only +2035, IceCube+Gen2 only +2035, all detectors +105 +106 +107 +Central energy of ν bump, Eν,bump +10−10 +10−9 +10−8 +10−7 +10−6 +ν bump height, E2 +ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] +Projected upper limits (95% C.L.): +Present-day best-fit +power law ∝ E−γ +ν +(γ = 2.5) +68% C.L. 2035, all detectors (proj.) +All limits: marginalized over bump width, αbump +IceCube HESE 7.5 yr (95% C.L.) +2025, IceCube only +2035, IceCube only +2035, IceCube+Gen2 only +2035, all detectors +FIG. 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D 59, 023002 (1999), arXiv:hep-ph/9807282. + diff --git a/09AyT4oBgHgl3EQfPfYd/content/tmp_files/load_file.txt b/09AyT4oBgHgl3EQfPfYd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa0dd7da6db321c153e6404e982d190bca379ada --- /dev/null +++ b/09AyT4oBgHgl3EQfPfYd/content/tmp_files/load_file.txt @@ -0,0 +1,2794 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf,len=2793 +page_content='Bump-hunting in the diffuse flux of high-energy cosmic neutrinos Damiano F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Fiorillo ∗ and Mauricio Bustamante † Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark (Dated: January 3, 2023) The origin of the bulk of the high-energy astrophysical neutrinos seen by IceCube, with TeV– PeV energies, is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' If they are made in photohadronic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', proton-photon, interactions in astrophysical sources, this may manifest as a bump-like feature in their diffuse flux, centered around a characteristic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We search for evidence of this feature, allowing for variety in its shape and size, in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 years of High-Energy Starting Events (HESE) collected by the IceCube neutrino telescope, and make forecasts using larger data samples from upcoming neutrino telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Present-day data reveals no evidence of bump-like features, which allows us to constrain candidate populations of photohadronic neutrino sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Near-future forecasts show promising potential for stringent constraints or decisive discovery of bump-like features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our results provide new insight into the origins of high-energy astrophysical neutrinos, complementing those from point-source searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' INTRODUCTION What is the origin of the bulk of the high-energy as- trophysical neutrinos discovered by IceCube [1–7], with TeV–PeV energies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' They are likely predominantly pro- duced by one or more populations of extragalactic sources capable of accelerating cosmic rays to EeV-scale energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Yet, so far, less than a handful of sources have been iden- tified [8–12]—though more conceivably will be [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Unquestionably, looking for individual sources is chal- lenging [15], due to the need to detect coincident electro- magnetic emission from them, incomplete catalogs, large trial factors, and low detection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To overcome these limitations, here we adopt a dif- ferent strategy: rather than resolving individual sources, we look, in a single swathe, for the population, or pop- ulations, of sources responsible for the bulk of the high- energy neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We inspect the diffuse neutrino energy spectrum, made up of the aggregated neutrino emission from all sources, for evidence of distinct features that may reveal contributions to it from tributary populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We consider two broad classes of candidate high-energy neutrino sources: those where neutrinos are made pri- marily in cosmic-ray interactions with ambient matter— i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', proton-proton (pp) sources [16–18]—and those where neutrinos are made primarily in cosmic-ray interac- tions with ambient radiation—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', photohadronic (pγ) sources [17, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In both, neutrinos come from the decay of the short-lived particles—pions and muons, mostly—born from these interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, they emit neutrinos with different energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Based on this, we use their spectra as proxies of their contributions to the diffuse neutrino flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Neutrinos from pp sources have a power-law spectrum, inherited from their parent cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Candidate pp sources include starburst galaxies [21–28], galaxy clus- ters [29–32], and low-luminosity active galactic nuclei (LL AGN) [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Neutrinos from pγ sources have instead a “bump-like” spectrum, centered around an energy de- termined by the properties of the interacting photons and cosmic rays [35–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Candidate pγ sources include gamma-ray bursts (GRBs) [36, 40–45], LL AGN [33, 34], radio-quiet AGN (RQ AGN) [46], radio-loud AGN (RL AGN) [47–49], BL Lacertae AGN (BL Lacs) [50, 51], flat- spectrum radio quasars (FSRQs) [50, 52–54], and tidal disruption events (TDEs) [55–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The above classification is admittedly approximate: in reality, most candidate source classes may produce neu- trinos via both pp and pγ interactions, though not nec- essarily in equal measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, we do not test pre- dictions of specific source models, but the presence of generic spectral features due to pp and pγ production in the neutrino data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Still, we do interpret our results, with caveats, in terms of population properties (Section V B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Present-day IceCube data are described well by a dif- fuse pure-power-law spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Φν ∝ E−γ, with γ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='37 [65] or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='87 [7], depending on the data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Naively, this would suggest that the diffuse flux is due to a single population of pp sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, it is widely understood that that conclusion would be premature: the large present-day uncertainty in the measured en- ergy spectrum might be hiding deviations from a pure power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To wit, while there is no marked prefer- ence for alternatives to a pure power law today, they are not strongly disfavored [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' More complex possibili- ties are not excluded either, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', a two-component model with pp sources dominating up to PeV energies and pγ sources dominating above [28, 66], or pγ sources opaque to gamma rays [67, 68] dominating below 60 TeV [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Motivated by these works, we perform a systematic search for the presence of power-law and bump-like dif- fuse flux components in present-day IceCube data, and make near-future forecasts using the combined exposure of upcoming neutrino telescopes, up to the year 2040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We model the power-law spectrum from the general class of pp sources and the bump-like spectrum from the general class of pγ sources with flexible parametrizations that capture the rich interplay of their relative contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For our present-day results, we use the recent 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year public sample of IceCube High-Energy Starting Events (HESE) [7], which have high astrophysical purity and en- ergy resolution, and the associated Monte-Carlo sample arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='00024v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='HE] 30 Dec 2022 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 10 20 30 40 80 120 Equivalent IceCube exposure [years] 2020 2025 2030 2035 Year 1 101 102 103 104 105 Strength of evidence for two components in ν flux, B B C Barely worth mentioning Substantial Strong Very strong Decisive IceCube (IC) IC–Gen2 (8 IC) Baikal-GVD (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 IC) KM3NeT (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='8 IC) P-ONE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='2 IC) TAMBO (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 IC) TRIDENT (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 IC) Using HESE only All detectors: IceCube HESE-detection efficiency All detectors IceCube+Gen2 only IceCube only A D 10−10 10−9 10−8 10−7 10−6 A 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 years exposure IceCube HESE (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 years) 10−10 10−8 10−8 10−7 All-flavor diffuse ν flux, E2 ν Φν [GeV cm−2 s−1 sr−1] B 2025 (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') Two-component fit One-component fit 10−10 10−8 10−8 10−7 C 2030 (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 105 106 107 Neutrino energy, Eν [GeV] 10−10 10−8 10−8 10−7 D 2035 (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Evidence for the existence of a PeV bump in the diffuse flux of high-energy astrophysical neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The discovery potential is quantified via a Bayes factor (Section III) that compares the evidence for a two-component flux fit—a power law plus bump—vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' a one-component flux fit—a power law—after marginalizing over all flux parameters (Section II D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Present-day results (snapshot A) are obtained using the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year public IceCube HESE sample [7, 64];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' the best-fit parameter values are in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Projections (snapshots B, C, D) are obtained using scaled-up event rates, adopting the present-day best-fit two-component flux as the true flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We assume that upcoming neutrino telescopes will have the same HESE-detection efficiency as IceCube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Left: Evolution of discovery potential with time, using combined detector exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Right: Best-fit and 68% allowed ranges of the one- and two-component flux fits for snapshots A–D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' A prominent PeV bump may be discovered decisively already by 2027, by combining IceCube, Baikal-GVD, and KM3NeT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (In contrast, constraining or discovering subdominant bumps will require adding more detectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') See Section IV for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' of simulated HESE events [64], which includes detector details and backgrounds, and which we reweigh to test our flux predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For our forecasts, given the absence of details on upcoming detectors, we assume for them HESE-detection capabilities with IceCube-like efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our goal below is double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' First, we show that, thanks to larger statistics, we may soon distinguish decisively between a single-component (pp only or pγ only) and a multi-component (pp and pγ) description of the diffuse neutrino flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Second, we show that, even if future obser- vations were to favor a dominant power-law diffuse flux from pp sources, sub-dominant bump-like contributions from pγ sources could still be discovered or constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The latter case would in turn constrain the properties of the pγ source population, independently of constraints from point-source searches [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 1 shows how our results address the first of these two goals (we defer details to Section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' It shows how the evidence in favor of a particular two-component dif- fuse flux—a power law and a PeV bump, hinted at by present-day data—may grow with time, assuming this is indeed the true flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Already by 2027, the com- bined exposure of IceCube, Baikal-GVD [70, 71], and KM3NeT [72–74] could decisively favor this explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 1 illustrates a larger point: growing statistics will allow us to probe progressively more inconspicuous fea- tures of the diffuse flux, offering powerful discrimination between competing source models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our methods (Section III) are not dissimilar from those used in collider physics to search for particle resonances: like them, we hunt for statistically significant bumps in an otherwise smooth landscape—in our case, in a power- law neutrino spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Discovering a bump would signal the existence of a population of pγ sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Not finding any would constrain their contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In both cases, the power of the method grows with statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In Section II, we review the current state of the diffuse flux of high-energy neutrinos and introduce parametriza- 3 tions for the power-law and bump-like flux components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In Section III, we describe the present-day IceCube data and the methods we use to compare them to our flux predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In Section IV, we focus on the case of a PeV bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In Section V, we focus on subdominant bumps in the TeV–PeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In Section VI, we list possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In Section VII, we summarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' THE DIFFUSE FLUX OF HIGH-ENERGY ASTROPHYSICAL NEUTRINOS The sources responsible for the diffuse flux of TeV–PeV astrophysical neutrinos seen by IceCube are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Yet, different neutrino production mechanisms, promi- nent in different candidate source classes, are expected to make neutrinos with different energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, we use the diffuse neutrino spectrum as proxy of the identity of the population, or populations, of neutrino sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Overview: one or more source populations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' At present, because the origin of the bulk of high- energy astrophysical neutrinos is unknown, models of their diffuse flux are many and varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Viable models must be able to explain the diffuse flux seen by Ice- Cube [7, 65], or a fraction of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' But, beyond that con- straint, there is significant room for variety in the predic- tions of the flux shape and size from various candidate astrophysical sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [14] for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Further, the diffuse neutrino flux could conceivably be the superposition of contributions from multiple source populations, each contributing a flux component with a differently shaped energy spectrum and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (Refer- ences [75–77] estimated the size of these contributions, though based on searches for point and stacked sources, rather than on the diffuse flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') Identifying these com- ponents in the diffuse flux—and, hence, identifying the contribution of multiple source classes—requires distin- guishing between their different spectral shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Later, in Sections IV and V, we show that the main challenge to do that is the paucity in high-energy neutrino data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Fortunately, this will be surmounted in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, we consider two broad classes of candidate sources that roughly map to two different neutrino pro- duction mechanisms: sources that make neutrinos via proton-proton interactions and sources that make neu- trinos via proton-photon interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Later, we look for their imprints in the diffuse flux of high-energy neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Neutrinos from pp vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' pγ sources Because the diffuse flux of TeV–PeV astrophysical neutrinos seen by IceCube is seemingly isotropic, the astrophysical sources responsible for it are likely pre- dominantly extragalactic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Their identity is presently unknown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' except for a few notable exceptions [8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' They are purportedly high-energy non-thermal astro- physical sources able to accelerate cosmic-ray protons and charged nuclei to energies of at least 100 PeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [78, 79] for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, in many mod- els, they are also sources of ultra-high-energy cosmic rays (UHECRs) and high-energy gamma rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For simplicity, we frame our discussion below in terms of UHECR pro- tons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' however, the mass composition of UHECRs is key to understanding the production of UHECRs and of the associated high-energy neutrinos [80–82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In the sources, diffusive shock acceleration may gener- ate UHECRs with a power-law spectrum ∝ E−γp p , where Ep is the proton energy and γp ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' UHECRs inter- act with ambient matter, in proton-proton (pp) interac- tions, or ambient radiation, in photohadronic (pγ) inter- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Both pp and pγ interactions make high-energy pions that, upon decaying, make the high-energy neutri- nos that IceCube detects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', π+ → µ++νµ, followed by µ+ → e++νe+¯νµ, and their charge-conjugated processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Each neutrino carries, on average, 5% of the energy of the parent proton, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Eν ≈ Ep/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, while both pp and pγ interactions can produce high-energy neutrinos, they may yield markedly different neutrino spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In pp interactions, deep inelastic scattering produces multiple π+ and π−, in roughly equal proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Be- cause UHECR protons collide with ambient protons that are comparatively at rest, the resulting neutrino spec- trum is entirely determined by the spectrum of the high- energy protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, the neutrino spectrum emitted by a pp source is a power law ∝ E−γ ν , where γ ≈ γp, up to a maximum neutrino energy Eν,cut = Ep,max/20, where Ep,max is the maximum energy to which the source can accelerate UHECR protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The latter depends on the properties of the source that drive particle acceleration, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', the size of the acceleration region, the bulk Lorentz factor in it, the intensity of the magnetic field, and the fraction of available of energy that is imparted to non- thermal protons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [39, 83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In pγ interactions, UHECR protons interact with am- bient photons whose spectrum is concentrated around a characteristic photon energy E⋆ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The value of E⋆ γ de- pends on the origin of the ambient photon field, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', synchrotron or synchrotron self-Compton emission by ac- celerated electrons or protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Pion production via the ∆(1232) resonance dominates around center-of-mass en- ergy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='232 GeV, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', p + γ → ∆+ → n + π+, and, at higher energies, deep inelastic scattering yields multi- ple π+ and π−, in roughly equal proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, the neutrino spectrum emitted by a pγ source stems from the interplay of the interacting protons and photons: to produce a ∆+ resonance, their energies must satisfy EpEγ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='2 GeV2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [20, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Hence, most ∆+-producing pγ interactions occur between photons of energy E⋆ γ and protons of energy E⋆ p ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='2 GeV2/E⋆ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' As a result, the neutrino spectrum is bump-like, con- centrated around the characteristic neutrino energy of Eν,bump ≈ E⋆ p/20 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='01 GeV2/E⋆ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (The high-energy 4 neutrino spectrum from certain classes of pp sources, like starburst galaxies [85], might be a power law with a spec- tral kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In those cases, it may also be approximated by a bump, centered at the spectral kink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') In our analysis, we do not model the intrinsic proper- ties of pp or pγ sources, particle acceleration, radiation processes, or specific shapes of the ambient photon field that are integral to building complete source models of neutrino production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Instead, for pp sources, we model directly the neutrino spectra that they emit as a power law ∝ E−γ ν augmented with an exponential suppression around Eν,cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For pγ sources, we model directly the neu- trino spectra that they emit as a bump-like flux, centered at Eν,bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This strategy allows us to describe many dif- ferent candidate pp and pγ source populations under a common, albeit simplified, framework (see Section VI for proposed refinements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We give details in Section II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The shapes of the neutrino spectra above are for in- dividual pp or pγ sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, the diffuse neutrino flux from a population of pp sources or pγ sources is ex- pected to approximately retain the shape of the energy spectra emitted by the individual sources that make up the population—a power-law flux or a bump-like flux, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In the diffuse flux, the spectral features of in- dividual sources are averaged by the spread in the source properties that affect UHECR acceleration and neutrino production—luminosity, density, magnetic field intensi- ties, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='—and are softened by the effect of cosmological expansion on the neutrino energies, and by the distri- bution of sources with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Nevertheless, the fun- damental difference between the diffuse neutrino energy spectra from a population of pp and pγ sources remains and is what motivates us to model them as two differ- ently shaped flux components, a power law and a bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' By varying the values of their shape parameters in fits to data (more on this later), we capture the interplay between them and, indirectly, the effects of spectral av- eraging and softening on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Overview of source candidates Presently, it is unknown whether the diffuse flux of high-energy astrophysical neutrinos seen by IceCube is due to a single population of pp sources, a single popula- tion of pγ sources, or a superposition of pp and pγ source populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For comparison, the unresolved diffuse flux of GeV–TeV gamma rays is likely due to various popu- lation of sources, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [86–88], including unre- solved blazars, star-forming galaxies, and radio galaxies, which, incidentally, may also be neutrino sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In con- trast, identifying the contributions from multiple source populations in the diffuse flux of high-energy neutrinos is hampered by low neutrino event rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Nevertheless, weak hints in present-day observations of the diffuse neu- trino flux suggest that different source populations may contribute at different energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We sketch them below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In the 10–100 TeV range, the flux of astrophysical neu- trinos seen by IceCube suggests an origin in pγ sources that are opaque to gamma rays [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' These sources must be opaque, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', must attenuate gamma rays via electron-positron pair production, in order for the flux of gamma rays co-produced with neutrinos not to ex- ceed the isotropic diffuse gamma-ray background seen by Fermi-LAT [67–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Various candidate pγ sources with potential high opacity have been proposed, includ- ing low-luminosity and choked GRBs [43, 89–91] and su- pernovae [92–94], and hidden cores of AGN [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Notably, in AGN corona models [95] neutrino production via pγ and pp might be comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Nevertheless, because our analysis uses detected events with energies above 60 TeV (Section III A)—to reduce the contamination of atmo- spheric backgrounds—it is largely insensitive to bumps that peak below this energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Around 100 TeV, the flux of astrophysical neutrinos seen by IceCube may originate in pp sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Exam- ples include cosmic reservoirs, like star-forming galax- ies [21, 24, 27, 28, 32, 96, 97] and galaxy clusters [30– 32, 98, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Cosmic reservoirs are believed to be cosmic- ray calorimeters: they confine cosmic rays for a long time, boosting their chances of interacting with interstellar ma- terial and making neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' They can explain the co- incidence observed between the energy generation rate of UHECRs and of high-energy neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, for some of these sources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', star-forming galaxies, it is challenging to model the acceleration of UHECRs and, therefore, the production of PeV-scale neutrinos [24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In the PeV range, pγ sources like blazars [50–54, 100], GRBs [36, 40, 42, 44, 101], and TDEs [55–63], may domi- nate neutrino production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This is expected because these sources are all candidate UHECR accelerators, and they are all known to contain eV–MeV photon fields that can act as targets for photohadronic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (There are also models of PeV-scale neutrino production via pp in- teractions of UHECRs on nuclei from the host galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') Thus, the picture that tentatively emerges is that a low-energy population of pγ sources may dominate neu- trino production below 100 TeV—though our analysis is largely insensitive to it—pp sources may dominate it up to a few hundred TeV, and a different population of pγ sources may dominate it at higher energies, up to a few PeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' References [28, 66, 102, 103] proposed multi- component flux models based on this tentative picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In short, above 60 TeV, where our analysis is sensitive, the diffuse neutrino flux may be a power law up to a few hundred TeV, followed by a bump centered at PeV en- ergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Indeed, in Sections IV and V, we find marginal evidence for this in present-day IceCube data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Still, as part of our analysis, we explore many alternative super- positions of a power law and bump flux components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Power-law and bump flux components Following the tenet of our work, laid out in Sec- tion II B, we forego modeling in detail the neutrino emis- sion from individual pp and pγ sources and computing the diffuse neutrino flux from the aggregated contributions of their populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Instead, we directly model the diffuse neutrino flux without recourse to any particular source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This strategy allows us to describe a vast number of possible superpositions of pp and pγ neutrino source populations within the same framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In practice, our goal is to assess the evidence in fa- vor of the existence of a bump in an otherwise feature- less power-law diffuse flux of high-energy astrophysical neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Discovering a bump would be evidence of a pγ source population (or of a pp source population with a spectral kink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Section II B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Hence, we model the diffuse flux as the superposition of two components: a power-law flux, representative of neutrino production in pp sources, and a log-parabola bump-like flux, represen- tative of neutrino production in pγ sources (or pp sources with a spectral kink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' By construction, the parametriza- tions that we adopt have the flexibility to capture the variety in the interplay between power laws and bumps of various shapes and relative sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, we describe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Later, when computing the evidence for a bump in Sections III–V, we vary the values of the flux parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The diffuse power-law flux component is E2 ν dΦPL dEνdAdtdΩ = Φ0,PL � Eν 100 TeV �2−γ e − Eν Eν,cut , (1) where Φ0,PL is a normalization parameter, γ is the spec- tral index, and Eν,cut is the neutrino cut-off energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Equation (1) describes the diffuse flux of neutrinos pro- duced in pp interactions of UHECRs that have a rel- atively soft spectrum ∝ E−γp p with γp ≳ 2, as ex- pected from diffusive shock acceleration [104, 105];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see- Section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, instead of modeling specific flux pre- dictions, we vary the values of Φ0,PL, γ, and Eν,cut in fits to present-day and projected samples of detected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The diffuse bump-like flux component is E2 ν dΦbump dEνdAdtdΩ = � E2 ν,bumpΦ0,bump � × exp � −αbump log2 � Eν Eν,bump �� ,(2) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', a log-parabola, where E2 ν,bumpΦ0,bump is a normal- ization parameter, Eν,bump is the energy at which the bump is centered, and αbump defines the width of the bump, which is approximately Eν,bump/α1/2 bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Most of the neutrinos are concentrated around energy Eν,bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The value of αbump controls whether the spectrum is wide around this energy—if αbump is small—or narrow— if αbump is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Equation (2) represents the diffuse flux of neutrinos produced in pγ interactions (or in pp inter- actions with a spectral kink);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, instead of modeling specific flux predictions, we vary the values of E2 ν,bumpΦ0,bump, αbump, and Eν,bump in fits to present-day and projected samples of detected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 2 compares our log-parabola bump-like flux, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2), with detailed models of the diffuse high-energy neutrino emission from various classes of sources, taken from the literature, both pγ—blazars [50], low-luminosity GRBs [106], and TDEs [107]—and pp sources—starburst galaxies [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' These models illustrate that, in reality, bumps may be asymmetric around Eν,bump and may fea- ture a plateau rather than a peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For the case of TDEs, for example, the flux at energies below the peak flattens out due to a contribution from pγ interactions on a sec- ond target of X-ray photons, which is not captured by our parametrization, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We leave searches for these fea- tures to future dedicated studies (Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 2 shows that, in all cases, the log-parabola bump-like flux, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2), is a reasonable fit to the flux models, especially close to the peak of the bump, where the flux component contributes the most to the rate of detected events, and especially for more symmetric model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This validates the use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2) in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' An alternative origin of a bump in the diffuse flux, from beyond the Standard Model, is from the decay of heavy dark matter particles between 100 TeV and 10 PeV into high-energy neutrinos [108–120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Dark-matter decay would yield a neutrino spectrum that peaks at an energy determined by the mass of the dark matter particle, with a width determined by its decay width and by the dis- tribution of its dark-matter density with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' While our analysis below focuses on bumps as coming from the neutrino production mechanism, it can be repurposed to perform searches for neutrino bumps from dark-matter decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Previous studies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [117, 118], have shown that including a bump-like high-energy neutrino flux component from the decay of PeV-scale dark matter can marginally improve fits to IceCube data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, we find a similar result, though motivated differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (A separate issue is that, for most dark matter decay chan- nels, gamma rays co-produced with neutrinos may be in tension with observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [121, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') Thus, our diffuse flux model is two-component, the superposition of the power-law and bump components, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (1) and (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', E2 νΦν(Eν) ≡ E2 ν �dΦPL(Eν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Φ0,PL, γ, Eν,cut) dEνdAdtdΩ + dΦbump(Eν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' E2 ν,bumpΦ0,bump, αbump, Eν,bump) dEνdAdtdΩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (3) The physical parameters of our model are Φ0,PL, γ, Eν,cut, E2 ν,bumpΦ0,bump, αbump, and Eν,bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Later, in our statistical analysis in Section III, we introduce addi- tional nuisance parameters, related to atmospheric neu- trino and muon backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Table I summarizes the free parameters of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We assume that the diffuse flux is made up of νe, νµ, ντ, ¯νe, ¯νµ, and ¯ντ in equal proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This is the canonical 6 105 106 107 Neutrino energy, Eν [GeV] 10−10 10−9 10−8 10−7 10−6 All-flavor diffuse ν flux, E2 νΦν [GeV cm−2 s−1 sr−1] IceCube 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-yr HESE (PRD 2021) Blazars (Palladino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', 2018) GRBs (Tamborra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', 2015) TDEs (Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', 2022) SBGs (Condorelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', 2022) Flux from source model Log-parabola bump-like approximation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Diffuse neutrino fluxes from representative source models of neutrino production via pγ and pp in- teractions vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' approximations using the bump-like flux from our work, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For blazars (mainly pγ), the flux is scenario 1 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [50], with constant baryon loading for all sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For gamma-ray bursts (GRBs, mainly pγ), the flux is from low-luminosity bursts, from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For tidal disruption events (TDEs, mainly pγ), the flux is from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [107], including interactions with optical and ultraviolet photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For starburst galaxies (SBGs, mainly pp), the flux is from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [85], from pp interactions of UHECRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For com- parison, we show the 68% allowed flux band from the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE analysis, assuming a pure power-law [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We do not test the source flux models shown in this figure, nor any specific source flux models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' we show them here merely as representative examples to validate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our log-parabola bump-like flux approximates the source flux models reasonably well, especially where they are highest, and especially if they are symmetric around their peak energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' expectation for high-energy neutrinos produced in pion decays (Section II B), after flavor oscillations have acted on them en route to Earth [123, 124], and is compati- ble with IceCube measurements of the flavor composi- tion [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Present uncertainties in the values of the neu- trino mixing parameters lead to uncertainties in the pre- dicted flavor composition of the neutrino flux, but they should be rendered negligible in the next decade by up- coming oscillation experiments [124], so we ignore them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 3 illustrates the role of the different flux param- eters on the shape of the neutrino diffuse flux, and singles out the impact that varying the width, αbump, has on the bump component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Also, the dip in the flux in-between the cut-off of the power-law component and the rise of the bump component is a feature that could reflect the transition from a pp to a pγ source population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 105 106 107 Neutrino energy, Eν [GeV] 10−10 10−9 10−8 10−7 10−6 All-flavor diffuse ν flux, E2 νΦν [GeV cm−2 s−1 sr−1] Power law Bump αbump = 10 αbump = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 Φ0,PL γ Eν,cut E2 ν,bumpΦ0,bump α−1/2 bump Eν,bump FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Illustration of the power-law and bump flux components used in our analysis, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (1)-(3), and ef- fect of their free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For this figure only, the values of the flux parameters are fixed to their best-fit values obtained in a two-component flux fit to the public 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE data [7, 64];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In our analysis, we allow the values of the parameters to vary in fits to data, ei- ther present-day or projected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Table I for a summary of the parameters and Section II D for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The IceCube Collaboration itself has explored various possible shapes of the diffuse neutrino spectrum when fitting to detected data, including their default pure power law, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', one without a high-energy cut-off, a dou- ble power law, a pure log-parabola, a segmented power law, and fluxes from different astrophysical models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [3–5, 7, 65, 126–130];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [28, 66, 102, 103] for independent analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Present-day statistics are in- sufficient to yield a conclusive preference for any of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, we reach the same conclusion when com- paring the present-day preference for a one-component flux model vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' a two-component flux model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' HUNTING FOR BUMPS We look for bump-like features in the diffuse flux of high-energy neutrinos by using IceCube High-Energy Starting Events (HESE), with high astrophysical purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We account for detector effects and the irreducible con- tamination from atmospheric neutrinos and muons by using the public IceCube Monte Carlo HESE sample to compute event rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We scan wide ranges of possible values of the flux model parameters (Table I) and, when computing evidence, account for the appearance of spu- 7 rious bump-like features (the “look-elsewhere effect”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' IceCube High-Energy Starting Events (HESE) IceCube is the largest high-energy neutrino telescope in operation: roughly 1 km3 of underground Antarctic ice instrumented with photomultipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' It is an optical Cherenkov detector: it collects the light made by radi- ating secondary particles in showers born from neutri- nos interacting in the ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The main interaction chan- nel is neutrino-nucleon deep inelastic scattering (DIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In it, a high-energy neutrino scatters off of a constituent parton of the nucleon—a quark or a gluon—and breaks up the nucleon in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The high-energy final- state particles—electrons, muons, tauons, and hadrons— initiate showers whose charged particles emit Cherenkov radiation that propagates through the ice and is recorded by the photomultipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' From the amount of light de- posited, and from its spatial and temporal distribution, IceCube reconstructs the neutrino energy, direction, and flavor, with varying degrees of precision [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In our analysis, we focus on IceCube High-Energy Starting Events (HESE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' These are events where the neu- trino interaction occurs inside the instrumented volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' They undergo a self-veto that reduces the contamina- tion from atmospheric muons, which would otherwise be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' By design, HESE samples are the most astro- physically pure out of all of the event samples selected by IceCube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [2, 3, 132, 133] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This, coupled to the fact that their energy resolution is high (more on this below), makes them the most suitable kind of events to look for features in the neutrino energy spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Later (Section VI), we comment on the use of the other main event sample, of through-going muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' When making projections that involve other detectors, we assume that they will also collect HESE samples, a capability that they will arguably likely have, and that their HESE-detection efficiency will be equal to that of IceCube, which is admittedly a necessary simplification, born from the absence of details on future detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In the TeV–PeV range, there are two main light-profile topologies of HESE events: cascades and tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Cas- cades are made mainly by charged-current DIS of νe or ντ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', νl + N → l + X, where l = e, τ, N is a nucleon, and X are final-state hadrons), and also by neutral- current DIS of all flavors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', νl + N → νl + X, where l = e, µ, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Tracks are made by charged-current DIS of νµ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', νµ + N → µ + X), where the final-state muon leaves an track of light in its wake, km-scale in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In a DIS interaction, on average, the final-state hadrons receive about 25% of the initial neutrino energy, and the final-state lepton receives 75% [134–136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, in cascades, essentially all of the neutrino energy is de- posited in the ensuing shower, which grants them good energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In tracks, because the track escapes the instrumented detector volume, energy resolution is somewhat poorer (but the muon energy can be approxi- mated by how much energy the track deposits inside the detector [131]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The energy resolution of HESE events is about 10% in the logarithm of the event energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Con- versely, because cascades have a roughly spherical light profile centered on the neutrino interaction vertex, their angular resolution may be as poor as tens of degrees, while tracks, because they are elongated, have sub-degree angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For details, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' At a few PeV, in addition, charged-current DIS of ντ may produce “double bangs” or “double cascades” [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In them, the neutrino-nucleon DIS produces a first cas- cade;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' the final-state tauons propagate away from the in- teraction vertex, decay, and produce a second cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Recently, IceCube identified the first two candidate dou- ble bangs [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, they are not captured by the public IceCube HESE Monte Carlo sample on which we base our analysis [7, 64] (Section III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Beside neutrino-nucleon DIS, high-energy neutrinos are also detected via neutrino-electron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This interaction channel is negligible except in a narrow energy range around 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='3 PeV—the Glashow reso- nance [139]—where ¯νe may produce an on-shell W boson, which enhances the expected event rate massively [140– 146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Recently, IceCube observed the first Glashow reso- nance candidate [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The public IceCube HESE Monte Carlo sample that we use (Section III B) does contain contributions from Glashow resonance, but it does not contain the dedicated analysis that was needed to dis- cover that one candidate, which was a partially contained shower, rather than a fully contained one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, IceCube HESE events are cascades, tracks, and double cascades;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' we keep this classification also when making forecasts for future detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Because we have assumed equal proportion of νe, νµ, and ντ in the flux (Section II D), we do not attempt to infer the flavor com- position from the relative numbers of events of different classes, like Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [123, 124, 128, 129, 138, 148–150] do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' After neutrinos reach the surface of the Earth, they propagate underground, for a length of up to the diame- ter of the Earth, until they reach IceCube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Inside Earth, they undergo DIS on nucleons [134–136], which damp- ens their flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The effect is stronger at higher energies, where the neutrino-nucleon cross section is larger, and for neutrinos that travel longer paths inside the Earth, which encounter a larger column depth of nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For ντ in particular, charged-current DIS produces tauons which decay back into ντ with lower energy, that partially counteract the dampening of its flux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' this is known as “ντ regeneration.” While this effect is present in our analy- sis, it is significant mainly at energies above 100 PeV, higher than the ones we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Overall, the propagation of high-energy neutrinos inside the Earth affects their flux in an energy-, direction-, and flavor-dependent manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [134–136, 151–154] for explicit examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The above effects are built into the public IceCube HESE Monte Carlo sample that we use in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The sample is generated assuming the neutrino-nucleon cross section from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [155], for the propagation of neu- 8 trinos inside Earth and their detection at IceCube, and the Preliminary Earth Reference Model [156] for the in- ternal matter density of Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For details, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' HESE public data and Monte Carlo Recently, the IceCube Collaboration made public the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE sample [7] and an accompanying Monte Carlo (MC) simulation of the performance of the detec- tor [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We build our analysis on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE sample contains 102 events in to- tal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In our analysis, we use only the 60 events with re- constructed shower deposited energy larger than 60 TeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' there are 41 cascades, 17 tracks, and 2 double cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Above 60 TeV, the irreducible contamination from at- mospheric neutrinos and muons that pass the HESE self- veto (Section III C) is small [132, 133, 157], since their fluxes decrease faster with energy than the flux of astro- physical neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Because of the event selection, most events are downgoing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', coming from the Southern Hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For details, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The HESE MC sample contains 821764 simulated HESE events, generated using the same detector simu- lation used in the analysis of the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE sample by the IceCube Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' They are initiated by all neutrino flavors, produce cascades, tracks, and double cascades, from all directions, and cover the energy range that is relevant for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Events in the MC sample were generated assuming a reference diffuse high-energy astrophysical neutrino flux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [64] and Section III E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In our analysis, we com- pute HESE events corresponding to different choices of the high-energy astrophysical neutrino flux by reweighing the events in the MC sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' we describe the procedure in Section III D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, our predicted event rates inher- ently include the detailed IceCube HESE response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Compared to the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year analysis by the IceCube Collaboration [7], we adopt a simplified treatment of three nuisance detector systematic uncertainties—the ef- ficiency of digital optical modules, the head-on efficiency, and the lateral efficiency—in order to reduce the time needed for our computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Whereas the IceCube anal- ysis allows the values of these parameters to float in fits to observed data, with narrow prior distributions, we keep their values fixed to their nominal expectations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', where their priors are maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (For the same reason, we also keep the shapes of the atmospheric background distributions fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Section III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') In Section III E, we verify that the impact of fixing their values is limited, by approximating closely the IceCube fit from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For each simulated event in the MC sample, we use its primary neutrino quantities—neutrino en- ergy, flavor, and zenith angle—and its reconstructed event quantities—reconstructed deposited energy, recon- structed zenith angle, and event topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In our statisti- cal analysis (Section III D), we compare predicted vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ob- served event rates using reconstructed quantities, since these are accessible experimentally, but reweigh events in the MC sample using primary neutrino quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Irreducible atmospheric backgrounds The HESE sample contains events initiated not only by astrophysical neutrinos, but also by the irreducible back- ground flux of atmospheric neutrinos and muons, created in cosmic-ray interactions in the atmosphere of the Earth, that escape the HESE self-veto [132, 133, 157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' There are three contributions to it—conventional atmospheric muons, conventional atmospheric neutrinos, and prompt atmospheric neutrinos—born from the decay of mesons and muons produced by the cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For all of them, we use the same flux prescriptions as the IceCube 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5- year HESE analysis [7], via their implementations in the HESE MC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, we sketch them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' for details, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7], especially Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='3, IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='4, and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='6 therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The conventional atmospheric muon flux comes from the decay of pions and kaons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Compared to the parent cosmic rays, the atmospheric muon spectrum is softer due to the energy losses of the pions and kaons prior to their decaying and of the muons themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The baseline muon flux prescription that we use comes from air-shower simulations made with CORSIKA [158], using the Hillas- Gaisser H4a cosmic-ray flux model [159] and the Sibyll 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='1 hadronic interaction model [160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The conventional atmospheric neutrino flux comes from the decay of pions, kaons, and muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Like the conventional atmospheric muon flux, because of energy losses, its spectrum is softer than that of the parent cos- mic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The baseline conventional neutrino flux pre- scription that we use is from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [161], obtained using the modified DPMJET-III generator [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The prompt atmospheric neutrino flux comes from the decay of charmed mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Because they are short-lived, they experience little to no energy losses before decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' As a result, the spectrum of prompt neutrinos that they produce is harder than that of conventional atmospheric neutrinos, and closer to that of the parent cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The baseline prompt neutrino flux prescription that we use is from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To date, the prompt neutrino flux remains unobserved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' still, we include its possible contri- bution to the HESE rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Accordingly, in all our fits to HESE data below, we find that the contribution of prompt atmospheric neutrinos is compatible with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Later, as part of our statistical analysis (Sec- tion III D 2), we let the normalization of the conventional muon flux, conventional neutrino flux, and prompt neu- trino flux float freely in fits to HESE data, like the Ice- Cube analysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7], and using the same priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Reference [7] included extra parameters that affect the shape, not just the normalization, of the energy spec- tra of the atmospheric backgrounds: the spectral index of the cosmic-ray spectrum, the ratio of kaons to pions produced, and the ratio of neutrinos to anti-neutrinos produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In our analysis, we keep these shape parame- 9 ters fixed at their nominal values, given in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='1 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7], to reduce the time needed for our computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This is justified because the atmospheric backgrounds are subdominant in the HESE event rate above 60 TeV, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', in the energy range of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Like for detector sys- tematics (Section III B), we verify in Section III E that the impact of fixing the shape parameters is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Statistical procedure Our analysis compares expected HESE event rates— induced by our two-component astrophysical neutrino flux model (Section II D) and by atmospheric back- grounds (Section III C)—against the public IceCube 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5- year HESE sample [7, 64] (Section III B), and against projected versions of it with larger statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To com- pute event rates for arbitrary flux choices, we reweigh the HESE MC sample and, when making projections, re-scale it by longer detector exposure times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To com- pare event-rate predictions with observations, we adopt a Bayesian approach, binned in reconstructed event en- ergy and direction (Section III B), and allow astrophys- ical and background flux parameters (Table I) to float freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, we describe this in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Astrophysical neutrinos The set of flux parameters introduced in Section II D, θ ≡ � Φ0,PL, γ, Eν,cut, E2 ν,bumpΦ0,bump, αbump, Eν,bump � , defines a specific realization of our two-component diffuse flux of high-energy astrophysical neutrinos, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In the fits to HESE data below, we let the value of each parameter float independently of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For a given realization of θ, we compute the expected mean number of HESE events due to the corresponding astrophysical neutrino flux by reweighing the sample of MC HESE events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' we explain the reweighing procedure below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' After reweighing, the mean number of astrophys- ical events in the i-th bin of reconstructed shower en- ergy, Edep, and the j-th bin of reconstructed direction, cos θrec z , is µν,ast ij,t (θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' we introduce our choice of binning later (Section III D 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We treat events of each topology (t) separately, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', cascades (c), tracks (tr), and double cascades (dc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We do the same for atmospheric events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The flux-reweighing procedure is as follows: from the public HESE data release [7, 64], we extract the weight wk ref,t associated with the k-th MC event of topology t, generated by a neutrino of energy Eν,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Events in the MC sample were originally generated assuming as reference flux the best-fit pure-power-law flux from the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE analysis [7], Φν,ref = Φ0,ref(Eν/100 TeV)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='87, with Φ0,ref = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='68 × 10−18 GeV−1 cm−2 s−1 sr−1, and exposure time Tref = 2635 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Given a new flux Φν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', our two-component model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (3), and exposure time T, the mean number of events of topology t is µν,ast ij,t (θ) = � k Φν(Eν,k, θ)T Φν,ref(Eν,k)Tref wk ref,t , (4) where the sum is restricted to MC events whose recon- structed deposited energy, Edep,k, falls within the i-th bin and whose reconstructed deposited direction, cos θrec z,k, falls within the j-th bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Atmospheric neutrinos and muons To account for the irreducible atmospheric background (Section III C), we extract from the IceCube HESE MC sample [64] the baseline number of conventional atmo- spheric neutrinos, N ν,c ij,t, prompt atmospheric neutrinos, N ν,pr ij,t , and atmospheric muons, N µ ij,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (In practice, we do this by setting the astrophysical flux to zero in the MC reweighing, and extracting the resulting event rates, which are purely atmospheric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') The baseline atmospheric event rates in the MC sample were produced using the MC generator of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [164];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Section III C for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We keep the shape of the atmospheric background event distributions fixed (Section III C), but allow their normalization constants, N ν,c, N ν,pr, and N µ, to float independently of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For a specific choice of their values, the number of background events of topology t is µatm ij,t (η) = N ν,cN ν,c ij,t + N ν,prN ν,pr ij,t + N µN µ ij,t , (5) where η ≡ (N ν,c, N ν,pr, N µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Likelihood function The mean number of HESE events of topology t in each bin, of astrophysical and atmospheric origin, is µij,t(θ, η) = µν,ast ij,t (θ) + µatm ij,t (η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (6) We use the same binning as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7]: NEdep = 21 bins evenly spaced in log10(Edep/GeV), between 60 TeV and 10 PeV, and Ncθrec z = 10 bins evenly spaced in cos θrec z , between -1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To compare our predicted HESE event rate, µij,t, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' the observed 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE sample or projected ver- sions of it, N data ij,t , we use a binned Poissonian likelihood, L (θ, η) = NEdep � i=1 Ncθrec z � j=1 {c,tr,dc} � t Lij,t(θ, η) , (7) where the likelihood in each bin, for event topology t, is Lij,t(θ, η) = µij,t(θ, η)N data ij,t N data ij,t !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' e−µij,t(θ,η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (8) 10 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Free model parameters, their priors, best-fit values and allowed ranges, from a fit to the IceCube 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE event sample [7, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Allowed parameter ranges are 68% one-dimensional marginalized credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Results are for fits with a pure power law (“Pure PL”), a power law with an exponential cut-off (“PL w/ cut-off”), and a power law with a cut-off plus a bump (“PL + B”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The former two serve as validation of our method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' we find parameter values similar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For the latter, we only show the values of the parameters that maximize the posterior, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (We keep some nuisance parameters of the original HESE analysis[7] fixed to their nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') See Sections III E and IV A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Parameter Prior Fit to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-yr IceCube HESE sample Symbol Units Description Pure PLa PL w/ cut-offb PL + Bc Physical parameters, θ Power law Φ0,PL GeV−1 cm−2 s−1 sr−1 Flux norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' at 100 TeV Log10-uniform ∈ [−20, −15] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='9+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='9+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 × 10−17 γ — Spectral index Uniform ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='3 Eν,cut GeV Cut-off energy Log10-uniform ∈ [4, 8] — 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='9+72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='4 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='7 × 105 Bump E2 ν,bumpΦ0,bump GeV cm−2 s−1 sr−1 Flux norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' at Eν,bump Log10-uniform ∈ [−10, −5] — — 3 × 10−8 αbump — Energy width of bump Uniform ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='1, 10] — — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='4 Eν,bump GeV Central energy of bump Log10-uniform ∈ [4, 7] — — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='4 × 106 Nuisance parameters, η N ν,c — Flux norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', convent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ν Gaussian, µ = 1,σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='96 N ν,pr — Flux norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', prompt atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ν Uniform ∈ [0, 10] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='17 N µ — Flux norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' µ Gaussian, µ = 1, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='05 a Mean value of the one-dimensional marginalized posterior ± 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The mean value coincides with the best-fit value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' b Mean value of the one-dimensional marginalized posterior ± 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The mean value coincides with the best-fit value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' c Best-fit, or maximum a posteriori, value of the full posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Because of correlations between parameters in the full posterior, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (9), this value does not coincide with the mean value when using the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The likelihood in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (7) accounts for the contribution of events in all energy and direction bins, and of all topolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The associated posterior probability distribution is P(θ, η) = L (θ, η) π (θ) π (η) Z , (9) where π(θ) and π(η) are the prior distributions for the astrophysical-flux parameters, θ, which are physical, and of the atmospheric-background parameters, η, which are nuisance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (9), the denominator is the evidence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', the posterior marginalized over all parameters, Z = � dθ � dη L (θ, η) π (θ) π (η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (10) We use UltraNest [165], an efficient importance nested sampler [166, 167], to maximize the posterior, find the best-fit and allowed ranges of parameter values, and com- pute the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Parameter priors and look-elsewhere effect Table I summarizes our choice of priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For the phys- ical parameters, θ, we adopt uniform priors over wide ranges to avoid introducing bias in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We use log- uniform priors for the flux normalization of the power-law and bump components, Φ0,PL and E2 ν,bumpΦν,bump, the energy of the exponential cut-off of the power law, Eν,cut, and the central energy of the bump, Eν,bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This allows them to more easily vary over wide ranges of values in order to capture a vast array of possibilities for the rela- tive contributions of the power-law and bump-like com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For the power-law spectral index, we restrict γ ≥ 2, as typically expected for pp sources with diffusive shock acceleration (Section II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For the energy width of the bump, αbump, we choose αbump > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='1, to avoid intro- ducing bumps so wide as to be mistaken for hard power laws over the entire energy range of our analysis, and αbump < 10, since narrower bumps are likely unrealistic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For the nuisance parameters, η, we adopt the same priors used in the IceCube 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE analysis [7], which are extracted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' They represent the uncertainty in the underlying models of cosmic-ray spec- trum and hadronic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For the prompt neutrino flux normalization, N ν,pr, we adopt a uniform prior up to 10, rather than a positive unbounded one as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Since our fits below are all compatible with N ν,pr = 0 (see Table I), our use of a more restrictive prior does not modify our results significantly compared to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In analogy with searches for resonances in collider data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' in searching for bump-like features in the diffuse high- energy neutrino spectrum we must account for the trials factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' or “look-elsewhere effect.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This is the decrease in the statistical significance with which the existence of a bump can be claimed due to the possibility of there be- ing spurious bump-like features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' mere random statistical fluctuations of the event rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' anywhere in the energy range that is relevant to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In a Bayesian approach like ours, integrating the likelihood over wide prior ranges in order to compute the evidence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (10), automatically accounts for the look-elsewhere effect by penalizing large prior volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Bump discovery Bayes factor We evaluate the preference for a two-component, power-law-plus-bump flux model (PL+B), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (3), vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' a one-component, power-law flux model (PL), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (1), via the Bayes factor B = ZPL+B ZPL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (11) We compute the evidence ZPL+B using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (9), and the evidence ZPL using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (9) with E2 ν,bumpΦ0,bump = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', with only the power-law flux component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The higher the value of B, the higher the preference of the data for the two-component flux model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Broadly stated, narrow bumps are hard to identify—unless they are very tall— because they only affect the event rate within a narrow energy window, while wide bumps are hard to identify be- cause they may resemble a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In-between these extremes, discovery may be more feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We adopt Jef- freys’ criteria to classify the preference qualitatively into barely worth mentioning, 100 ≤ B < 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' substantial, 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 ≤ B < 101;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' strong, 101 ≤ B < 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' very strong, 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 ≤ B < 102;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' and decisive, B ≥ 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our likelihood, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (7), is valid but approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Because our predicted astrophysical HESE event rates are obtained by reweighing the HESE MC sample (Sec- tion III D 1), Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [169] proposed using a more sophisti- cated, though computationally expensive, likelihood pre- scription that accounts for random fluctuations intrinsic to the MC sample itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, in our analysis, we forego this after verifying, in Section III E below, that our approach reproduces closely the best-fit values and al- lowed intervals reported in the analysis performed by the IceCube Collaboration [7] using a one-component power- law-flux fit to the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Validation: power-law fits to present-day data As validation of our method, we fit a pure power law and a power law with exponential cut-off to the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE event sample, as in the IceCube analysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Table I shows the best-fit and 68% confidence inter- vals for the free parameters in each case (“Pure PL” and “PL w/ cut-off”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In both cases, our results approximate those of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For the power law with exponential cut-off, the best-fit value of Eν,cut is at a few PeV, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [7], but has a large uncertainty, so it should be treated only as a weak suggestion, which we do below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' A PEV BUMP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' First, we apply our methods above to the present-day, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We find marginal preference (B ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='7) for a one-component flux model—a power law flux with a cut-off at a few hundred TeV— vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' a two-component flux model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Then we adopt the best-fit two-component flux that we find—a steep power law with a bump centered at roughly 1 PeV—as template for a possible real two-component flux scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We use it to forecast what detector exposure would be needed to discover a PeV bump, which would require combining contributions of several neutrino telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Bump-hunting in present-day IceCube data Applying the statistical procedure introduced in Sec- tion III D to the present-day, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE sample, we find a value for the Bayes factor, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (11), of B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Following Jeffreys’ criteria, this rep- resents mild preference for a one-component power-law flux, to explain the data vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' a two-component power-law- plus-bump flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Table I shows the best-fit values of the model parameters and their allowed ranges in each case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', “PL w/cut-off” vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' “PL + B”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Appendix B contains details on the full posterior for the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 1 (also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 3) shows the present-day best-fit two-component flux: a steep power law, with γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='75 and cut-off at Eν,cut = 170 TeV, followed by a prominent, relatively wide bump centered at Eν,bump = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='4 PeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This flux explains the dearth of HESE events between 300 TeV and 1 PeV (see snapshot A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1) by this being the energy range where the power-law flux from a population of pp sources vanishes and before the bump- like flux from a population of pγ sources becomes ap- preciable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' A PeV bump could be indicative, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', of blazars [50], low-luminosity GRBs [106], or TDEs [107] as sources of PeV-scale neutrinos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2 Although we find evidence against a two-component flux model to explain the present-day HESE data, in what follows we entertain the possibility that instead the present-day best-fit two-component flux is borderline pre- ferred, for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' First, the present-day preference against the two-component flux is only marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Small changes to the priors, data, or analysis methods, could conceivably change the value of B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='1 that we find into B ≳ 1, which would represent no preference between the one-component and two-component flux models, or marginal preference for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Second, our preference for a two-component flux with a PeV bump is compati- ble with similar results from previous works [28, 66, 130], obtained using different methods or event samples (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [118] for an origin in dark matter decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, below we adopt our best-fit two-component flux to forecast the near-future prospects of discovering a PeV bump, using larger HESE samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (Later, in Section V, we consider bumps centered at other energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 12 10−1 100 101 102 Astrophysical ν: Power-law flux Atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' muons Atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ν, conventional Atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ν, prompt Astrophysical ν Astrophysical ν: Power-law flux 105 106 Deposited energy, Edep [GeV] 10−1 100 101 102 Astrophysical ν: Power-law + PeV bump flux −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 Reconstructed direction, cos θrec z Astrophysical ν: Power-law + PeV bump flux Expected HESE event rate (2035, all detectors, proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Projected HESE event rates by the year 2035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The combined detector exposure is due to all the neutrino telescopes expected at that time (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1): Baikal-GVD [70, 71], IceCube, IceCube-Gen2 [170], KM3NeT [72–74], P-ONE [171], TAMBO [172], and TRIDENT [173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Events are distributed in reconstructed deposited energy (left) and reconstructed direction (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Top: Assuming a power law with a high-energy cut-off, with flux parameters fixed at their present-day best-fit values (“PL w/ cut-off” in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Bottom: Assuming a power law with a high-energy cut-off plus a PeV bump, with flux parameters fixed at their present-day best-fit values (“PL + B” in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The HESE-detection efficiency of upcoming detectors is assumed to be equal to that of IceCube today [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' their combined exposure by 2035 is equivalent to 159 years of IceCube exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Modeling near-future neutrino telescopes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Assumptions about future neutrino telescopes Following Section IV A, we forecast the discovery prospects of our best-fit two-component flux (Table I) based on larger HESE event sample made possible by upcoming TeV–PeV neutrino telescopes, currently in op- eration, construction, and planning stages [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Because detailed information about their detection capabilities, or simulations of them, are not publicly available at the time of writing, and because all of them are in-water or in-ice optical Cherenkov detectors (with the exception of TAMBO [172], see below), we model each as a re-scaled version of IceCube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' While this simple procedure admit- tedly does not capture the differences between detector designs, photomultiplier efficiency, backgrounds, atten- uation and scattering length of light in water and ice, systematic errors, and analysis techniques, it allows us to produce informed estimates of upcoming event rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 1 shows the effective volume of each detector, relative to IceCube, and their tentative start dates, which may change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' By 2030, we expect nearly an order-of- magnitude increase in the combined detector exposure to high-energy astrophysical neutrinos, thanks to the continuing operation of IceCube and the completion of Baikal-GVD [70, 71] and KM3NeT [72, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' After 2030, we expect a faster growth of the event rate thanks to the construction of new detectors IceCube-Gen2 [170], P-ONE [171], TAMBO [172], and TRIDENT [173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To compute future event samples of a detector, we re- scale the number of events in the IceCube MC sample by a factor equal to the size of the detector relative to IceCube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We only account for the contribution of a detec- tor after it has reached its full target size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' by doing this, we ignore possible contributions from partially finished detector configurations, which may be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Given the commonalities between detectors, we safely assume that they will all be capable of detecting HESE or HESE-like events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (This is less clear for TAMBO, which is the only detector among the ones that we consider that is a surface array of water Cherenkov tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, TAMBO, whose science case is specific to multi-PeV ντ detection, represents only a small contribution to the to- tal event rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') Further, we assume that their efficiency to detect HESE events will be the same as in IceCube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This is likely an optimistic assumption, which implies that the bump discovery prospects that we find later are, 13 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This assumption could be revisited in revised fore- casts, as details on upcoming detectors become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, we sketch the relevant features of each detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (Combining multiple neutrino telescopes at differ- ent locations into a global monitoring system, like PLEνM [174], would also increase the field of view to high-energy neutrinos and significantly boost the chances of discovering point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [174] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Overview of near-future neutrino telescopes Baikal-GVD [70, 71], the successor of Baikal NT- 200 [175], is an in-water detector currently under con- struction in Lake Baikal, Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' It has been operat- ing in partial configuration since 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' in 2022, its effec- tive volume was about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='35 km3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Recently, it reported the detection [176] of a high-energy astrophysical neu- trino from the TXS 0506+056 blazar previously observed by IceCube [8, 9], and of the IceCube diffuse flux of high-energy astrophysical neutrinos, with a significance of about 3σ [177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We assume a start date for the full Baikal-GVD of 2025, with an effective volume of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 km3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' IceCube-Gen2 [170] is the envisioned upgrade of Ice- Cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We consider its in-ice optical array, composed of 120 new detector strings, that will extend the effective volume of IceCube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Because the new strings will be more sparsely deployed than in IceCube, the HESE detection efficiency of IceCube-Gen2 might be lower;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' this is not captured by our forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (There is an additional en- visioned radio-detection component that targets the dis- covery of ultra-high-energy neutrinos [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') We assume a start date for the full IceCube-Gen2 optical array of 2030, with an effective volume of 8 km3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' KM3NeT [72, 74], the successor of ANTARES [179], is an in-water detector currently under construction in the Mediterranean Sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Its high-energy component, ARCA, targets high-energy astrophysical neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Of the 230 detection units planned at ARCA, 19 units are already deployed and operating in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' It is expected that a building block of 115 units will be able to measure the diffuse flux detected by IceCube in about a year of ob- servation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We assume a start date for the full KM3NeT of 2025, with an effective volume of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='8 km3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' P-ONE [171], the Pacific Ocean Neutrino Experiment, is an in-water detector, currently under planning and pro- totyping, to be deployed in the Cascadia Basin, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' P-ONE will have 70 detector strings with 20 detector modules each, instrumented over a cylindrical volume with radius 1 km and height 1 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The first prototype string is expected to be deployed in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We assume a start date for the full P-ONE of 2030, with an effective volume of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='2 km3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' TAMBO [172], the Tau Air-Shower Mountain Based Observatory, is a proposed surface array of water Cherenkov tanks to be located in a canyon in Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' It tar- gets Earth-skimming ντ with energies of 1–100 PeV that interact on one side of the canyon and produce a high- energy tauon whose decay triggers a particle shower that is detected on the opposite of the canyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The detection strategy of TAMBO is different from IceCube, and its en- ergy range, while overlapping, extends to higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, because detailed simulations are unavailable at the time of writing, we model it as a small version of IceCube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We assume a start date for the full TAMBO of 2030, with a target effective volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 km3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' TRIDENT [173], The tRopIcal DEep-sea Neutrino Telescope, is a proposed in-water detector to be located in the South China Sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' TRIDENT is expected to be able to detect a transient neutrino source like TXS 0506+056 [8, 9] with 10σ significance and the steady- state neutrino source NGC 1068 [12] within two years of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We assume a start date for the full TRIDENT of 2030, with a target effective volume of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 km3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Projected discovery prospects In the near future, the increased event statistics pro- vided by the combined exposure of the above detec- tors will enhance our ability to discriminate between a one-component and a two-component diffuse flux model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To quantify this, below we forecast and compare future HESE event rates for both flux models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For benchmark- ing, we assume that the true diffuse flux is the present- day best-fit two-component flux found in Section IV A—a steep power law followed by a PeV bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We follow the same procedure detailed in Section IV A to compute the projected Bayes factor, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (11), that compares the evi- dence for the benchmark two-component flux vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' the evi- dence for the one-component flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We do this for increas- ing values of the IceCube-equivalent combined detector exposure, as delineated in Section IV B, from halfway through the year 2017—the end of data-taking of the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year HESE data sample—to the year 2035;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1 To produce our forecasts, we assume that the future observed event rates coincide with the expected event rates, which amounts to using an Asimov data sam- ple [180] to find representative results for the Bayes fac- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In a real future event sample, Poisson fluctuations would naturally be present, which could bias the value of the Bayes factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' By using an Asimov data sample, we obtain the median value of the logarithm of the ev- idence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (10), for each flux model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', ZPL+B and ZPL in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' If the distribution of the Bayes factor is Gaussian, as expected from the central limit theorem, this median value coincides with the expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Fur- ther, for growing detector exposure, the relative size of the Poisson fluctuations in the observed event sample shrinks by a factor of 1/ √ N, where N is the total num- ber of observed events, so that their impact on the Bayes factor wanes at longer exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 4 shows, as illustration, the event rates ex- pected in 2035 assuming as true flux the present-day best-fit one-component flux and best-fit two-component flux (“PL w/ cut-off” and “PL + B” in Table I, respec- 14 tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The combined detector exposure corresponds to roughly 159 years of equivalent IceCube HESE exposure (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1) and is due to all of the neutrino telescopes that we consider (Section IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The energy distributions of the events for the one-component and two-component cases are noticeably different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' As expected, for the latter there is a visible excess of events in the PeV region due to its PeV bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In contrast, the angular distributions of events are nearly identical, since they are mostly driven by the isotropy of the high-energy astrophysical neutrino fluxes and by neutrino absorption inside Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Differ- ences in the energy spectra between the two cases only affect the angular distributions indirectly, by changing the intensity of neutrino attenuation inside Earth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' these differences are small in the TeV–PeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 1 shows how the Bayes factor grows with com- bined detector exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Its rate of growth increases when new detectors are added to the combined expo- sure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1, this is seen as a kink in the slope of the Bayes factor curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' As expected, because of growing event rates, the longer the exposure, the clearer the sep- aration between the evidence for the one-component and two-component flux fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We illustrate the growing sepa- ration via four snapshots of the best-fit and 68% allowed bands of the fluxes, A–D, from present-day to 2035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We conclude that the combined exposure of IceCube, Baikal-GVD, and KM3NeT may provide decisive evi- dence in favor of a two-component flux with a PeV bump already by 2027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (This is contingent on future detec- tors having IceCube-like HESE-detection capabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Section IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') Alternatively, IceCube plus IceCube-Gen2 may achieve the same by 2031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In any case, a prominent population of pγ sources of PeV neutrinos could be dis- coverable in the diffuse flux within only a few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' HUNTING FOR TEV–PEV BUMPS Section IV explored the discovery of a prominent PeV bump in the diffuse high-energy neutrino flux, which is only marginally disfavored by present-day HESE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Next, we use the same statistical methods to explore the more general case of constraining or discovering a bump of varying size anywhere in the TeV–PeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Constraining subdominant bumps If future HESE observations were to still favor a one- component power-law description of the diffuse flux, we could place upper limits on the height of a coexistent bump component, which must be necessarily subdom- inant so that it does not disrupt the preference for a power-law description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We compute the limits as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For given values of the position of the bump, Eν,bump, which we vary in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5, and of its width, which we keep fixed at the representative value of αbump = 1 in the main text, we compute the posterior under the two-component 105 106 107 Central energy of ν bump, Eν,bump 10−10 10−9 10−8 10−7 10−6 ν bump height, E2 ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] Projected upper limits (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ): Present-day best-fit power law ∝ E−γ ν e−Eν/Eν,cut (γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='75, Eν,cut = 4 PeV) 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') All limits: bump width αbump = 1 IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 2025, IceCube only 2035, IceCube only 2035, IceCube+Gen2 only 2035, all detectors FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Upper limits on the height of a bump in the dif- fuse flux of high-energy astrophysical neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The bump flux component, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2), is centered at energy Eν,bump, has height E2 ν,bumpΦν,bump, and width αbump = 1, and is over- laid on a power-law flux ∝ E−γ ν eEν/Eν,cut, with parameter values given by the best fit to the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE sample [7] (“PL w/ cut-off” in Table I), shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Section II D and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 3 for the definitions of the flux com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Today, IceCube limits the height of a bump centered at a few hundred TeV to be, at most, comparable to the size of the dominant power-law component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In the future, the upper limit may be tightened to tens of percent of the power-law com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 6 shows how this translates into constraints on candidate neutrino source populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Section III for the statistical analysis and Section V A for details on this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' flux model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (9), and marginalize it over all the free model parameters (see list in Table I), except for the bump height, E2 ν,bumpΦ0,bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We integrate the resulting one-dimensional marginalized posterior to find the 95% credible interval on the bump height, for each value of the bump position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Differently from our previous results, in drawing constraints on the bump height we adopt a flat linear prior on it, rather than a logarithmic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (Oth- erwise, because the posterior is flat for arbitrarily low values of the bump height, limits drawn using a logarith- mic prior would differ depending on our arbitrary choice of the lower end of the logarithmic prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') Figure 5 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Present-day limits, based on the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE sample, disfavor especially the presence of relatively wide bumps (αbump = 1) cen- tered around 200 TeV, where event statistics are higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For all values of Eν,bump, the limit lies above the present- day best-fit power-law flux, meaning that a sizable con- tribution to the diffuse flux from a population of photo- hadronic sources cannot presently be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The lim- 15 its are weaker for bumps centered at lower energies, where the atmospheric background is higher, and at higher en- ergies, where statistics are poorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The weakening above 500 TeV reflects the fact that a two-component flux with a bump between hundreds of TeV and a few PeV is only marginally disfavored in present-day data (Section IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 5 shows limits for αbump = 1, but marginal- izing over αbump yields comparable results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C2 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' If the dominant power-law component is harder, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', ∝ E−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 ν , the limits weaken at low energies and strengthen at high energies, but the overall conclu- sions are unchanged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' D1 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The limits are expected to strengthen with more statis- tics, made available by the continued operation of Ice- Cube and by upcoming detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We forecast limits us- ing larger combined detector exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To do this, we as- sume that the true diffuse flux coincides with the present- day best-fit power-law flux, “PL w/ cut-off” in Ta- ble I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For upcoming detectors, we use the same IceCube- equivalent exposures as in Section IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We choose two reference years, 2025, using IceCube only, and 2035, us- ing IceCube only, IceCube plus IceCube-Gen2, and the combination of all detectors available by then (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 5 shows that future HESE data may finally limit the bump height to be a fraction of the size of the domi- nant power-law component, especially at energies below 1 PeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The limits strengthen roughly as the square root of the ratio of future combined exposure to present-day exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Unlike present-day limits, they do not weaken above 500 TeV because they are obtained from Asimov event samples generated assuming a power-law flux and, therefore, are by design inconsistent with the presence of a bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 5 shows that by 2035, IceCube could limit the height of a bump with αbump = 1 and centered at 100 TeV to be 86% of the present-day best-fit power- law component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' combined with IceCube-Gen2, 66%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' and, combining all detectors, 47%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The above findings reveal promising power, accessible by 2035 and with IceCube alone, to constrain a poten- tial dominant contribution of photohadronic sources at around 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' With the help of future detectors, con- straints may improve by about a factor of 2 by 2035 and apply also to bumps at PeV energies, contingent on having IceCube-like HESE-detection capabilities (Sec- tion IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, we discuss what these limits entail for the properties of candidate source populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Constraints on source populations In Section II, we motivated the existence of a bump-like component in the diffuse flux as coming from a popula- tion of sources that make neutrinos via pγ interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below, we translate the upper limits that we found in Section V A on the bump height into upper limits on the local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', redshift z = 0) high-energy neutrino lumi- nosity density of candidate pγ source populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The translation depends on the values of the bump parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' As benchmark, we pick Eν,bump = 1 PeV for the central energy of the bump and αbump = 1 for its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Appendix A shows how we relate the size of the diffuse neutrino flux to the local neutrino luminosity density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We show results for steady-state sources only, though similar results can be obtained for transient sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 6 shows results using present-day, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year Ice- Cube HESE data, and the same projections of combined detector exposure as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [181], we consider three different possibilities for the redshift evo- lution of the source luminosity density, representative of different candidate source classes: no evolution, evolu- tion following the star-formation rate (SFR) [182], and strong, FSRQ-like evolution [183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Each source class is assumed to be independently dominant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', to saturate the local high-energy neutrino luminosity density [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Present-day point-source limits from IceCube [170, 181] already disfavor FSRQs, BL Lacs, and galaxy clus- ters as the dominant source class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In contrast, our present-day limits from bump search are too weak to con- strain any of the candidate source classes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 6 as the dominant pγ emitter of PeV neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This is consistent with our finding in Section V A that present-day data al- low for a bump taller than the power-law component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' By 2035, the situation evolves favorably for our limits from bump search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' There, our limits match the power of point-source limits drawn from ten years of IceCube- Gen2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' If there is indeed no evidence for a PeV bump, our limits using the combined exposure of IceCube plus IceCube-Gen2 could put to test the independent domi- nance, as PeV pγ sources, of all the source classes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In fact, using IceCube alone already provides nearly the same power (although, if IceCube-Gen2 is present, its contribution quickly becomes dominant after 2035).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The combined detector exposure of all detectors by 2035 affords even more stringent limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Since Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5 shows that the projected limits on the bump height strengthen for bumps centered at hundreds of TeV, we expect the corresponding limits on the lu- minosity density of pγ sources that emit those bumps to strengthen, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Similarly, the limits on the lumi- nosity density for wider and narrower bumps, and for a harder power-law flux component, trace the limits on bump height shown in Figures C2 and D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Discovering bumps In Section V B, we placed limits on the height of bumps in the diffuse flux if no evidence for them is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Now we answer a related question: if a bump exists, how large should the detector exposure be to discover it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Like in Section V B, we take the true flux to be the present-day best-fit power-law flux (“PL w/ cut-off” in Table I), but now we add a subdominant bump to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We vary the bump position, Eν,bump, and height, E2 ν,bumpΦν,bump, and, like before, we fix the bump width at a representative value of αbump = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For each choice of 16 1040 1042 1044 1046 1034 1035 1036 1037 1038 Lum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' density, z = 0 [erg s−1 Mpc−3] Projected upper limits (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ): All limits: bump width αbump = 1 No redshift evolution 2025, IceCube only 2035, IceCube only 2035, IceCube+Gen2 only 2035, all detectors 1040 1042 1044 1046 Source luminosity [erg s−1] Point sources (IceCube, 10 yr) Point sources (IC-Gen2, 10 yr) SFR redshift evolution IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 1040 1042 1044 1046 Strong redshift evolution FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Upper limits on the local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', redshift z = 0) high-energy neutrino luminosity density of steady-state source candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our new limits apply to pγ source populations that emit a diffuse neutrino spectrum with a bump centered at Eν,bump = 1 PeV and with width αbump = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' they are interpretations of the limits from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Within a population, all sources are identical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' they have the same neutrino luminosity in their rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We show candidate source classes without distinction between mostly pp and mostly pγ sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In each panel, the neutrino luminosity density evolves with redshift differently: no evolution (left), star-formation rate (SFR) evolution (center), and strong (FSRQ-like) evolution (right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For each source class, its local luminosity density is chosen to saturate the present-day high-energy neutrino flux [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our limits put this assumption to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Limits from searches for point neutrino sources are from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our limits show that by 2035 the combined exposure IceCube plus IceCube-Gen2, or of all available detectors, could constrain the source luminosity density of pγ to a fraction of what is needed to saturate the diffuse flux at 1 PeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Section V B for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' parameter values, we compute the Bayes factor for bump discovery, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (11), following the methods in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 7 shows the results computed at the same snap- shots of combined detector exposure used in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For comparison, we include the present-day best-fit power-law flux and its 68% allowed band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our results mirror what we found for the bump constraints in Sec- tion V B: discovering a subdominant bump component that is smaller than the dominant power-law component will require the combined exposure of all the detectors available by 2035 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Further, it will only be possible if the bump is located in the energy region with higher statistics, around 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Appendix D shows results obtained using instead a harder power-law flux, with γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5, and no cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 8 illustrates the projected 68% allowed flux bands obtained from one-component and two-component fits to a specific realization of the true flux, picked from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 7: a power law with a subdominant bump centered at 141 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Broadly stated, the one-component and two- component explanations can be discriminated between when their allowed flux bands shrink to a size compa- rable to the difference between the true power-law flux and the true power-law-plus-bump flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Because such a difference is tiny, this is only possible with the combined detector exposure expected by 2035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' FUTURE DIRECTIONS Using other bump shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='— We searched for log- parabola bumps in the diffuse flux as generic proxies of the different bump shapes expected from different source classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Future dedicated searches for the im- prints of specific photohadronic source classes could use alternative bump shapes predicted by source models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Varying systematic detector parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='— In our anal- ysis, we varied the normalization of the atmospheric neu- trino and muon backgrounds, but fixed other parameters associated to their shape and to detector systematics to their nominal expectations (Section III B), in order to re- duce the time needed for our large parameter space scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Nevertheless, the IceCube HESE MC sample allows for varying them as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Doing so would naturally reduce the sensitivity of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Yet, the fact that in the analysis performed by the IceCube Collaboration [7] most of these parameters affect the fits only weakly might be indicative of their possibly limited effect on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Using other event types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='— So far, our analysis has used only HESE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Using other event types would come at the expense of introducing a larger atmospheric background and poorer energy reconstruction, but may be worth it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Including the IceCube 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year sample of through-going muons [65] would increase the statistics massively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Reference [174] shows an example of char- 17 10−9 10−8 10−7 10−6 Present-day best-fit power law ∝ E−γ ν e−Eν/Eν,cut (γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='75, Eν,cut = 4 PeV) 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2025, IceCube only (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') Upper limit (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ), IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr 10−9 10−8 10−7 10−6 Bump height E2 ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] 2035, IceCube only (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 10−9 10−8 10−7 10−6 2035, IceCube+Gen2 only (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 105 106 107 Central energy of ν bump, Eν,bump [GeV] 10−10 10−9 10−8 10−7 10−6 All panels: bump width αbump = 1 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 Bayes factor for bump discovery, log10 B Substantial Strong Very strong Decisive FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Projected discovery potential of a bump in the diffuse flux of high-energy neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The bump flux component, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2), is centered at energy Eν,bump, has height E2 ν,bumpΦν,bump, and width αbump = 1, and is overlaid on a power-law flux ∝ E−γ ν eEν/Eν,cut, with parameter values given by the best fit to the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE sample [7] (“PL w/ cut-off” in Table I), shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The Bayes factor that quantifies the discovery potential, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (11), is ob- tained in a two-component flux fit to projected event distribu- tions, and is marginalized over the power-law flux parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure 5 shows a corresponding plot of bump constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' from top to bottom, the snapshots here are the same as in that figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Decisive discovery of a subdominant bump may be achieved by 2035, using IceCube-Gen2 or, more prominently, using all planned upcoming neutrino telescopes available at the time (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Section III for the statistical analysis and Section V C for details on this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The white star (⋆) marks the bump flux parameters chosen to make Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' acterizing the diffuse flux using through-going muons in 10−9 10−8 10−7 10−6 2025, IceCube only (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') True power law True bump 10−9 10−8 10−7 10−6 All-flavor diffuse ν flux, E2 ν Φν [GeV cm−2 s−1 sr−1] 2035, IceCube only (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') True two-component flux ( ) Two-component fit (68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') One-component fit (68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 10−9 10−8 10−7 10−6 2035, IceCube+Gen2 only (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 105 106 107 Neutrino energy, Eν [GeV] 10−10 10−9 10−8 10−7 10−6 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Illustration of the separation between a one-component vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' a two-component fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We assume as the true flux, picked from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 7 (marked with ⋆ therein), a bump with normalization E2 ν,bumpΦν,bump = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='6 × 10−8 GeV cm−2 s−1 sr−1, width αbump = 1, and centered at energy Eν,bump = 141 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' From top to bottom, the snap- shots and corresponding combined detector exposure here are the same as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Section III for the statistical analysis and Section V C for details on this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' PLEνM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Including the sample of medium-energy starting events (MESE) [126] would allow us to look for bumps below 60 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This is particularly interesting in view of the suggested photohadronic origin of medium-energy neutrinos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [32, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Using priors informed by point-source and stacked- source searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='— To avoid introducing bias to our re- sults above, we adopted flat, uninformed priors for the flux parameters (Section III D and Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Yet, point- 18 source and stacked-source searches carried out in parallel may provide complementary limits, hints, and discover- ies on individual sources and source classes that could be interpreted as informed priors on the power-law and bump parameters of our analysis, strengthening it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Considering more flux components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='— Reference [66] considered, in addition to pp and pγ neutrino flux com- ponents of extragalactic origin, similar to ours, a pp neu- trino component of Galactic origin, subdominant to the other components and contributing mainly below about 1 PeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' So far, the contribution of Galactic neutrinos to the diffuse flux is limited to be at most a few tens of per- cent of the total [15, 184–187], but this may change with more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Further, ANTARES recently reported the de- tection of TeV neutrinos from the Galactic Ridge [188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, future versions of our analysis could include a Galactic component, which may induce a directionally dependent excess of events towards the Galactic Center in the low-energy range of the event sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' SUMMARY AND OUTLOOK Despite important experimental advances, the origin of the bulk of the TeV–PeV astrophysical neutrinos dis- covered by IceCube remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Recent success in discovering point neutrino sources [8–12], while out- standing, accounts for only a small fraction of the to- tal number of neutrinos detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, we have ex- plored a parallel strategy to probing their origin: to glean from the shape of the diffuse energy spectrum of high- energy neutrinos—made up of the contributions of all high-energy neutrino sources—insight into the identity of dominant, co-dominant, and subdominant classes of neutrino source populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Motivated by previous analyses that looked for dif- ferently shaped diffuse energy spectra [3–5, 7, 65, 126– 130] or contributions of multiple source populations to it [28, 66], we performed a systematic search in the energy spectrum of present-day IceCube data and made fore- casts based on expected future data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We looked for fea- tures that could be imprinted on the diffuse spectrum by two broad source classes: sources that make neutrinos via proton-proton (pp) interactions—like starburst galaxies and galaxy clusters—and sources that make neutrinos via photohadronic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', proton-photon (pγ) interactions— like active galactic nuclei, gamma-ray bursts, and tidal disruption events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Generally, the former are expected to yield a power-law flux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' the latter, a bump-like flux con- centrated around a characteristic energy (Section II B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The strength of our analysis is triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' First, we use the same observed and mock data as the IceCube Collabora- tion uses in their own analysis [7, 64], including detailed detector resolution and geometry, and atmospheric neu- trino and muon backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Second, because we adopt flexible spectral shapes for the power-law and bump-like fluxes, we probe many different shapes and relative sizes of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Third, we extend our analysis to the expected combined exposure of multiple upcoming neutrino detec- tors, to deliver on the full potential of our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' As observed data, we use the recent IceCube 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5- year public HESE (High-Energy Starting Event) sam- ple [7, 64], because of its high purity in astrophysical neutrinos (Section III A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To test different shapes of the diffuse spectrum, we used the public HESE Monte Carlo event sample provided by the IceCube Collaboration [64] (Section III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our statistical analysis is Bayesian, and uses wide, unbiased priors for the model parameters to avoid introducing bias (Section III D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Overall, we find that hunting for bumps in the diffuse high-energy neutrino flux may indeed reveal important insight about a photohadronic origin of the neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below we summarize our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Bump-hunting could test whether PeV neutrinos are made by the same population of pp sources that make 100-TeV neutrinos, or by a separate population of pho- tohadronic sources, a scenario that has been proposed before [28, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We find that present-day HESE data are best described by a power-law diffuse flux, though that description is only marginally preferred over an al- ternative one containing in addition a PeV bump (Sec- tion IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' If this bump is truly present, we find that it could be decisively discovered already by 2027 using the combined exposure of IceCube, Baikal-GVD [70, 71], and KM3NeT [72, 74], or by 2031 using the combined expo- sure of IceCube and IceCube-Gen2 [170] (Section IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Even if the diffuse neutrino flux were dominated by a population of pp sources producing a power-law flux, a second population of photohadronic sources could still produce a subdominant bump-like flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Present-day HESE data only place weak constraints on the contri- bution of this second population (Section V A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' By 2035, however, the combined exposure of neutrino telescopes available at the time may limit the contribution of pho- tohadronic sources to the diffuse flux at 100 TeV to be no more than a few tens of percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This would imply upper limits on the local high-energy neutrino luminos- ity density of photohadronic source populations, based on the spectral shape of their flux alone (Section V B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In contrast, discovering a subdominant bump in HESE data, with decisive evidence, will be comparatively more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Only subdominant bumps centered around 100 TeV are likely to be discovered, and only using the combined exposure of multiple detectors (Section V C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our results demonstrate the power to test the pos- sible photohadronic origin of high-energy astrophysical neutrinos by looking for bump-like features in the dif- fuse flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our results are complementary to those from point-source and stacked-source searches, but obtained independently of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In the coming years, they might reveal not just the existence of a population of photo- hadronic neutrino sources, but possibly also its identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 19 ACKNOWLEDGEMENTS We thank Kohta Murase for illuminating discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' DF and MB are supported by the Villum Fonden un- der project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 29388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This work used resources provided by the High Performance Computing Center at the Uni- versity of Copenhagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Appendix A: Connection between bump parameters and astrophysical source parameters Since the diffuse flux of high-energy neutrinos is the ag- gregated contribution of all neutrino sources, the bump- like diffuse flux component, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2) in the main text, is the combination of the individual bumps emitted by all photohadronic sources in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Connecting the bump parameters—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', height, Eν,bump, width, αbump, and position, Eν,bump—to the parameters that describe the population of sources—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', the neutrino luminosity density and the local number density of sources—allows us to translate the limits that we have obtained on the former into limits on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Below we describe our procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 6 in the main text shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Our approach is approximate and based on simple physical considerations, the main one of which is that all sources emit neutrinos with the same spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' the only difference between them is the redshift at which they are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We leave refinements, such as using a luminosity-dependent redshift evolution, for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The comoving neutrino spectrum emitted by any one source in the population is E2 ν dNν dEνdt = Lνω(Eν) , (A1) where Lν is the total neutrino luminosity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', the lumi- nosity integrated over all energies, and ω describes the shape of the neutrino spectrum, normalized so that � ∞ 0 dEν Eν ω(Eν) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (A2) In what follows, we focus on a bump-like spectral shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The diffuse neutrino energy spectrum at Earth is E2 ν dΦν dEνdΩ = Lνnν 4π � ∞ 0 dz ρ(z) H(z)(1 + z)2 ω [Eν(1 + z)] , (A3) where H is the Hubble parameter, ρ is the number density of sources, normalized so that ρ(0) = 1, and nν is the local source number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We assume a ΛCDM cosmology, with the Hubble constant H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='4 km s−1 Mpc−1, and adimensional energy density parameters Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='315, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='685 [189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (A3), the product Lνnν is the local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', at z = 0) high-energy neutrino luminosity density that ap- pears in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The bump width, αbump, in the dif- fuse spectrum at Earth is determined by the two factors: the intrinsic spread in energy of the bump in ω and the spread in redshift of the sources in the population, given by ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Connected to the latter, in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (A3), ω is evaluated at an energy (1+z) times higher than at Earth to account for cosmological expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' On the one hand, if ω is a very narrow bump peaked at comoving energy E⋆ ν, then the width of the bump in the diffuse spectrum is entirely determined by the spread in redshift of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In this case, the diffuse flux can be approximated as E2 ν dΦν dEνdΩ = Lνnν 4π E⋆ ν Eν φ �E0 E − 1 � , (A4) where φ(z) ≡ ρ(z)/[H(z)(1 + z)2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Using for ρ the star- formation rate from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [182], we have verified that the function φ has a bump structure that can be fitted by our log-parabola bump, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2) in the main text, with a width αbump ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Therefore, we conclude that bumps much narrower than that one, with αbump ≫ 2, are not realizable by photohadronic sources, due to the intrinsic spread in their redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Of course, this conclusion de- pends on the choice of redshift evolution of the source number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Accounting for the spread in other pa- rameters of the sources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', the comoving neutrino lumi- nosity or the comoving peak energy, would only increase the width of the bump in the diffuse spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' On the other hand, for wide bumps in the diffuse spec- trum, with αbump ≪ 2, we can assume that the spread mostly comes almost completely from the intrinsic width of ω, since by itself it is larger than the spread induced by the redshift distribution of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In the main text, we adopt this approximation already when we produce results for αbump = 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Do- ing this allows us to connect the diffuse spectrum to the emitted spectrum by the simpler relation E2 ν dΦν dEνdΩ = nνLν 4π ω(2Eν) � ∞ 0 dz ρ(z) H(z)(1 + z)2 , (A5) where we have assumed SFR evolution, so that contri- butions mostly come from sources at z ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For other choices of redshift evolution, the relation between the peak energy of the diffuse spectrum and the peak en- ergy of the individual source spectrum changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' How- ever, evaluating the diffuse flux at its peak value, and assuming for ω the same log-parabola form, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (2), that we use for the diffuse flux, but normalized according to the condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (A2), we can still obtain the connection between the diffuse bump normalization and the intrinsic source luminosity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', E2 ν,bumpΦν,bump = nνLν 4π �αbump π � ∞ 0 dz ρ(z) H(z)(1 + z)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (A6) Numerically, this is E2 ν,bumpΦν,bump = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='13 × 10−7 GeV cm−2 s−1 sr−1 × nν 10−6 Mpc−3 Lν 1043 erg s−1 √αbump ξz , (A7) 20 where ξz is defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (5) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [190], and is equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='8 for SFR evolution, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='6 for no redshift evolution, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='4 for strong, FSRQ-like evolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (A7) to produce Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Appendix B: Details of the posterior probability distribution In the main text (Section III D), we described our sta- tistical procedure to fit the present-day, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE sample [7, 64] using a two-component flux model composed of a power law and a bump, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Here we provide more details on the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' In our fits, we scan over a nine-dimensional parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Table I shows the free parameters: three phys- ical parameters describe the power-law flux component (Φ0,PL, γ, Eν,cut), three physical parameters describe the bump-like flux component (E2 ν,bumpΦ0,bump, αbump), and three nuisance parameters vary the normalization of the atmospheric backgrounds (N µ, N ν,c, N ν,pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure B1 shows the resulting 1σ and 2σ contours of the posterior in the planes of each pair of parameters, marginalized over all the remaining ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Results are for present-day IceCube data and for 2035 forecasts, using the combined exposure of all future detectors (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For the present-day results, in the planes involving the bump parameters, the posterior is multimodal, since the data can be explained either by a soft power law with no bump, by a relatively harder power law and a bump at hundreds of TeV, or by a power law with a cut-off at hundreds of TeV and a PeV bump;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The latter leads to the combination of parameters that max- imizes the posterior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', the maximum a posteriori esti- mator in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Qualitatively, this solution is a multi- component flux model on par with those proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [28, 66], where the power-law component was associated with SBGs and the bump component was as- sociated with photohadronic sources such as blazars or TDEs (see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [102, 103]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, the region of parameter space corresponding to this maximum a posteriori solution is tiny, since it re- quires a tuning between the power-law and the bump pa- rameters such that the bump takes over from the power law after its cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For this reason, the marginalized two-dimensional posterior in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' B1 favors instead an explanation of the data with a single power-law compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This is evidence by the mean value of the posterior corresponding to a low value of log10 E2 ν,bumpΦbump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure B1 also shows the projected contours in 2035, taking the true flux as given by the the present-day max- imum a posteriori solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', a power law followed by a PeV bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The contours shrink significantly, which al- lows a remarkably precise measurement of the bump pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, the planes involving the power-law parameters show evident degeneracy, mainly because our true flux includes a power law cut-off at a few hundred TeV, which could just as well be explained by a very soft power law without a cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Appendix C: Limits on subdominant bumps of different widths In the main text (Section V A), we showed limits on the height of subdominant bumps, assuming a fixed bump width of αbump = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Here we show how the limits change with the bump width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure C1 shows present-day and projected limits on the height of subdominant wide (αbump = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5) and nar- row bumps (αbump = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For wide bumps, the limits weaken for bumps centered in the high-statistics energy region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', around Eν ≈ Eν,bump ≈ 200 TeV, compared to the limits obtained for αbump = 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' This is because wide bumps introduce less sharply defined spec- tral features into the diffuse flux, spread out over a wide energy range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' they are, therefore, harder to spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' For nar- row bumps, in contrast, the limits strengthen for bumps centered in the high-statistics region, because they intro- duce sharper spectral features that are easier to spot us- ing high statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, for narrow bumps the limits weaken at the lowest energies, because the HESE sam- ple that we use contains only events with energy above 60 TeV (Section III B), which makes the analysis sensi- tive to bumps centered at low energies only if they are wide enough to affect also higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure C2 shows analogous limits after the posterior has been marginalized over the bump width, αbump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Compared to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5 and C1, the limits are signifi- cantly weakened at low values of Eν,bump, because narrow bumps are essentially undetectable if centered below the 60-TeV cut in the HESE sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' On the other hand, the main conclusion that we had found in the main text for αbump = 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5 is fortified: by 2035, the combined detector exposure may limit the contribution of a popula- tion of photohadronic sources to a fraction of the diffuse flux from 100 TeV to 1 PeV, regardless of the bump width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Appendix D: Limits and discovery of subdominant bumps assuming a harder power-law spectrum In Section V and Appendix C, we placed limits on and computed the discovery potential to subdominant bumps under the assumption that the true diffuse neutrino spec- trum is the best-fit one-component flux from present-day data (“PL w/ cut-off” in Table I), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=', γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='75 and Eν,cut ≈ 10 PeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' However, a spectrum this soft may be difficult to reconcile with theory expectations of realis- tic cosmic-ray acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Thus, here we show how the limits would change if the true diffuse spectrum were in- stead a harder power law ∝ E −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 ν , with no cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' To find the normalization of the new power law, we fit it to the public 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5-year IceCube HESE sample [7, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' We perform two-component fits to the data, present and future, using the same methods as before (Section V), to 21 0 1 l10Φ0,PL 0 1 2 3 N ν,pr 0 1 2 N ν,c 0 1 2 N µ 0 5 10 αbump 4 5 6 7 l10Eν,bump −2 0 2 l10E2 ν,bumpΦ0,bump 5 6 7 8 l10Eν,cut 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 γ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 γ 5 6 7 8 l10Eν,cut −2 0 2 l10E2 ν,bumpΦ0,bump 4 5 6 7 l10Eν,bump 0 5 10 αbump 0 1 2 N µ 0 1 2 N ν,c Mean value, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 years exposure Maximum a posteriori estimator, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 years exposure 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 0 1 2 3 N ν,pr IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Posterior probability distribution using present-day HESE data and projected HESE data in the year 2035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Each panel shows the two-dimensional posterior, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (9) in the main text, for a pair of parameters, marginalized over all the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' The units for the dimensional parameters are as follows: log10 � ΦPL/ � 10−18 GeV−1 cm−2 s−1 sr−1�� , log10(Eν,cut/GeV), log10 � E2 ν,bumpΦbump/ � GeV cm−2 s−1 sr−1�� , and log10(Eν,bump/GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Projected results assume as true flux the one described by the present-day maximum a posteriori estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' place upper limits on the bump height and to compute the bump discovery Bayes factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure D1 shows the resulting limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Compared to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5 and C2, they are significantly weaker for bumps centered at high energies, due to the larger number of events there that can be explained solely by a hard power- law flux component, which falls more slowly with energy than in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Regardless, the main conclusion that we found in the main text, based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C2, holds: by 2035, the combined exposure of all detectors may limit the contribution of a population of photohadronic sources to a fraction of the diffuse flux in the 100–500 TeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 22 105 106 107 Central energy of ν bump, Eν,bump 10−10 10−9 10−8 10−7 10−6 ν bump height, E2 ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] Projected upper limits (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ): Present-day best-fit power law ∝ E−γ ν e−Eν/Eν,cut (γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='75, Eν,cut = 4 PeV) 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') All limits: bump width αbump = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 2025, IceCube only 2035, IceCube only 2035, IceCube+Gen2 only 2035, all detectors 105 106 107 Central energy of ν bump, Eν,bump 10−10 10−9 10−8 10−7 10−6 ν bump height, E2 ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] Projected upper limits (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ): Present-day best-fit power law ∝ E−γ ν e−Eν/Eν,cut (γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='75, Eν,cut = 4 PeV) 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') All limits: bump width αbump = 2 IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 2025, IceCube only 2035, IceCube only 2035, IceCube+Gen2 only 2035, all detectors FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Upper limits on the height of a bump in the diffuse flux of high-energy neutrinos, for varying bump width, αbump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5, which assumed αbump = 1, but for wide bumps (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5, left) and narrow bumps (αbump = 2, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Section V A in the main text and Appendix C for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Figure D2 shows the resulting discovery potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 105 106 107 Central energy of ν bump, Eν,bump 10−10 10−9 10−8 10−7 10−6 ν bump height, E2 ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] Projected upper limits (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ): Present-day best-fit power law ∝ E−γ ν e−Eν/Eν,cut (γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='75, Eν,cut = 4 PeV) 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') All limits: marginalized over bump width, αbump IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 2025, IceCube only 2035, IceCube only 2035, IceCube+Gen2 only 2035, all detectors FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Upper limits on the height of a bump in the diffuse flux of high-energy neutrinos, marginalized over the bump width, αbump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Same as Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5 and C1, which assumed fixed values of αbump = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5, 1, 2, but marginal- izing the posterior distribution function over αbump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Sec- tion V A in the main text and Appendix C for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Comparec to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 7, the main change is at low values of Eν,bump, where the harder power law predicts fewer events and, therefore, a bump in the spectrum would stand out more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Here also the main conclusion that we found in the main text, based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 7, holds: by 2035, the combined exposure of all detectors may dis- cover a bump whose height is comparable to or smaller than the diffuse flux in the 100–500 TeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 23 105 106 107 Central energy of ν bump, Eν,bump 10−10 10−9 10−8 10−7 10−6 ν bump height, E2 ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] Projected upper limits (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ): Present-day best-fit power law ∝ E−γ ν (γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5) 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') All limits: bump width αbump = 1 IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 2025, IceCube only 2035, IceCube only 2035, IceCube+Gen2 only 2035, all detectors 105 106 107 Central energy of ν bump, Eν,bump 10−10 10−9 10−8 10−7 10−6 ν bump height, E2 ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] Projected upper limits (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ): Present-day best-fit power law ∝ E−γ ν (γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5) 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') All limits: marginalized over bump width, αbump IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 2025, IceCube only 2035, IceCube only 2035, IceCube+Gen2 only 2035, all detectors FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Upper limits on the height of a bump in the diffuse flux of high-energy neutrinos, assuming the true spectrum to be a hard power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Same as Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5 and C2, which assumed as true spectrum a power law with γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='75 and Eν,cut ≈ 10 PeV, but now assuming as true spectrum a power law with γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 and no cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Left: Assuming a bump width of αbump = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' this should be compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Right: Marginalizing over αbump;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' this should be compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Section V A in the main text and Appendix D for details.' metadata={'source': 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neutrinos: A Snowmass white paper, JHEAp 36, 55 (2022), arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='08096 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' (IceCube), Time-Integrated Neu- 24 10−9 10−8 10−7 10−6 Present-day best-fit power law ∝ E−γ ν (γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5) 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 2025, IceCube only (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') Upper limit (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' ), IceCube HESE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 yr 10−9 10−8 10−7 10−6 Bump height E2 ν,bumpΦν,bump [GeV cm−2 s−1 sr−1] 2035, IceCube only (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 10−9 10−8 10−7 10−6 2035, IceCube+Gen2 only (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 105 106 107 Central energy of ν bump, Eν,bump [GeV] 10−10 10−9 10−8 10−7 10−6 All panels: bump width αbump = 1 2035, all detectors (proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='0 Bayes factor for bump discovery, log10 B Substantial Strong Very strong Decisive FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Projected discovery potential of a bump in the diffuse flux of high-energy neutrinos, assuming the true spectrum to be a hard power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' 7, which assumed as true spectrum a power law with γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='75 and Eν,cut ≈ 10 PeV, but now assuming as true spectrum a power law with γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content='5 and no cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' See Section V C in the main text and Appendix D for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfPfYd/content/2301.00024v1.pdf'} +page_content=' trino Source Searches with 10 Years of IceCube Data, Phys.' metadata={'source': 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search engines have marked everyone’s life by transforming +how one searches and accesses information. Search engines give +special attention to the user interface, especially search engine +result pages (SERP). The well-known “10 blue links” list has evolved +into richer interfaces, often personalized to the search query, the +user, and other aspects. More than 20 years later, the literature has +not adequately portrayed this development. We present a study +on the evolution of SERP interfaces during the last two decades +using Google Search as a case study. We used the most searched +queries by year to extract a sample of SERP from the Internet +Archive. Using this dataset, we analyzed how SERP evolved in +content, layout, design (e.g., color scheme, text styling, graphics), +navigation, and file size. We have also analyzed the user interface +design patterns associated with SERP elements. We found that +SERP are becoming more diverse in terms of elements, aggregating +content from different verticals and including more features that +provide direct answers. This systematic analysis portrays evolution +trends in search engine user interfaces and, more generally, web +design. We expect this work will trigger other, more specific studies +that can take advantage of our dataset. +CCS CONCEPTS +• Human-centered computing → Interaction design process +and methods; Human computer interaction (HCI); • Infor- +mation systems → Users and interactive retrieval; Web search +engines. +KEYWORDS +Search engines, SERP features, Web interfaces, Web design, Evolu- +tion +ACM Reference Format: +Bruno Oliveira and Carla Teixeira Lopes. 2023. The Evolution of Web Search +User Interfaces - An Archaeological Analysis of Google Search Engine +Result Pages. In ACM SIGIR Conference on Human Information Interaction +and Retrieval (CHIIR’23), March 19–23, 2023, Austin, TX, USA. ACM, New +York, NY, USA, 14 pages. https://doi.org/10.1145/3576840.3578320 +Permission to make digital or hard copies of part or all of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for third-party components of this work must be honored. +For all other uses, contact the owner/author(s). +CHIIR’23, March 19–23, 2023, Austin, TX, USA +© 2023 Copyright held by the owner/author(s). +ACM ISBN 979-8-4007-0035-4/23/03. +https://doi.org/10.1145/3576840.3578320 +1 +INTRODUCTION +The wealth of information available on the Web make search en- +gines an essential tool nowadays [1]. Web search engines have +evolved a lot, and their user interface is no exception. Web search +engines’ front end is gaining importance in a scenario where rel- +evance is becoming more difficult to evaluate by users, and their +perception becomes more influenced by the user experience [5, +p. 512]. +Search interfaces support tasks from query formulation to select- +ing and understanding search results [5, p. 26-40]. Typically, web +search engines have a home page containing a search entry form +in which the user types a query. Retrieval results are usually dis- +played as vertical lists on Search Engine Results Pages (SERP). For +its richness, our study focuses on these pages. SERP have started +as simple “10 blue links” pages. Although search engines have kept +a consistent format of presenting search results, the information in +SERP goes way beyond these links. The design of query interfaces +and retrieval results display is an active area of research and exper- +imentation. Although works provide an in-depth analysis of search +user interfaces, such as the one from Hearst [22], the temporal +evolution of SERP is understudied. +Portraying the evolution of SERP contributes to preserving the +history of web search user interfaces. Moreover, assuming these +interfaces follow the general trends in web user interfaces, it con- +tributes to the overall study of web interfaces. Broadly, it depicts +trends in web design. This evolutionary analysis is also helpful +for the interactive information retrieval community to understand +better how the search interfaces have evolved in content, layout, +and navigation and build upon this for further and deeper analysis. +This study conducts an evolutionary analysis of Google’s SERP +during the last two decades of Google Search, analyzing the overall +user interfaces over time and their elements. We chose Google for +its popularity in the web search engine landscape. For this analysis, +we have captured from the Internet Archive 5,653 Google SERP +from 2000 to 2020. +This work makes several contributions. First, we present a sys- +tematization of the elements that appear or have appeared in a SERP, +defining each and providing visual examples. This systematization +can be helpful in future studies that have the SERP as their focus +and allows the establishment of common terminology. Second, we +analyze the evolution of the overall SERP and each element. Third, +we propose a methodology that can be used to study different types +of web search user interfaces (e.g., mobile ones) or user interfaces +in other contexts. Fourth, this paper makes two resources available +arXiv:2301.08613v1 [cs.IR] 20 Jan 2023 + +CHIIR’23, March 19–23, 2023, Austin, TX, USA +Bruno Oliveira and Carla Teixeira Lopes +to the community: a dataset1 containing the screenshot and files +associated with each extracted capture and a website2 summarizing +the analysis. These resources can also be the input of further studies, +inclusively done by researchers without advanced technological +skills. +2 +RELATED WORK +Research in search user interfaces focuses on their design and +evaluation, either in broad or focused on query formulation, the +presentation of search results, or even query reformulation. There +are also works focused on personalization, information visualiza- +tion, or domain-specific search interfaces such as mobile, social, or +multimedia. There are whole books and monographs dedicated to +this subject [22, 35, 43, 48, 61, 63]. Despite such research, given our +focus, we only describe works focused on analyzing the anatomy +of SERP and, eventually, its evolution. This section does not cover +works proposing or evaluating search interface components. +Höchstötter and Lewandowski [23] address SERP composition +and count the various elements’ appearances. To the best of our +knowledge, this was the first work to analyze the entire structure of +the SERP. Besides, the authors examined the retrieved results, their +sources, and types (e.g., organic results, advertisements, shortcuts). +When the authors wrote this paper, advanced features in SERP +were not widespread, which was not the case when Moran and +Goray [40] studied the anatomy of SERP, defining the terminology +for SERP elements. Nielsen Norman Group uses this terminology in +several articles [36, 40–42]. In their ‘Search Patterns’ book, Morville +and Callender [43], apart from addressing the anatomy of the search +process and related behavior, also list elements and principles of +interaction design, illustrating many user interface design patterns +around search websites. +To the best of our knowledge, no works systematically collect +and analyze SERP interfaces over time. +3 +METHODOLOGY +The first stage of our work involves building a sample of SERP inter- +faces over time, a process described in Section 3.1 and Section 3.2. +After collecting this sample, our attention focused on its analysis +and automation, as described in Section 3.3. +3.1 +What SERP have we captured? +Google Search currently has 91.4% of the market share [10], a lead- +ership that goes back to 2002 [27, 46]. In this context, we decided +to focus our analysis on this search engine. This study will address +desktop versions of Google Search from 2000 until 2020. A compar- +ative analysis with the seconded ranked search engine, Microsoft +Bing, is done in another work [45] and is available on the study’s +website. +The Internet Archive keeps snapshots and the respective HTML +version of web pages over time. Its collection contains 588 billion +web pages [3]. Internet Archive provides the Wayback CDX Server +API, which allows complex querying, filtering, and analysis of cap- +tures. While filtering by URL, we can use a wildcard (*) at the end of +1Available at https://doi.org/10.25747/991g-f765 +2https://bedgarone.github.io/serpevolution/ +the URL to specify the latter as a prefix and receive entries beyond +the specified URL (e.g., www.google.com/search?q=cookies*). +We found more than 195 thousand captures of Google SERP +during two decades using the API. This large number of SERP and +existing resource restrictions led us to devise a method to iden- +tify a smaller set of SERP. To increase the likelihood of reaching +pages with SERP element diversity, we have used a set of 129 most +searched queries in the last 20 years, retrieved from Google Trends +during the same period. This set3 contains the first search query +from each available category, such as People, Health, or Electron- +ics. These queries include relevant terms often searched by users +and trigger features in SERP. Hence, it is highly likely that SERP +interfaces derived from these queries are richer and, thus, more +relevant for this study than those generated by random searches. +We decided to append these queries with the ‘*’ wildcard while +submitting them to the API to obtain more captures. +We noticed that some years had no captures using the most +searched queries, which coincided with periods in which there +were few captures from Google Search’s domain. Hence, in those +years, we collected all the available captures (all method in Table 1). +We also noticed that the last two years had much more captures +(>10 thousand). Therefore, in 2019 and 2020, we restricted the URL +submitted to the API to those containing queries shorter than 37 +characters. Considering that an English word has, on average, about +five characters [44, 65], the 37 characters are the equivalent of 6 +words plus spaces between them, which is more than the two to +three words that a query typically has [6, 24, 25]. This restriction +excludes more specific queries that are probably less useful to the +plurality of interfaces. Our sample has 5.653 captures. The last +column of Table 1 has ordered lists with the number of captures +per year. +Table 1: Method used to collect the captures, maximum +length of the query (search URL), the width of the screen- +shot, and the number of captures extracted per year. +method +max. length +width +#captures per year +2000 - 2002 +queries +- +800px +200, 3, 23 +2003 +queries +- +1024px +231 +2004 - 2008 +all +- +1024px +12, 0, 200, 0, 26 +2009 +queries +- +1024px +11 +2010 +all +- +1024px +78 +2011 +queries +- +1024px +7 +2012 - 2018 +queries +- +1366px +57, 975, 30, +89, 172, 192, 548 +2019 +queries +37 char +1366px +171 +2020 +queries +37 char +1920px +2628 +3.2 +How have we captured SERP? +We used Python and Selenium Webdriver to visit each capture +online, check if the capture was valid, save the HTML version, and +3Available at https://bedgarone.github.io/serpevolution/mostsearchedqueries + +The Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP +CHIIR’23, March 19–23, 2023, Austin, TX, USA +CNN +AMAZON +TRUMP +... +Organic result +found within +6 seconds? +Load capture +in Browser +Remove +banners +Extract +source and files +Wayback +Machine CDX API +FOR-EACH capture +Set of +queries +Get SERP timestamp +and original URL +Generate SERP +archived URL +NO +YES +web.archive.org +/web/ ++ +TIMESTAMP ++ +ORIGINAL URL +{TIMESTAMP, +ORIGINAL URL} +{TIMESTAMP, +ORIGINAL URL} +{TIMESTAMP, +ORIGINAL URL} +Skip capture +Figure 1: Extracting captures procedure +generate a screenshot. The capture process is shown in Figure 1. +The original URL is the URL of the original SERP (e.g., google.com/ +search?q=photography), while the archived URL is the URL of its +archived version (e.g., web.archive.org/web/20160125203434/www. +google.com/search?q=photography). +Some captures with an HTTP OK status code were not consid- +ered valid. Some are inexistant, showing a contradictory message +of URL not captured, while others are defective (e.g., incomplete +interfaces without search results). To automatically check the va- +lidity of each capture, we try to find a result entry, the element +that cannot lack in a SERP. Captures from SERP tabs other than the +general first page, identified with “tbm” in the URL, were discarded +for being outside this work’s scope. We raise a timeout exception +after 6 seconds, the time we empirically considered sufficient to +load the capture in the browser. In these situations, the program +will skip the capture. Before downloading the page, we still remove +graphical elements from Internet Archive, such as its information +and donation bars. Some of the captures present other distracting +banners and overlapping parts of the interface, such as the ones +related to cookie consent. We removed all the identified ones and +extracted the source and associated files. +The process concludes with generating full-height screenshots +of every HTML version opened in another browser instance in +headless mode. We produced screenshots considering the most +popular screen size at the time of the capture, as stated by the +statistics [55]. We only considered the width, shown in Table 1, +because SERP height is highly variable. The dataset with all the +extracted captures is available online4. +3.3 +How have we analyzed SERP? +The analysis process included two main stages, as shown in Figure 2. +First, we have extracted a sample of captures from the primary +dataset to identify SERP elements. For each month with captures in +the primary dataset, we manually looked at the screenshots of that +month’s captures and selected the capture with the most features. +In the end, this set included 117 captures, with which we visually +identified SERP elements. We manually analyzed each element’s +source code, looking for identifiers to locate the element in a later +4https://doi.org/10.25747/991g-f765 +automated process. Element identifiers consist of HTML classes, ids, +tags, or a combination of these using CSS selectors (e.g., ‘.knowledge- +panel’, ‘#tads’ or ‘#newsbox’). All the encountered identifiers were +logged and listed on the website5. +FOR-EACH +element +Identify SERP +elements +Build set of +SERP +Seek identifiers in +captures’ HTML +Log detected +identifiers +Search element +in the dataset +Log element’s first timestamp +Screenshot +element +Color +interface +Load capture +in Broswer +.#id +.id +#id > .id +Figure 2: Detection and analysis of elements procedure +In the second stage, we automated the detection of these elements +over time, allowing the exploration of a more significant number of +cases. Finding an element with these identifiers triggers a function +that stores the date of the element’s appearance in a log file. We +imposed no limit of captures per month to register the element’s +appearance, as the computation permits a full dataset scan in an +acceptable time. The function also receives the element’s upper-left +corner coordinates, width, and height, generating and saving its +image in the element’s folder. Contrary to the element’s timestamp, +we imposed a limit of 15 captures per month while screenshotting +to reduce and make the scan time feasible. We estimate that 15 +monthly samples are enough to capture the possible changes of an +element. +Following a similar procedure, we automatically used the iden- +tifiers to detect and color the web page’s targeted areas. We used +Python, Selenium Webdriver, and BeautifulSoup to scrape every +HTML capture to identify and generate transparency-colored im- +ages for each category of elements. We generated these images in +a headless browser with a 1920px width, regardless of the capture’s +width. This simplification does not affect the final result because the +elements in the interfaces do not move dynamically as the width +increases or decreases. We have not constrained the height in this +generation process. Due to the size of the dataset, we imposed a +limit of 15 elements per month. +We overlayed all the individual images from single captures for +each category of elements, which allows the overlay to enhance the +most common areas while leaving the others almost unnoticeable. +The overlaying process uses the upper-left corner as the reference +for image alignment. Navigation & user inputs includes elements +in and next to the footer, where common areas were not evident +due to the page height’s variability. In this case, to generate and +correctly overlap the footers of the interface, we considered a height +value of 600px for the footer, cropping it from the bottoms of the +interface. The 600px height was estimated after a visual analysis of +SERP pages and included a margin of error to ensure we covered all +elements under study. Thus, the orange result displayed in Figure 3 +is trimmed in the middle and combines those two capturing steps. +We will analyze the positioning of every category of elements in +5https://bedgarone.github.io/serpevolution/elements + +... +HtmL +FILES.. +HTMl +FILES<> +HTML<> +HTMLCHIIR’23, March 19–23, 2023, Austin, TX, USA +Bruno Oliveira and Carla Teixeira Lopes +the next section. An animated version of each result in this Figure +is available on the website6, permitting us to observe how the +positioning of the elements’ categories in SERP changed over time. +Figure 3: Transparency-colored overlaying results for each +category +4 +EVOLUTION OF SERP ELEMENTS +We present each element’s description and analyze its period of +presence and positioning in SERP (also displayed on the website7). +Moreover, we analyze each element’s evolution regarding content, +graphics, navigation, and their relation with user interface design +patterns. These patterns are problem-oriented and generally repeat- +able solutions to usability problems in interface and interaction +design [18, 57]. +4.1 +Visual identity & search statistics +Considering Google’s visual identity, each logo version has kept its +position and size with rare variations. Figure 3 shows the overall +position of the logo in red. Some interfaces are exceptions, offering +a more significant left margin and a right-shifted logo and search +statistics. Figure 3, in gray, shows how search statistics appear +consistently below the search query or navigation bar, either left- +aligned, right-aligned, or justified. Statistics included the number of +results seen per page in the first decade. Later, Google removed this +information and kept only the estimated number of results. Details +about the logo evolution and the content about search statistics can +be seen online8. +4.2 +Navigation & User Inputs +Figures 4 and 5 display the main stages of how the search box and its +surroundings evolved. The left-aligned query bar has also marked +its place at the top of the page. Yet, in the early phases, it also +appeared at the bottom, as seen in the 2000 and 2006 screenshots of +Figure 5. The Input Prompt design pattern has always been applied +to User Inputs. +We notice a change in the width of the entry form after 2006, +which may suggest an encouragement to the formulation of longer +queries [6, 19]. This change is in line with experimental evaluations +where query length is positively related to effectiveness in the IR +6https://bedgarone.github.io/serpevolution/layout +7https://bedgarone.github.io/serpevolution/elements +8https://bedgarone.github.io/serpevolution/design +Figure 4: SERP headers neatly from 2000, 2001, 2006, 2012, +2017 and 2020 +Figure 5: SERP footers neatly from 2000, 2006, 2012, 2017 and +2020 +task [6, 14]. Since query formulation influences effectiveness more +than algorithmic factors [14], it makes sense to encourage users +to do so. We can also notice the appearance of a way to specify + +800 +1024 +1366 +1920 +800 +1024 +1366 +1920 +800 +1024 +1366 +1920 +800 +1024 +1366 +1920 +600 +768 +Visual identity +1080 +&results statistics +800 +010241366 +1920 +1080 +Navigation& +Organic +Sponsored +Features +user inputs +results +resultsGoogle +mp3 pure +10 results +All Languages +GooqleSearch +I'm Feeling Lucky +Search Tips +LanguageQptions +EmailTheseResults +Sponsored Links +ClickheretofindMP3files.Listen,ShareandStore themforfree! +http://www.myplay.com +SignupTodayforamyplayLockerandget3GBofFreeStorage! +Music - 1o0's of songs to choose from! Stop searching. Start listening: +www.firstlook.com +Wheretomorrow'smusicis heardfirst!Firstlook.com +Googleresults1-10ofabout23,997formp3pure.Searchtook 0.34seconds +Category: +Arts>Music>Sound Files>MP3>Link Lists +Google +AdvancedSearch +Preferences +LanguageTools +Search Tips +Spathiphyllum +GoogleSearch +I'm Feeling Lucky +Web +Images +GroupsDirectory +SearchedthewebforSpathiphyllum +Results1-10ofabout7,160.Searchtook0.10seconds +Category: +Science>Biology>...>Magnoliophyta>Liliopsida>Araceae>Spathiphyllum +Sign in +Google +Web +Images +VideoNew! +News +Maps +more " +"tsunami" +Search +Advanced Sesrch +Preferences +Web +Results 1 - 10 of about 85,900,000 for "tsunami" [definition]. (0.28 seconds) +Search Images Videos Maps News Shopping Gmail More +Google +ipod touch +Search +About 325,000,000 results +Advanced search +Search Images Maps Play YouTube News Gmail Drive More +Google + sparky +α +All +Images +Videos +News +Shopping +Maps +Books +Gougle +scoliosis +X +Q +Q All +Images +国News + Videos + Books +: More +Settings +ToolsGoo0000000ogle +ResultPage: +12345678910 +Next +Space Racer +GooqleSearch +Search within results +Try your query on: AltaVista Deja Excite HotBot Infoseek Lycos Yahoo! +Google Web Directory - Cool Jobs - Advertise with Us! - Add Google to your Site - Google in your Language - All About Google +Result Page: +12345678910 +Next +"tsunami" +Search +Search within results I Language ToolsI Search Tips I Dissatisfied? Help us improve +Google Home -Advertising.Programs - Business Solutions -About Google +2345 +8910 +Next +Search Help +Give us feedback +Google Home +Advertising Programs +Business Solutions +Privacy &Terms +About Google +1 2 34 5 6 78 9 10 +Next +Advanced search +Search Help +Send feedback +Google Home +Advertising Programs +Business Solutions +Privacy +Terms +About Google +234567 +8910 +NextThe Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP +CHIIR’23, March 19–23, 2023, Austin, TX, USA +the query via spoken commands in the latest screenshot. This ap- +pearance aligns with the increasing attention conversational search +interfaces have received [2]. Although works show promising re- +sults in incorporating visual elements in query formulation [53], +there is no sign of them in Google Search. +The search box and the current query are always visible for the +searcher, as recommended [63], since 2019. This visibility occurs +even when the user scrolls down and the top of the SERP is no +longer visible. Before 2019, the query was always available in the +search box, but this box disappeared if the user scrolled down. +Because we cannot interact with past SERP versions, we cannot +analyze interaction features such as auto-complete or auto-correct. +In 2000, as seen in Figure 4, the main buttons for this category +were ‘Google Search’ and ‘I’m feeling lucky’. It was possible to +select, directly in a dropdown near the search box, how many +results should be shown, the language intended for the results, and +an option to send the retrieved results by email. In the screenshots +of 2000 and 2001 presented in Figure 4, the Category Hierarchy +feature is also visible showing one or more categories related to the +query, probably obtained from Google Web Directory. This feature, +available until 2004, located these categories in more general areas +using breadcrumbs, a design pattern that linearly specifies hierarchy +levels leading to the current subject or page [57, 58]. At the bottom +of the interface, there was also an option to search within the results +and links to trigger the query on other Web search engines. Other +lesser relevant links would point to SERP experience (e.g., language, +search tips). +Although nonexistent for several years, it is possible to notice in +Figure 3, in orange, the significant presence of the left navigation +bar during some years of SERP history. This left column in orange +is naturally interrupted in the middle area of the interface, trimmed +as explained in Section 3.3. +In 2001, Google introduced the tabs bar in the shape of Module +Tabs used when content is groupable and there is no room for +everything. Modules of content are divided into small tabbed areas +with only one visible at a time, allowing the user to click on tabs +to reveal other modules [57, 58]. Tabs don’t need to be literal tabs +and don’t have to be at the top of the stack of modules [57]. The +modules were Web, Images, Groups, and Directory. In 2002, Google +added the News tab. In 2006, the tabs were replaced by simple links +above the search query. This year, the Video, Maps, Froogle, and More +tabs were introduced. In 2008, the tabs bar went to the very top of +the page, and the Shopping and Gmail tabs were included. In 2010, +the left sidebar was introduced, complementing the interface with +other tabs and information such as location and results filtering. In +2012, Google removed the bottom query bar. In 2013, the tabs bar +was displaced underneath the main query bar, maintaining some +links and changing others. At the same time, in the left sidebar, only +the results’ filtering options were available before Google removed +this sidebar in 2014. Finally, in 2015, the tabs bar below the query +bar became the only existing one. +There is also a usual space for sign-in and user account informa- +tion on the right side of the screen. At the bottom of the page, two +areas are noticeable: pagination, aligned to the center of the results +container, and the footer, at the very end, covering all the width. +These areas are visible in Figure 3. The Pagination design pattern +has been applied to Navigation since the beginning of SERP. Pagi- +nation breaks up the long results list into different SERP, loading +them one at a time [57]. +4.3 +Organic Results +Organic results are retrieved based on the document’s content +and the overall retrieval algorithm. Regarding their positioning, +in Figure 3 (blue), there is a more substantial presence of colored +frames in the area where results are typically included (henceforth +called results container), with a greater focus on the visible area. By +visible area, we mean the interface area that can be seen without +scrolling. Information that needs scrolling to be accessed is in a +scrolling area. Over time, it is noticeable that SERP pages have +increased their height due to the vertical decrease of color intensity, +revealing results in lower page regions. Part of the results is slightly +shifted to the right, referring to interfaces with a left-side navigation +bar. It is possible to observe two other very tenuous sets. One with +more centered results since the interface from 2010 to 2012 adjusted +elements’ position according to the width and central axis of the +screen rather than a left alignment. Some frames cover the entire +interface width, not because the content was that large, but because, +in the early days, HTML divisions (div) typically spanned across the +complete width of the viewport because Document Object Model +(DOM) trees were less deep. The viewport consists of the visible +area of an interface on a display device. +Organic results can be regular or enriched. The structure of +regular results, seen in Figure 6, started as a basic block of the +page title, snippet, and URL links for similar or/and cached pages +for the result. In 2013, Google hid these links in a dropdown, only +visible by its arrow icon until now. In 2018, Google introduced a +link to translate the result. +Figure 6: Regular result from 2003 (left) and 2020 (right) +The enriched results, seen in Figure 7, are variations of regu- +lar results, with extra elements below the title, snippet, and URL, +giving some additional information to the user. This element can +have greater visibility and, in turn, a higher click rate [8, 21, 47]. +It appeared for the first time in 2008, lasting until now. Its posi- +tioning is consistently at the top of the results container. Initially, +the extra content included two columns of site links pointing to +sub-pages of the result’s domain. These links, named quicklinks, are +navigational aid that attempts to take the users to the content they +want quickly [8]. In 2009, it started highlighting structured data, +such as reviews and ratings for products and services, based on +experiments that showed that users find value in this new data [29]. +In 2010, Google enhanced each site link with a short description. +In 2016, a search bar was introduced so that the user could search +the result’s website directly from SERP. +In both regular and enriched results, query term highlighting +with boldfacing has been applied since 2006 to improve the usability +of search results listings [11, 32, 38]. The dominant colors are blue +for the title, green for the URL, and black/grey for the snippet. + +Holisticopia:WyomingMassageTherapy +www.reddit.com > Coronavirus > new +Foxy Lady Salon 113o S ... Torrington, Banner Health Massage Therapy 625 Albany ... +www.holisticopia.com/qeo/WvominaMassaqe+Therapv.htm -34k -Cached-Similar paqes +r/Coronavirus: In December 2019,a novel coronavirus strain (SARS-CoV-2)emergec +city of Wuhan, China. This subreddit seeks to monitortheCHIIR’23, March 19–23, 2023, Austin, TX, USA +Bruno Oliveira and Carla Teixeira Lopes +Figure 7: Enriched result from 2009 (left) and 2016 (right) +About 2011, the URL moved above the snippet. Google underlined +the title until 2014. In 2020, the URL changed its color to gray and +moved above the title. The new position of the URL may have +been influenced by the importance of URL and domain names in +evaluating the credibility of a result [34, 50]. In 2019, the URL started +being displayed as a breadcrumb, which may have been motivated +by research that concluded that long and complex URL negatively +impact clickthroughs [11]. +In both types of organic results, the snippet length has not visibly +changed, with most of the snippets having two lines, the suggested +size for informing the user, and including as many results as pos- +sible in the visible area [63]. As most of the queries used for data +collection are associated with informational tasks, we cannot con- +clude if Google adjusts the length of the snippet to the type of +query (e.g., informational, navigational) as suggested by existing +research [15]. +4.4 +Sponsored Results +As seen in Figure 8, textual ads are short advertisements that +appear alongside organic results. These entries are focused on com- +mercial intent, combining relevance with revenue, and are usually +manually crafted, a standard in the advertisement industry [16]. +This element has been present in SERP since the first day, lasting +until now. This element’s content and evolution are identical to the +ones of Regular Results, except these were initially marked with a +specific tag, ‘Sponsored link’. In 2012, some of the results started +to include quicklinks, although less noticeable than in Enriched re- +sults. Until 2014, a different background color distinguished these +elements from organic results. Scarce occurrences in 2017 also had +a colored background. In 2014, the tag ‘Ad’ substituted the previous +one, having a yellow background color, while from 2016 onwards, +it was green. In 2020, some ads could have the shape of an actual +Enriched result. The URL also changed its color to gray and moved +above the title. +Figure 8: Textual ads from 2002 (top), 2014 (left) and 2020 +(right) +The other type of ads correspond to shopping content and are, +therefore, called shopping ads, seen in Figure 9. These are acti- +vated when the query is commercial [47]. These are very striking +results, listing various information about each product. This el- +ement appeared for the first time in 2013, lasting until now. Its +positioning is usually at the top of the results container and, more +frequently, at the right of that container. Shopping ads used to be +exclusive to the right sidebar, displaying one to four results in a +matrix. In 2018, each result was embedded in an individual card. +Until 2019, query terms were highlighted in bold. In 2020, this ele- +ment appeared in the results container in a carousel of cards with +more width and less height. The Cards design pattern has been +applied to this element since 2019, and the Carousel since 2020. +Cards display content composed of distinct parts, generally about +a single subject, to form one coherent piece of content designed +to expose information efficiently [56, 57]. It is usual for cards to +accompany other cards, carrying similar content but addressing +different subjects. The Carousel is a horizontal strip of simple cards, +letting the user scroll horizontally to view them and encouraging +the inspection of the following items [57, 58]. Since the element’s +beginning, the Thumbnail Grid has been applied, enabling a quick +overview of images by shrinking the original ones [51, 57]. +In Figure 3, it is possible to identify two major advertisement +areas in SERP: top ads and right ads, following a typical SERP +layout [5, p. 489]. Elements in the right sidebar move horizontally +over time because some interfaces force this sidebar to be responsive +to the screen’s width. Top ads are the most common positions +for ads within a soft vertical variation. Older interfaces placed +advertisements on a fully colored div that occupied almost the +entire viewport width for the same reason described in Section 4.3. +Compared with regular results and other features, we can see in +Figure 3 that sponsored results are more concentrated in the visible +area, probably motivated by the fact that searchers rarely scroll [63]. +Figure 9: Shopping ads from 2013 (left), 2019 (center) and +2020 (right) +4.5 +Features +SERP features complement organic and sponsored results, attempt- +ing to provide answers to the query without just pointing to web- +sites that might deliver that information. +Figure 3, in green, shows features spread around the interface, +mainly in the visible area and upper half of the scrolling area. Many +features are similar to regular results but with more significant +height. These features also share a place in the sidebar with adver- +tisements, recently becoming more present than the latter. Most +frames with large height values correspond to the well-known +Knowledge Panel. A horizontal area is noticeable, generally assigned +to the Carousel and the Carousel Grid. + +Recent Earthguakes in California and Nevada -Index Map +WelcometoIKEA.com-IKEA +Includes an index map and zoom views, with location and magnitude information. Updated +www.ikea.com/-IKEA +hourly +Featuring Scandinavian modern style furniture and accessories. 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The +Video Pack feature will not be considered for its similarity with the +Image Pack. The same happens with Direct Answer Results, Local +Pack, People Also Ask, Carousel, Carousel Grid, Recipe Cards, Twitter +Pack, Category Hierarchy, and Covid-19 Left Panel for their shorter +presence. More information about these features is available in +another paper [45] and online9. +Featured Snippets, seen in Figure 10 are answer boxes in which +Google responds to a question-related query based on information +taken from a page [59]. This element appeared for the first time in +2016, lasting until now. Its positioning is consistently at the top of +the results container. Although these snippets resemble enriched +results, they differ in the type and position of extra content. In +featured snippets, the additional content appears before the result’s +title and not after, as happens in enriched results. The content of a +featured snippet was initially made of a short paragraph with an- +swering information. It evolved to a general layout, used until now, +consisting of a larger paragraph, a thumbnail at the upper-right +corner, and the title and link to the information’s source, to where +it is possible to navigate. The answer also started to be returned as +an ordered list or table. Instead of a thumbnail, a video or a carousel +of images may also accompany it. In 2018, when available, Google +introduced the date of the source’s publication after the information +paragraph. The underlined style of the title was removed in 2017, +and the green URL was changed to a gray breadcrumb URL in 2020, +as in the organic and sponsored results. Text styling is used in both +paragraph and title, enhancing relevant words in bold. Studies have +found that these snippets help make more accurate decisions when +containing the correct information [7]. +Figure 10: Featured Snippets from 2016 (top), 2017 (left) and +2020 (right) +The Knowledge Panel, seen in Figure 11, is perhaps the high- +light of SERP features. It is a dynamic feature that provides direct +information in various formats within the same panel, pointing to +related content. The contents range from text to images, ratings, +social profiles, factual information, and similar search topics [47], +helping the user to understand a particular subject quickly and +facilitating a more in-depth search [17]. This element appeared for +the first time in 2014, lasting until now. Its positioning is always +at the right of the results container. The basic structure consists +of a panel with a top thumbnail of the subject, vertically followed +by a title, a website link if applicable, a resume paragraph usually +by Wikipedia, a structured list of direct information, and a block +9https://bedgarone.github.io/serpevolution/elements +of People also search for. During the following years, Google intro- +duced other content highly dependable on the search topic and +variable in coverage and quality [37]. It is common to have this +panel populated by Wikipedia information [60]. The graphics of +this element was stable over time, following the improvements +in Google’s interface design. The Grid of Equals and Thumbnail +Grid design patterns have been applied to this element since its +beginning. Grid of Equals is a pattern to display items in a grid or +matrix, each following a standard template, linking to respective +pages [58]. The Cards, Carousel, and Module Tabs design patterns +were applied to this element in 2018. +Figure 11: Knowledge Panel from 2016 (left) and 2018 (right) +The Image Pack, seen in Figure 12, presents a set of images +taken from various sources in Google’s index in searches where +visual content is valuable [59]. This element appeared for the first +time in 2006, but only after 2010 did it frequently appear, lasting +until now. Its positioning is highly variable throughout the results +container. Most of the time, the content was a title associating +images with the search query and a block of image thumbnails. In +2014, Google included a link for ‘more images’ and a link to report +pictures. In 2019, Google introduced a bar of image categories. It +could have simple buttons or buttons with a thumbnail associated +with its category. The graphics started with a considerable presence +of blue colors, typical in Google’s early interfaces when images +had a blue border. In 2014, Google removed this border, but the +main change was in 2019 when images started to be in a carousel. +In 2018, the layout of a matrix appeared for the first time. These +changes increased the element’s area in the last two years. In 2020 +the title, as usual in most elements, turned dark gray. The carousel +of image categories changed its shape to a line that expands to a +matrix in the form of progressive disclosure using the Collapsible +Panels pattern. The Grid of Equals design pattern has been applied +to this element since 2018, while Thumbnail Grid, naturally, since +its beginning. Arguello et al. [4] examined vertical results, such as +images, in aggregated search. They found that these results had +more clicks in more complex tasks and that users were divided in +their preferences for vertical search displays. + +Kourtney Kardashian and Scott Disick were partners for 9 years (until 2015) +Kourtney Kardashian produced and Scott Disick appears on Keeping Up with the +Kardashians +Closing ceremony to celebrate Brazil 2014 in +style. Before the 2014 FIFA World Cup TM Final +gets underway at the Maracana in Rio de +Janeiro on Sunday, a special closing ceremony +involvingaround1,ooopeoplewillcelebratethe +sport as the tournament nears its unmissable +climax . +Space architecture, in its simplest definition, is the theory and practice of designing +and building inhabited environments in outer space. ... Space architecture borrows +Closing ceremony to celebrate Brazil 2014 in style - FIFA.com +from multiple forms of niche architecture to accomplish the task of ensuring human +2014-in-stvle-2404018.btml +beings can live and work in space +en.wikipedia.org>wiki>Space_architecture +Space architecture - WikipediaCNN +Cable channel - cnn.com +The Cable News Network is +an American basic cable +and satellite television channel that is +owned by the Turner Broadcasting System +news channel was founded in 1980 by +division of Time Warner. The 24-hour cable +American media proprietor Ted Turner. +images +Wikipedia +ScottGreenstein +Customer service: 1 (404) 827-1500 +Headquarters: Atlanta, GA +Film producer +Founder: Ted Turner +Founded: June 1, 1980, Atlanta, GA +Scott Greenstein is president and chief content +Parent organization: Turner +Broadcasting System +officer of Sirius XM Radio. He leads the programming +and advertising sales of the largest radio company +Profiles +by revenue and one of the largest subscription media +f +in +companies in the worid. Wikipedia +Twitter +Linkedin +Born: 1959 (age 61 years), Freehold, NJ +Facebook +Employer: SiriusXM Satellite Radio +TV shows +Movies: The Enqlish Patient +CW +NEWDAY +Profiles +Live +Twitter +CNN Live +Anthony +New Day +Today +Parts Unk.. +Bourdain +Sinoe 2013 +2001 2006 +People also searchfor +Sinoe 2013 +View 5+ more +riusxm! +People also search for +BBC +NEWS +James E. +Mel +Barry Diller +saul +Meyer +Karmazin +Zaentz +BBC +MSNBC +ESPN + Claim this knowiedge panel +FeedbackCHIIR’23, March 19–23, 2023, Austin, TX, USA +Bruno Oliveira and Carla Teixeira Lopes +Figure 12: Image Pack from 2010 (left) and 2020 (right) +The Top Stories, seen in Figure 13, are blocks of three or more +recent news considered relevant to the query, recently placed in +the form of a carousel [47]. Each story is now presented with a +thumbnail, publisher, and timestamp. This element appeared for the +first time in 2004, but only after 2011 it started to appear frequently, +lasting until now. Its positioning is mainly in the visible area of the +results container. The element’s content started with a vertical list +of at most four news titles, each followed by the source’s name and +how long ago it was published. In 2006, a journal icon was placed at +the left of the list, and a link to ‘today’s top stories’ was introduced. +In 2013, the icon was substituted by a thumbnail for the first news +result, being the most important news in the element. The latter was +complemented with an extract of the news, while the rest stayed the +same. In 2020, leading to a considerable increase in the element’s +area, the graphics was majorly altered to display the results in +a carousel of cards. However, the content was simplified to only +present, for each result, a thumbnail, title, source, and how long ago +it was published. As usual, the color scheme was mainly blueish +and filled with blue borders and underlines. These were removed +after 2020 with the softer gray colors for additional information +but still blue titles. The Streams and Feeds design pattern has been +applied to this element since its beginning. It defines a pattern to +list time-sensitive items chronologically, combining the sources in +one place [58]. However, in this element, the relation to the query +appears to be more relevant than publication time since it is no +longer possible to observe any chronological order. The Cards and +Carousel design patterns have been applied since 2020. +Figure 13: Top Stories from 2006 (left-top), 2013 (left-bottom) +and 2020 (right) +Related Searches, seen in Figure 14, is a common element on +SERP pages from a very early age and offers suggestions for related +searches, i.e., queries that are in some way related to the current +query and may be good candidates for follow-on queries. These +suggestions can be helpful to support exploration or provide query +statements that express information needs in different ways [62]. +Usually, these suggestions are generated based on search log data, +either picking queries that frequently follow the current query [28] +or clustering queries based on results’ clicking [13]. Each link takes +the user to the respective SERP. This element appeared for the first +time in 2008, lasting until now, except for 2010. Its positioning is +always at the bottom of the results container. The content was +diversified regarding how many suggestions would appear and its +layout. Each suggestion of search is a hyperlinked title pointing to +its respective SERP. Initially, it was organized in a matrix of columns. +In 2011, Google reduced this schema to two columns, which could +be displayed in just one column for suggestions with longer text. +Until mid-2020, suggestions were blue and, until 2014, underlined. +Google applied a search icon to each entry in 2020. Later, a new +version changed the graphics, making each entry a button with a +solid gray background, a search icon, and a title in black. This latter +change contributed to a recent increase in the element’s average +area. +Figure 14: Related Searches from 2008 (left - top), 2017 (left +- bottom) and 2020 (right) +5 +AGGREGATED ANALYSIS +This section analyzes SERP’s evolution from an aggregated per- +spective. +5.1 +Elements’ lifetime +As shown, SERP have always had a large variety of elements, each +with its evolution and active periods, as described in Section 4. Ac- +cording to the framework proposed by Wilson [63], input features +were analyzed in Section 4.2, with the search box being the main +one. To support control, besides the input features, we also iden- +tified the related searches feature. All the other analyzed features +are informational. It is important to note that, given the nature of +this study, it was not possible to analyze dynamic features such as +interactive querying or personalizable features. +Black cells in Table 2 identify the years we detected elements +in our dataset. For all years in-between black cells, we have con- +ducted a manual search in other sources to avoid false negatives. +In these cases, we have searched specific websites10 for evidence +of elements in such years. If found, we paint the cell grey. It is +noticeable how SERP features have emerged in the last decade, +contributing to a matrix full of element possibilities in recent years. +Almost every SERP element includes well-known design patterns. +A visual timeline with screenshots of these elements’ presence in +SERP is available online11. +5.2 +Design pattern application +Table 3 maps each element with the patterns proposed by Tidwell +et al. [57], along with the start date of that appliance. Older SERP +elements make later use of design patterns for individual improve- +ment. In contrast, some contemporary elements may have arisen +10https://searchengineland.com, https://googlesystem.blogspot.com and Internet +Archive +11https://bedgarone.github.io/serpevolution/timeline/2010 + +ImagestorParisweather +paris france +weather forecast +climate +temperature +eiffel tower +average rainfall +accuwea +Images for"complex event processing" +< +<国Topstories +News results for"american idol" - View today's top stories +'ldol Winner Hicks Sues Former Producer - Washington Post - 9 hours ago +LIVE +LIVE +LIVE +'ldolFinalist OK After Vegas Robbery - FOX News - 14 hours ago +Life after 'ldol' - MLive.com - 19 hours ago +NIA +LIVECOVERAGE + CNN.com +News forfacebook +Che New ork Cimes +OCBSNEWS +Facebook Surpasses $30_Ahead of Q4 Earnings +Electionresults2020: +Live Trump vs Biden +Pennsylvania2020 +Wall Street Journal (blog) - 1 hour ago - 239 related articles +LivenewsonTrump +Tracker: Presidential +election results +Shares of Facebook Inc. breached the $30 mark on Wednesday, the first +Biden and electoral +Election +time in six months. +votes +CBC.ca +As Facebook shares hit $30, ETFs with holdingS_gain +MarketWatch (blog) - 2 hours ago - 2 related articles +10 mins ago +24mins ago +19 mins ago +Facebook rolling out Timeline changes, including redesign +Chicago Tribune - 8 hours ago - 28 related articles 》 +< +7 +ViewallSearches related to: "logo design" +Related searches +online logo design +design your own logo +logo design tips +logo design ideas +who was prince henry the +Q +what did prince henry +navigator +discover +a +prince henry the navigator +a +how did prince henry the +Searches related to amazon diversity +school +navigator die +amazon diversity officer +amazon black employee network +a +prince henry the navigator facts +a +prince henry the navigator fun +amazon diversity 2016 +amazon workforcelogin +facts +amazon diversity report 2016 +amazon supplier diversity program +a +prince henry the navigator +prince henry the navigator +amazon supplier diversity registration +amazonworkforcephonenumber +accomplishments +timelineThe Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP +CHIIR’23, March 19–23, 2023, Austin, TX, USA +Table 2: Presence of SERP elements from 2000 to 2020. In black the years in which the element appeared in our dataset. In +grey are the years in which the element’s existence is documented elsewhere. +2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 +Visual Identity +Search Statistics +Navigation +User Inputs +Regular Results +Enriched Results +Textual Ads +Shopping Ads +Knowledge Panel +Featured Snippets +Direct Answer +Local Pack +Image Pack +Video Pack +Top Stories +Carousel +Carousel Grid +People Also Ask +Related Searches +Twitter Pack +Recipe Cards +Category Hierarchy +Covid-19 Left Panel +from the need to apply a design pattern solution whose traces are +evident from the element’s beginning. The website12 lists design +patterns with images and elements that use them. +5.3 +Highlights +We identify the most relevant changes along the upper part of a +two-decade timeline of SERP in Figures 15 and 16. These changes +correspond to the entry of new elements or significant changes +in input and navigation options. This timeline of changes is also +available online13, enhanced with images and the option to filter +the entries for navigation changes or element additions. +As stated before, the first stage of the analysis used a visual +selection from a monthly sample from the dataset. Figures 15 and +16 also display, in their lower part, how Google’s SERP interfaces +evolved in terms of design. In these timelines, we include the main +versions of the SERP interface based on significant changes. Apart +from how the overall interfaces visually evolved, similar timelines +about visual identity, search statistics, and navigation are available +on the website14. +Google’s initial interfaces had few depth levels in their HTML +DOM. The first interface design, traced in 2000, differentiated the +sponsored results with a colored background. A second search +12https://bedgarone.github.io/serpevolution/patterns +13https://bedgarone.github.io/serpevolution/timeline +14https://bedgarone.github.io/serpevolution/design +query bar existed at the bottom of the page, and the user could +change the number of results presented. Google removed this bar in +the following interfaces. The one traced from 2000 to 2004 revealed +a right block of results, exclusive to sponsored ones. It marked the +appearance of the first bar with tabs directing the user to other +types of content (e.g., images and news). The fourth interface design, +traced from 2010 to 2012, was not left-oriented but varied in a spaced +manner depending on the screen’s width. It introduced a sidebar +on the left, containing tabs to manage the results, but some of these +tabs were duplicated due to the navbar’s tabs bar mentioned in +Section 4.2. +Significant aesthetic changes occurred in 2012. The fifth interface +design relates to the launch of the Knowledge Graph, with the +right column being divided between it and sponsored results. Some +modifications were found earlier in the dataset during those years, +the sixth interface design. However, a design close to the current +one began at the end of 2018, the seventh interface design. As noted +in some elements’ graphics, this interface focused on modernizing +its lines. +5.4 +User interface area +We calculated the area of all screenshots in the dataset to analyze its +evolution over time. Figure 17 shows the development of interface +area per month (dots) and per year (line), measured in pixels. Each +entry in the chart corresponds to the average area per month for all + +CHIIR’23, March 19–23, 2023, Austin, TX, USA +Bruno Oliveira and Carla Teixeira Lopes +Table 3: Design Patterns and time of their appliance to SERP elements. Cells without a dash after the year represent single +years. +Organizing Navigation +Layout +Lists +Input +Streams +and Feeds +Bread- +crumbs +Grid of +Equals +Module +Tabs +Accor- +dion +Collapsi- +ble +Panels +Cards +Thumb- +nail +Grid +Carousel +Pagina- +tion +Input +Prompt +Visual Identity +Search Statistics +Navigation +2000-2020 +User Inputs +2000-2020 +Regular Results +2019- +Enriched Results +2020- +2020- +Textual Ads +Shopping Ads +2019- +2020- +Knowledge Panel +2014- +2018 +2018 +2014- +2016- +Featured Snippets +2016- +2020- +Direct Answer +2018- +2018- +Local Pack +Image Pack +2018- +2019- +2006- +2019- +Video Pack +2015- +2015- +Top Stories +2004- +2020- +2020- +Carousel +2015- +2016- +2015- +2015- +Carousel Grid +2017- +2017- +2017- +People Also Ask +2016- +Related Searches +Twitter Pack +2017- +2017- +Recipe Cards +2020- +2020- +Category Hierarchy +2000-2004 +Covid-19 Left Panel +Right sidebar with sponsored results +Top Stories introduced +SERP tabs bar introduced (Images, Groups, Directory) +Tabs bar placed above the query bar +Image Pack introduced +Left sidebar introduced +Local Pack introduced +Related Searches introduced +Tabs bar placed on navbar +News tab added +Videos and Maps tabs added +Shopping and Gmail tabs added +2010 +2009 +2008 +2007 +2006 +2005 +2004 +2003 +2002 +2001 +2000 +Figure 15: Highlights of SERP overall evolution (top) and Interfaces’ visual evolution (bottom) from 2000 to 2010 +captures in the dataset. Months without values are months without +captures in the dataset, as indicated in Table 1. Results show an +increase close to exponential due to the appearance of SERP fea- +tures that have added extra content to SERP, thus, making them + +Googlem +GoogleThe Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP +CHIIR’23, March 19–23, 2023, Austin, TX, USA +Shopping ads introduced +Bottom query bar removed +Carousel introduced +Left sidebar removed +Knowledge panel introduced +Ads loose their colored background +Tabs bar placed under search query +Direct Answers introduced +Feature Snippets introduced +Covid-19 left panel +2017 +2016 +2015 +2014 +2013 +2012 +2011 +2018 +2019 +2020 +Figure 16: Highlights of SERP overall evolution (top) and Interfaces’ visual evolution (bottom) from 2011 to 2020 +more extensive over time. We can also notice this evolution in the +animated overlaying of elements on the website15. +px (103) +Figure 17: Interface area evolution, measured in squared +pixel units +5.5 +Files’ size and number +We made a similar approach to study the variation in file size re- +garding the entire dataset. Figure 18 represents the SERP captures’ +file size evolution and the number of files in its folder. For each +capture, we summed the size of each associated file and averaged it +per month and, consequently, year. We did not consider embedded +files in the HTML but files linked through the associated files folder. +Size results accompany the development of the interface area, +as seen in Figure 17, expressing a steep rise in the last few years. +This increase cannot be related to a surge in the associated files, +as later values share similar values with the first years of SERP +existence. Nevertheless, results suggest that SERP sought to reduce +the number of related files, achieving this aim during the first +decade. This number started to rise again because interfaces evolved +15Available at https://bedgarone.github.io/serpevolution/layout +and demanded more images and graphics, which can increase the +files needed to load a SERP. Besides, protocol advancements such +as HTTP pipelining or SPDY may have contributed to the increase +of these associated assets. +Bytes (103) +Average nr. of fles +Figure 18: Source code size (left - line - circumference), +Source code + Files folder size (left - dashed - plus sign) and +average number of files associated (right) over time +6 +DISCUSSION +SERP are no longer pages with “10 blue links”. We have shown that +the range of SERP elements has been growing continuously but +steeper in the last few years. Although this increase might con- +tribute to cluttering the page and falling into the “more is less” trap, +we expect SERP to continue evolving quickly [5, p. 512]. Interfaces +have kept track of web development’s evolution, as shown by the +regular adoption of design patterns. +The Category Hierarchy stands out as the only feature used in the +first years and later discontinued. The decrease in web directories’ +popularity may have been the reason for this. +Although organic results have been keeping a relatively stable +format, SERP have become more diversified over time, providing +increasingly sophisticated navigational aids to enhance results in- +terpretation, relevance assessment, and user satisfaction [8, 21, 54]. + +Goole m +Google +cnn +sgin +at Archi +YouTube +NNG +Z> +Gogle +G +CNN +ND +CNN +The 10 Best Paris Tou +CNN(@CN)ITVCHIIR’23, March 19–23, 2023, Austin, TX, USA +Bruno Oliveira and Carla Teixeira Lopes +These features complement organic and sponsored results and at- +tempt to infer users’ needs and quickly satisfy them even if not ex- +plicitly mentioned in their search queries [8] following a “universal +search” vision [39]. As shown, most SERP elements are informa- +tional, providing information about results. +The growth in SERP elements almost exponentially increased +interface area since modern pages need more vertical space. These +pages are getting heavier. Surprisingly, given the number of current +features that use non-textual content, the average number of files +is not much more significant than in 2000. Notwithstanding, we +noticed a higher dispersion in the average number of files in more +recent years, as seen in Figure 18. +Aggregating results from heterogeneous sources - verticals - and +presenting them in a single interface – aggregated search – has +become standard practice [66]. We could notice this in the Image +Pack, Video Pack, Local Pack, and Top Stories features. Other features +like the Featured Snippet assemble information in different formats +extracted from one source. Since 2020, information on the Web +has dramatically increased in quantity and diversity. Videos are +nowadays much more popular than they were at the beginning +of the century, and content from new platforms such as location +technology (e.g., Google Maps) or social media (e.g., Twitter) has +emerged. This evolution naturally affected the need for the change +of the SERP elements. The fact that graphical information is pro- +cessed before the textual information [23] might also explain the +recurrent appearance of images and videos in features. On the other +hand, the evolution of SERP is strongly informed and influenced +by users’ search behavior. Users scarcely look at results other than +the first ones [26] and tend to reformulate the query if they cannot +find promising results at the top of the list [20]. They rarely look at +results, such as videos or news, in their respective SERP ‘tabs’ [52]. +This behavior may also motivate search engines to include aggre- +gations of other types of results [23], not for the diversity, because +most users will not reach them outside the first results page. +The SERP are also affected by the interests of the search engine +providers who provide users not only with relevant results but +also with results of their interest. This reality gained prominence +when the European Commission concluded that Google abused its +market dominance by the way it presented sponsored results [12]. +Although not a focus of analysis of this work, it is frequent to see +Google showing results from its maps service, YouTube results in +the video container, shopping results from its shopping ads service, +and blurring the lines between organic and sponsored results. These +decisions have a higher impact on users with less search engine +knowledge, who are more likely to trust and use Google [49]. +SERP features often allow the user to interact with the contents +of a web page directly from the SERP [21, 43]. This cannibalizes +clicks [9] and might mean that users get satisfied without clicking +on search results, which was defined by Li et al. [33] as “good +abandonment”. Studies have found that features that provide direct +answers improve user engagement on SERP, reduce user effort, +and promote user satisfaction [64]. Besides contributing to user +satisfaction, these features also encourage user engagement and, +thus, revenue [21, 31]. +Our results reinforce the idea that evaluation measures solely +based on the list of “10 blue links” must be rethought based on +the SERP we have today. The standard practices of aggregating +results from heterogeneous verticals and including features that +provide direct answers on SERP have implications for how users +interact with search systems and, therefore, on their evaluation. +The cannibalization of clicks requests evaluations that consider +other types of interactions with the SERP. Challenges emerge in the +way users’ feedback is explored, either explicitly from user studies +or implicitly from weblogs. Work has already been conducted to +rethink evaluation in the context of aggregated search pages [66] +and good abandonment scenarios [30]. +7 +CONCLUSIONS AND FUTURE WORK +Using Google as a case study, we studied how SERP user interfaces +evolved over two decades. While existing research has relied on +the actual states of these interfaces, we have updated and improved +the analysis with an evolution perspective, addressing old and new +elements, their positioning, size, and patterns. We extracted and +provide a dataset with 5,000+ SERP captures, including HTML +versions and screenshots. +We showed that SERP are becoming more diverse in terms of +elements, aggregating content from different verticals and including +more features that provide direct answers. These changes affect +user behavior that, more often, abandon the page satisfied, the +so-called “good abandonment”. +In the future, we want to analyze other web search engines’ SERP +and compare results. We would also like to explore the evolution +of SERP in mobile environments. Here, we would like to know if +the increase in the SERP area found in this work results in a more +significant differentiation of user interfaces between desktop and +mobile environments. As stated previously, features that require +interaction with the SERP were not analyzed here because our page +captures from the Internet Archive don’t allow such interaction. +Given the importance of such features, we would like to explore +them in current SERP versions. Studies that analyze SERP charac- +teristics by types of queries (e.g., informational, navigational) and +user studies comparing old and contemporary SERP would also be +interesting. +ACKNOWLEDGMENTS +The Master in Informatics and Computing Engineering and the +Department of Informatics Engineering of the Faculty of Engineer- +ing of the University of Porto supported this work by funding the +registration fee. +REFERENCES +[1] Khalil Amjad and Fadi Alrub. 2013. A comparison of search engine’s features and +mechanisms. Advanced Database Systems 2131-8070580-sec1 (December 2013), +1–8. +[2] Avishek Anand, Lawrence Cavedon, Hideo Joho, Mark Sanderson, and Benno +Stein. 2020. Conversational Search (Dagstuhl Seminar 19461). Dagstuhl Reports +9, 11 (2020), 34–83. https://doi.org/10.4230/DagRep.9.11.34 +[3] The Internet Archive. 2021. 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Association for Computing Machinery, New York, NY, +USA, 115–124. https://doi.org/10.1145/2348283.2348302 + diff --git a/2tFAT4oBgHgl3EQfkx2Z/content/tmp_files/load_file.txt b/2tFAT4oBgHgl3EQfkx2Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9cd135372724e7731a6787abf32fcce8629369c8 --- /dev/null +++ b/2tFAT4oBgHgl3EQfkx2Z/content/tmp_files/load_file.txt @@ -0,0 +1,1359 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf,len=1358 +page_content='The Evolution of Web Search User Interfaces - An Archaeological Analysis of Google Search Engine Result Pages Bruno Oliveira Faculty of Engineering of the University of Porto Porto, Portugal up201605516@edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='pt Carla Teixeira Lopes Faculty of Engineering of the University of Porto and INESC-TEC Porto, Portugal ctl@fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='pt ABSTRACT Web search engines have marked everyone’s life by transforming how one searches and accesses information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Search engines give special attention to the user interface, especially search engine result pages (SERP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The well-known “10 blue links” list has evolved into richer interfaces, often personalized to the search query, the user, and other aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' More than 20 years later, the literature has not adequately portrayed this development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We present a study on the evolution of SERP interfaces during the last two decades using Google Search as a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We used the most searched queries by year to extract a sample of SERP from the Internet Archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Using this dataset, we analyzed how SERP evolved in content, layout, design (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', color scheme, text styling, graphics), navigation, and file size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We have also analyzed the user interface design patterns associated with SERP elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We found that SERP are becoming more diverse in terms of elements, aggregating content from different verticals and including more features that provide direct answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This systematic analysis portrays evolution trends in search engine user interfaces and, more generally, web design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We expect this work will trigger other, more specific studies that can take advantage of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' CCS CONCEPTS Human-centered computing → Interaction design process and methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Human computer interaction (HCI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' • Infor- mation systems → Users and interactive retrieval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Web search engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' KEYWORDS Search engines, SERP features, Web interfaces, Web design, Evolu- tion ACM Reference Format: Bruno Oliveira and Carla Teixeira Lopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Evolution of Web Search User Interfaces - An Archaeological Analysis of Google Search Engine Result Pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In ACM SIGIR Conference on Human Information Interaction and Retrieval (CHIIR’23), March 19–23, 2023, Austin, TX, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' ACM, New York, NY, USA, 14 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1145/3576840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='3578320 Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Copyrights for third-party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' CHIIR’23, March 19–23, 2023, Austin, TX, USA © 2023 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' ACM ISBN 979-8-4007-0035-4/23/03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1145/3576840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='3578320 1 INTRODUCTION The wealth of information available on the Web make search en- gines an essential tool nowadays [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Web search engines have evolved a lot, and their user interface is no exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Web search engines’ front end is gaining importance in a scenario where rel- evance is becoming more difficult to evaluate by users, and their perception becomes more influenced by the user experience [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 512].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Search interfaces support tasks from query formulation to select- ing and understanding search results [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 26-40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Typically, web search engines have a home page containing a search entry form in which the user types a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Retrieval results are usually dis- played as vertical lists on Search Engine Results Pages (SERP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' For its richness, our study focuses on these pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' SERP have started as simple “10 blue links” pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Although search engines have kept a consistent format of presenting search results, the information in SERP goes way beyond these links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The design of query interfaces and retrieval results display is an active area of research and exper- imentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Although works provide an in-depth analysis of search user interfaces, such as the one from Hearst [22], the temporal evolution of SERP is understudied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Portraying the evolution of SERP contributes to preserving the history of web search user interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Moreover, assuming these interfaces follow the general trends in web user interfaces, it con- tributes to the overall study of web interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Broadly, it depicts trends in web design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This evolutionary analysis is also helpful for the interactive information retrieval community to understand better how the search interfaces have evolved in content, layout, and navigation and build upon this for further and deeper analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This study conducts an evolutionary analysis of Google’s SERP during the last two decades of Google Search, analyzing the overall user interfaces over time and their elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We chose Google for its popularity in the web search engine landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' For this analysis, we have captured from the Internet Archive 5,653 Google SERP from 2000 to 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This work makes several contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' First, we present a sys- tematization of the elements that appear or have appeared in a SERP, defining each and providing visual examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This systematization can be helpful in future studies that have the SERP as their focus and allows the establishment of common terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Second, we analyze the evolution of the overall SERP and each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Third, we propose a methodology that can be used to study different types of web search user interfaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', mobile ones) or user interfaces in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Fourth, this paper makes two resources available arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='08613v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='IR] 20 Jan 2023 CHIIR’23, March 19–23, 2023, Austin, TX, USA Bruno Oliveira and Carla Teixeira Lopes to the community: a dataset1 containing the screenshot and files associated with each extracted capture and a website2 summarizing the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These resources can also be the input of further studies, inclusively done by researchers without advanced technological skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2 RELATED WORK Research in search user interfaces focuses on their design and evaluation, either in broad or focused on query formulation, the presentation of search results, or even query reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' There are also works focused on personalization, information visualiza- tion, or domain-specific search interfaces such as mobile, social, or multimedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' There are whole books and monographs dedicated to this subject [22, 35, 43, 48, 61, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Despite such research, given our focus, we only describe works focused on analyzing the anatomy of SERP and, eventually, its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This section does not cover works proposing or evaluating search interface components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Höchstötter and Lewandowski [23] address SERP composition and count the various elements’ appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' To the best of our knowledge, this was the first work to analyze the entire structure of the SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Besides, the authors examined the retrieved results, their sources, and types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', organic results, advertisements, shortcuts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' When the authors wrote this paper, advanced features in SERP were not widespread, which was not the case when Moran and Goray [40] studied the anatomy of SERP, defining the terminology for SERP elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Nielsen Norman Group uses this terminology in several articles [36, 40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In their ‘Search Patterns’ book, Morville and Callender [43], apart from addressing the anatomy of the search process and related behavior, also list elements and principles of interaction design, illustrating many user interface design patterns around search websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' To the best of our knowledge, no works systematically collect and analyze SERP interfaces over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 3 METHODOLOGY The first stage of our work involves building a sample of SERP inter- faces over time, a process described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' After collecting this sample, our attention focused on its analysis and automation, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1 What SERP have we captured?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Google Search currently has 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='4% of the market share [10], a lead- ership that goes back to 2002 [27, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In this context, we decided to focus our analysis on this search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This study will address desktop versions of Google Search from 2000 until 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' A compar- ative analysis with the seconded ranked search engine, Microsoft Bing, is done in another work [45] and is available on the study’s website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Internet Archive keeps snapshots and the respective HTML version of web pages over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Its collection contains 588 billion web pages [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Internet Archive provides the Wayback CDX Server API, which allows complex querying, filtering, and analysis of cap- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' While filtering by URL, we can use a wildcard (*) at the end of 1Available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='25747/991g-f765 2https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/ the URL to specify the latter as a prefix and receive entries beyond the specified URL (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/search?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='q=cookies*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We found more than 195 thousand captures of Google SERP during two decades using the API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This large number of SERP and existing resource restrictions led us to devise a method to iden- tify a smaller set of SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' To increase the likelihood of reaching pages with SERP element diversity, we have used a set of 129 most searched queries in the last 20 years, retrieved from Google Trends during the same period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This set3 contains the first search query from each available category, such as People, Health, or Electron- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These queries include relevant terms often searched by users and trigger features in SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Hence, it is highly likely that SERP interfaces derived from these queries are richer and, thus, more relevant for this study than those generated by random searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We decided to append these queries with the ‘*’ wildcard while submitting them to the API to obtain more captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We noticed that some years had no captures using the most searched queries, which coincided with periods in which there were few captures from Google Search’s domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Hence, in those years, we collected all the available captures (all method in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We also noticed that the last two years had much more captures (>10 thousand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Therefore, in 2019 and 2020, we restricted the URL submitted to the API to those containing queries shorter than 37 characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Considering that an English word has, on average, about five characters [44, 65], the 37 characters are the equivalent of 6 words plus spaces between them, which is more than the two to three words that a query typically has [6, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This restriction excludes more specific queries that are probably less useful to the plurality of interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Our sample has 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='653 captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The last column of Table 1 has ordered lists with the number of captures per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Table 1: Method used to collect the captures, maximum length of the query (search URL), the width of the screen- shot, and the number of captures extracted per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' method max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' length width #captures per year 2000 - 2002 queries 800px 200, 3, 23 2003 queries 1024px 231 2004 - 2008 all 1024px 12, 0, 200, 0, 26 2009 queries 1024px 11 2010 all 1024px 78 2011 queries 1024px 7 2012 - 2018 queries 1366px 57, 975, 30, 89, 172, 192, 548 2019 queries 37 char 1366px 171 2020 queries 37 char 1920px 2628 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2 How have we captured SERP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We used Python and Selenium Webdriver to visit each capture online, check if the capture was valid, save the HTML version, and 3Available at https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/mostsearchedqueries The Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP CHIIR’23, March 19–23, 2023, Austin, TX, USA CNN AMAZON TRUMP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Organic result found within 6 seconds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Load capture in Browser Remove banners Extract source and files Wayback Machine CDX API FOR-EACH capture Set of queries Get SERP timestamp and original URL Generate SERP archived URL NO YES web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org /web/ + TIMESTAMP + ORIGINAL URL {TIMESTAMP, ORIGINAL URL} {TIMESTAMP, ORIGINAL URL} {TIMESTAMP, ORIGINAL URL} Skip capture Figure 1: Extracting captures procedure generate a screenshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The capture process is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The original URL is the URL of the original SERP (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/ search?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='q=photography), while the archived URL is the URL of its archived version (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/web/20160125203434/www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/search?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='q=photography).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Some captures with an HTTP OK status code were not consid- ered valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Some are inexistant, showing a contradictory message of URL not captured, while others are defective (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', incomplete interfaces without search results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' To automatically check the va- lidity of each capture, we try to find a result entry, the element that cannot lack in a SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Captures from SERP tabs other than the general first page, identified with “tbm” in the URL, were discarded for being outside this work’s scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We raise a timeout exception after 6 seconds, the time we empirically considered sufficient to load the capture in the browser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In these situations, the program will skip the capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Before downloading the page, we still remove graphical elements from Internet Archive, such as its information and donation bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Some of the captures present other distracting banners and overlapping parts of the interface, such as the ones related to cookie consent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We removed all the identified ones and extracted the source and associated files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The process concludes with generating full-height screenshots of every HTML version opened in another browser instance in headless mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We produced screenshots considering the most popular screen size at the time of the capture, as stated by the statistics [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We only considered the width, shown in Table 1, because SERP height is highly variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The dataset with all the extracted captures is available online4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='3 How have we analyzed SERP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The analysis process included two main stages, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' First, we have extracted a sample of captures from the primary dataset to identify SERP elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' For each month with captures in the primary dataset, we manually looked at the screenshots of that month’s captures and selected the capture with the most features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In the end, this set included 117 captures, with which we visually identified SERP elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We manually analyzed each element’s source code, looking for identifiers to locate the element in a later 4https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='25747/991g-f765 automated process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Element identifiers consist of HTML classes, ids, tags, or a combination of these using CSS selectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', ‘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='knowledge- panel’, ‘#tads’ or ‘#newsbox’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' All the encountered identifiers were logged and listed on the website5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' FOR-EACH element Identify SERP elements Build set of SERP Seek identifiers in captures’ HTML Log detected identifiers Search element in the dataset Log element’s first timestamp Screenshot element Color interface Load capture in Broswer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='#id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='id #id > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='id Figure 2: Detection and analysis of elements procedure In the second stage, we automated the detection of these elements over time, allowing the exploration of a more significant number of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Finding an element with these identifiers triggers a function that stores the date of the element’s appearance in a log file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We imposed no limit of captures per month to register the element’s appearance, as the computation permits a full dataset scan in an acceptable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The function also receives the element’s upper-left corner coordinates, width, and height, generating and saving its image in the element’s folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Contrary to the element’s timestamp, we imposed a limit of 15 captures per month while screenshotting to reduce and make the scan time feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We estimate that 15 monthly samples are enough to capture the possible changes of an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Following a similar procedure, we automatically used the iden- tifiers to detect and color the web page’s targeted areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We used Python, Selenium Webdriver, and BeautifulSoup to scrape every HTML capture to identify and generate transparency-colored im- ages for each category of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We generated these images in a headless browser with a 1920px width, regardless of the capture’s width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This simplification does not affect the final result because the elements in the interfaces do not move dynamically as the width increases or decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We have not constrained the height in this generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Due to the size of the dataset, we imposed a limit of 15 elements per month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We overlayed all the individual images from single captures for each category of elements, which allows the overlay to enhance the most common areas while leaving the others almost unnoticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The overlaying process uses the upper-left corner as the reference for image alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Navigation & user inputs includes elements in and next to the footer, where common areas were not evident due to the page height’s variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In this case, to generate and correctly overlap the footers of the interface, we considered a height value of 600px for the footer, cropping it from the bottoms of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The 600px height was estimated after a visual analysis of SERP pages and included a margin of error to ensure we covered all elements under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Thus, the orange result displayed in Figure 3 is trimmed in the middle and combines those two capturing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We will analyze the positioning of every category of elements in 5https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/elements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' HtmL FILES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='. HTMl FILES<> HTML<> HTMLCHIIR’23, March 19–23, 2023, Austin, TX, USA Bruno Oliveira and Carla Teixeira Lopes the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' An animated version of each result in this Figure is available on the website6, permitting us to observe how the positioning of the elements’ categories in SERP changed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 3: Transparency-colored overlaying results for each category 4 EVOLUTION OF SERP ELEMENTS We present each element’s description and analyze its period of presence and positioning in SERP (also displayed on the website7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Moreover, we analyze each element’s evolution regarding content, graphics, navigation, and their relation with user interface design patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These patterns are problem-oriented and generally repeat- able solutions to usability problems in interface and interaction design [18, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1 Visual identity & search statistics Considering Google’s visual identity, each logo version has kept its position and size with rare variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 3 shows the overall position of the logo in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Some interfaces are exceptions, offering a more significant left margin and a right-shifted logo and search statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 3, in gray, shows how search statistics appear consistently below the search query or navigation bar, either left- aligned, right-aligned, or justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Statistics included the number of results seen per page in the first decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Later, Google removed this information and kept only the estimated number of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Details about the logo evolution and the content about search statistics can be seen online8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2 Navigation & User Inputs Figures 4 and 5 display the main stages of how the search box and its surroundings evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The left-aligned query bar has also marked its place at the top of the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Yet, in the early phases, it also appeared at the bottom, as seen in the 2000 and 2006 screenshots of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Input Prompt design pattern has always been applied to User Inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We notice a change in the width of the entry form after 2006, which may suggest an encouragement to the formulation of longer queries [6, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This change is in line with experimental evaluations where query length is positively related to effectiveness in the IR 6https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/layout 7https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/elements 8https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/design Figure 4: SERP headers neatly from 2000, 2001, 2006, 2012, 2017 and 2020 Figure 5: SERP footers neatly from 2000, 2006, 2012, 2017 and 2020 task [6, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Since query formulation influences effectiveness more than algorithmic factors [14], it makes sense to encourage users to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=" We can also notice the appearance of a way to specify 800 1024 1366 1920 800 1024 1366 1920 800 1024 1366 1920 800 1024 1366 1920 600 768 Visual identity 1080 &results statistics 800 010241366 1920 1080 Navigation& Organic Sponsored Features user inputs results resultsGoogle mp3 pure 10 results All Languages GooqleSearch I'm Feeling Lucky Search Tips LanguageQptions EmailTheseResults Sponsored Links ClickheretofindMP3files." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Listen,ShareandStore themforfree!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='myplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com SignupTodayforamyplayLockerandget3GBofFreeStorage!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=" Music - 1o0's of songs to choose from!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Stop searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Start listening: www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='firstlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content="com Wheretomorrow'smusicis heardfirst!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Firstlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com Googleresults1-10ofabout23,997formp3pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Searchtook 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content="34seconds Category: Arts>Music>Sound Files>MP3>Link Lists Google AdvancedSearch Preferences LanguageTools Search Tips Spathiphyllum GoogleSearch I'm Feeling Lucky Web Images GroupsDirectory SearchedthewebforSpathiphyllum Results1-10ofabout7,160." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Searchtook0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='10seconds Category: Science>Biology>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='>Magnoliophyta>Liliopsida>Araceae>Spathiphyllum Sign in Google Web Images VideoNew!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' News Maps more " "tsunami" Search Advanced Sesrch Preferences Web Results 1 - 10 of about 85,900,000 for "tsunami" [definition].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='28 seconds) Search Images Videos Maps News Shopping Gmail More Google ipod touch Search About 325,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='000 results Advanced search Search Images Maps Play YouTube News Gmail Drive More Google sparky α All Images Videos News Shopping Maps Books Gougle scoliosis X Q Q All Images 国News Videos Books : More Settings ToolsGoo0000000ogle ResultPage: 12345678910 Next Space Racer GooqleSearch Search within results Try your query on: AltaVista Deja Excite HotBot Infoseek Lycos Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Google Web Directory - Cool Jobs - Advertise with Us!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' - Add Google to your Site - Google in your Language - All About Google Result Page: 12345678910 Next "tsunami" Search Search within results I Language ToolsI Search Tips I Dissatisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Help us improve Google Home -Advertising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Programs - Business Solutions -About Google 2345 8910 Next Search Help Give us feedback Google Home Advertising Programs Business Solutions Privacy &Terms About Google 1 2 34 5 6 78 9 10 Next Advanced search Search Help Send feedback Google Home Advertising Programs Business Solutions Privacy Terms About Google 234567 8910 NextThe Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP CHIIR’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' March 19–23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' USA the query via spoken commands in the latest screenshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This ap- pearance aligns with the increasing attention conversational search interfaces have received [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Although works show promising re- sults in incorporating visual elements in query formulation [53], there is no sign of them in Google Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The search box and the current query are always visible for the searcher, as recommended [63], since 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This visibility occurs even when the user scrolls down and the top of the SERP is no longer visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Before 2019, the query was always available in the search box, but this box disappeared if the user scrolled down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Because we cannot interact with past SERP versions, we cannot analyze interaction features such as auto-complete or auto-correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2000, as seen in Figure 4, the main buttons for this category were ‘Google Search’ and ‘I’m feeling lucky’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It was possible to select, directly in a dropdown near the search box, how many results should be shown, the language intended for the results, and an option to send the retrieved results by email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In the screenshots of 2000 and 2001 presented in Figure 4, the Category Hierarchy feature is also visible showing one or more categories related to the query, probably obtained from Google Web Directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This feature, available until 2004, located these categories in more general areas using breadcrumbs, a design pattern that linearly specifies hierarchy levels leading to the current subject or page [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' At the bottom of the interface, there was also an option to search within the results and links to trigger the query on other Web search engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Other lesser relevant links would point to SERP experience (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', language, search tips).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Although nonexistent for several years, it is possible to notice in Figure 3, in orange, the significant presence of the left navigation bar during some years of SERP history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This left column in orange is naturally interrupted in the middle area of the interface, trimmed as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2001, Google introduced the tabs bar in the shape of Module Tabs used when content is groupable and there is no room for everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Modules of content are divided into small tabbed areas with only one visible at a time, allowing the user to click on tabs to reveal other modules [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Tabs don’t need to be literal tabs and don’t have to be at the top of the stack of modules [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The modules were Web, Images, Groups, and Directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2002, Google added the News tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2006, the tabs were replaced by simple links above the search query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This year, the Video, Maps, Froogle, and More tabs were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2008, the tabs bar went to the very top of the page, and the Shopping and Gmail tabs were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2010, the left sidebar was introduced, complementing the interface with other tabs and information such as location and results filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2012, Google removed the bottom query bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2013, the tabs bar was displaced underneath the main query bar, maintaining some links and changing others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' At the same time, in the left sidebar, only the results’ filtering options were available before Google removed this sidebar in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Finally, in 2015, the tabs bar below the query bar became the only existing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' There is also a usual space for sign-in and user account informa- tion on the right side of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' At the bottom of the page, two areas are noticeable: pagination, aligned to the center of the results container, and the footer, at the very end, covering all the width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These areas are visible in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Pagination design pattern has been applied to Navigation since the beginning of SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Pagi- nation breaks up the long results list into different SERP, loading them one at a time [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='3 Organic Results Organic results are retrieved based on the document’s content and the overall retrieval algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Regarding their positioning, in Figure 3 (blue), there is a more substantial presence of colored frames in the area where results are typically included (henceforth called results container), with a greater focus on the visible area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' By visible area, we mean the interface area that can be seen without scrolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Information that needs scrolling to be accessed is in a scrolling area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Over time, it is noticeable that SERP pages have increased their height due to the vertical decrease of color intensity, revealing results in lower page regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Part of the results is slightly shifted to the right, referring to interfaces with a left-side navigation bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It is possible to observe two other very tenuous sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' One with more centered results since the interface from 2010 to 2012 adjusted elements’ position according to the width and central axis of the screen rather than a left alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Some frames cover the entire interface width, not because the content was that large, but because, in the early days, HTML divisions (div) typically spanned across the complete width of the viewport because Document Object Model (DOM) trees were less deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The viewport consists of the visible area of an interface on a display device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Organic results can be regular or enriched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The structure of regular results, seen in Figure 6, started as a basic block of the page title, snippet, and URL links for similar or/and cached pages for the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2013, Google hid these links in a dropdown, only visible by its arrow icon until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2018, Google introduced a link to translate the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 6: Regular result from 2003 (left) and 2020 (right) The enriched results, seen in Figure 7, are variations of regu- lar results, with extra elements below the title, snippet, and URL, giving some additional information to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This element can have greater visibility and, in turn, a higher click rate [8, 21, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It appeared for the first time in 2008, lasting until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Its posi- tioning is consistently at the top of the results container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Initially, the extra content included two columns of site links pointing to sub-pages of the result’s domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These links, named quicklinks, are navigational aid that attempts to take the users to the content they want quickly [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2009, it started highlighting structured data, such as reviews and ratings for products and services, based on experiments that showed that users find value in this new data [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2010, Google enhanced each site link with a short description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2016, a search bar was introduced so that the user could search the result’s website directly from SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In both regular and enriched results, query term highlighting with boldfacing has been applied since 2006 to improve the usability of search results listings [11, 32, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The dominant colors are blue for the title, green for the URL, and black/grey for the snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Holisticopia:WyomingMassageTherapy www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com > Coronavirus > new Foxy Lady Salon 113o S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Torrington, Banner Health Massage Therapy 625 Albany .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='holisticopia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/qeo/WvominaMassaqe+Therapv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='htm -34k -Cached-Similar paqes r/Coronavirus: In December 2019,a novel coronavirus strain (SARS-CoV-2)emergec city of Wuhan, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This subreddit seeks to monitortheCHIIR’23, March 19–23, 2023, Austin, TX, USA Bruno Oliveira and Carla Teixeira Lopes Figure 7: Enriched result from 2009 (left) and 2016 (right) About 2011, the URL moved above the snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Google underlined the title until 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2020, the URL changed its color to gray and moved above the title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The new position of the URL may have been influenced by the importance of URL and domain names in evaluating the credibility of a result [34, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2019, the URL started being displayed as a breadcrumb, which may have been motivated by research that concluded that long and complex URL negatively impact clickthroughs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In both types of organic results, the snippet length has not visibly changed, with most of the snippets having two lines, the suggested size for informing the user, and including as many results as pos- sible in the visible area [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' As most of the queries used for data collection are associated with informational tasks, we cannot con- clude if Google adjusts the length of the snippet to the type of query (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', informational, navigational) as suggested by existing research [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='4 Sponsored Results As seen in Figure 8, textual ads are short advertisements that appear alongside organic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These entries are focused on com- mercial intent, combining relevance with revenue, and are usually manually crafted, a standard in the advertisement industry [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This element has been present in SERP since the first day, lasting until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This element’s content and evolution are identical to the ones of Regular Results, except these were initially marked with a specific tag, ‘Sponsored link’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2012, some of the results started to include quicklinks, although less noticeable than in Enriched re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Until 2014, a different background color distinguished these elements from organic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Scarce occurrences in 2017 also had a colored background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2014, the tag ‘Ad’ substituted the previous one, having a yellow background color, while from 2016 onwards, it was green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2020, some ads could have the shape of an actual Enriched result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The URL also changed its color to gray and moved above the title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 8: Textual ads from 2002 (top), 2014 (left) and 2020 (right) The other type of ads correspond to shopping content and are, therefore, called shopping ads, seen in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These are acti- vated when the query is commercial [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These are very striking results, listing various information about each product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This el- ement appeared for the first time in 2013, lasting until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Its positioning is usually at the top of the results container and, more frequently, at the right of that container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Shopping ads used to be exclusive to the right sidebar, displaying one to four results in a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2018, each result was embedded in an individual card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Until 2019, query terms were highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2020, this ele- ment appeared in the results container in a carousel of cards with more width and less height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Cards design pattern has been applied to this element since 2019, and the Carousel since 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Cards display content composed of distinct parts, generally about a single subject, to form one coherent piece of content designed to expose information efficiently [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It is usual for cards to accompany other cards, carrying similar content but addressing different subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Carousel is a horizontal strip of simple cards, letting the user scroll horizontally to view them and encouraging the inspection of the following items [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Since the element’s beginning, the Thumbnail Grid has been applied, enabling a quick overview of images by shrinking the original ones [51, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In Figure 3, it is possible to identify two major advertisement areas in SERP: top ads and right ads, following a typical SERP layout [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 489].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Elements in the right sidebar move horizontally over time because some interfaces force this sidebar to be responsive to the screen’s width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Top ads are the most common positions for ads within a soft vertical variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Older interfaces placed advertisements on a fully colored div that occupied almost the entire viewport width for the same reason described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Compared with regular results and other features, we can see in Figure 3 that sponsored results are more concentrated in the visible area, probably motivated by the fact that searchers rarely scroll [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 9: Shopping ads from 2013 (left), 2019 (center) and 2020 (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='5 Features SERP features complement organic and sponsored results, attempt- ing to provide answers to the query without just pointing to web- sites that might deliver that information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 3, in green, shows features spread around the interface, mainly in the visible area and upper half of the scrolling area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Many features are similar to regular results but with more significant height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These features also share a place in the sidebar with adver- tisements, recently becoming more present than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Most frames with large height values correspond to the well-known Knowledge Panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' A horizontal area is noticeable, generally assigned to the Carousel and the Carousel Grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Recent Earthguakes in California and Nevada -Index Map WelcometoIKEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com-IKEA Includes an index map and zoom views, with location and magnitude information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Updated www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='ikea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/-IKEA hourly Featuring Scandinavian modern style furniture and accessories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Include storage options quake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='usgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='gov/recenteqs/ - 11k - Cached - Similar_pages lighting, decor products, kitchen appliances and pet care Earthquake list Recent Earthquakes for 115-34 Real-time Earthguake Maps FAQs Results from ikea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com Q Los Angeles Disclaimer 2-degree map Big earthquake list IKEASeattleHomePage Ikea More results from usas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='aov IKEA Seattle Home Page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='IKEA IKEA Home furnishings, kitchens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Seattle Store Information .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' appliances, sofas, beds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='.The Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP CHIIR’23, March 19–23, 2023, Austin, TX, USA Due to space limitations, we selected a subset of features based on their lifetime and distinctiveness to be analyzed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Video Pack feature will not be considered for its similarity with the Image Pack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The same happens with Direct Answer Results, Local Pack, People Also Ask, Carousel, Carousel Grid, Recipe Cards, Twitter Pack, Category Hierarchy, and Covid-19 Left Panel for their shorter presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' More information about these features is available in another paper [45] and online9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Featured Snippets, seen in Figure 10 are answer boxes in which Google responds to a question-related query based on information taken from a page [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This element appeared for the first time in 2016, lasting until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Its positioning is consistently at the top of the results container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Although these snippets resemble enriched results, they differ in the type and position of extra content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In featured snippets, the additional content appears before the result’s title and not after, as happens in enriched results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The content of a featured snippet was initially made of a short paragraph with an- swering information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It evolved to a general layout, used until now, consisting of a larger paragraph, a thumbnail at the upper-right corner, and the title and link to the information’s source, to where it is possible to navigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The answer also started to be returned as an ordered list or table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Instead of a thumbnail, a video or a carousel of images may also accompany it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2018, when available, Google introduced the date of the source’s publication after the information paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The underlined style of the title was removed in 2017, and the green URL was changed to a gray breadcrumb URL in 2020, as in the organic and sponsored results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Text styling is used in both paragraph and title, enhancing relevant words in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Studies have found that these snippets help make more accurate decisions when containing the correct information [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 10: Featured Snippets from 2016 (top), 2017 (left) and 2020 (right) The Knowledge Panel, seen in Figure 11, is perhaps the high- light of SERP features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It is a dynamic feature that provides direct information in various formats within the same panel, pointing to related content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The contents range from text to images, ratings, social profiles, factual information, and similar search topics [47], helping the user to understand a particular subject quickly and facilitating a more in-depth search [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This element appeared for the first time in 2014, lasting until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Its positioning is always at the right of the results container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The basic structure consists of a panel with a top thumbnail of the subject, vertically followed by a title, a website link if applicable, a resume paragraph usually by Wikipedia, a structured list of direct information, and a block 9https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/elements of People also search for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' During the following years, Google intro- duced other content highly dependable on the search topic and variable in coverage and quality [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It is common to have this panel populated by Wikipedia information [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The graphics of this element was stable over time, following the improvements in Google’s interface design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Grid of Equals and Thumbnail Grid design patterns have been applied to this element since its beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Grid of Equals is a pattern to display items in a grid or matrix, each following a standard template, linking to respective pages [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Cards, Carousel, and Module Tabs design patterns were applied to this element in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 11: Knowledge Panel from 2016 (left) and 2018 (right) The Image Pack, seen in Figure 12, presents a set of images taken from various sources in Google’s index in searches where visual content is valuable [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This element appeared for the first time in 2006, but only after 2010 did it frequently appear, lasting until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Its positioning is highly variable throughout the results container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Most of the time, the content was a title associating images with the search query and a block of image thumbnails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2014, Google included a link for ‘more images’ and a link to report pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2019, Google introduced a bar of image categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It could have simple buttons or buttons with a thumbnail associated with its category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The graphics started with a considerable presence of blue colors, typical in Google’s early interfaces when images had a blue border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2014, Google removed this border, but the main change was in 2019 when images started to be in a carousel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2018, the layout of a matrix appeared for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These changes increased the element’s area in the last two years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2020 the title, as usual in most elements, turned dark gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The carousel of image categories changed its shape to a line that expands to a matrix in the form of progressive disclosure using the Collapsible Panels pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Grid of Equals design pattern has been applied to this element since 2018, while Thumbnail Grid, naturally, since its beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Arguello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' [4] examined vertical results, such as images, in aggregated search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' They found that these results had more clicks in more complex tasks and that users were divided in their preferences for vertical search displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Kourtney Kardashian and Scott Disick were partners for 9 years (until 2015) Kourtney Kardashian produced and Scott Disick appears on Keeping Up with the Kardashians Closing ceremony to celebrate Brazil 2014 in style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Before the 2014 FIFA World Cup TM Final gets underway at the Maracana in Rio de Janeiro on Sunday, a special closing ceremony involvingaround1,ooopeoplewillcelebratethe sport as the tournament nears its unmissable climax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Space architecture, in its simplest definition, is the theory and practice of designing and building inhabited environments in outer space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Space architecture borrows Closing ceremony to celebrate Brazil 2014 in style - FIFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com from multiple forms of niche architecture to accomplish the task of ensuring human 2014-in-stvle-2404018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='btml beings can live and work in space en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org>wiki>Space_architecture Space architecture - WikipediaCNN Cable channel - cnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com The Cable News Network is an American basic cable and satellite television channel that is owned by the Turner Broadcasting System news channel was founded in 1980 by division of Time Warner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The 24-hour cable American media proprietor Ted Turner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' images Wikipedia ScottGreenstein Customer service: 1 (404) 827-1500 Headquarters: Atlanta, GA Film producer Founder: Ted Turner Founded: June 1, 1980, Atlanta, GA Scott Greenstein is president and chief content Parent organization: Turner Broadcasting System officer of Sirius XM Radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' He leads the programming and advertising sales of the largest radio company Profiles by revenue and one of the largest subscription media f in companies in the worid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Wikipedia Twitter Linkedin Born: 1959 (age 61 years), Freehold, NJ Facebook Employer: SiriusXM Satellite Radio TV shows Movies: The Enqlish Patient CW NEWDAY Profiles Live Twitter CNN Live Anthony New Day Today Parts Unk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='. Bourdain Sinoe 2013 2001 2006 People also searchfor Sinoe 2013 View 5+ more riusxm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' People also search for BBC NEWS James E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Mel Barry Diller saul Meyer Karmazin Zaentz BBC MSNBC ESPN Claim this knowiedge panel FeedbackCHIIR’23, March 19–23, 2023, Austin, TX, USA Bruno Oliveira and Carla Teixeira Lopes Figure 12: Image Pack from 2010 (left) and 2020 (right) The Top Stories, seen in Figure 13, are blocks of three or more recent news considered relevant to the query, recently placed in the form of a carousel [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Each story is now presented with a thumbnail, publisher, and timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This element appeared for the first time in 2004, but only after 2011 it started to appear frequently, lasting until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Its positioning is mainly in the visible area of the results container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The element’s content started with a vertical list of at most four news titles, each followed by the source’s name and how long ago it was published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2006, a journal icon was placed at the left of the list, and a link to ‘today’s top stories’ was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2013, the icon was substituted by a thumbnail for the first news result, being the most important news in the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The latter was complemented with an extract of the news, while the rest stayed the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2020, leading to a considerable increase in the element’s area, the graphics was majorly altered to display the results in a carousel of cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' However, the content was simplified to only present, for each result, a thumbnail, title, source, and how long ago it was published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' As usual, the color scheme was mainly blueish and filled with blue borders and underlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These were removed after 2020 with the softer gray colors for additional information but still blue titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Streams and Feeds design pattern has been applied to this element since its beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It defines a pattern to list time-sensitive items chronologically, combining the sources in one place [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' However, in this element, the relation to the query appears to be more relevant than publication time since it is no longer possible to observe any chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Cards and Carousel design patterns have been applied since 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 13: Top Stories from 2006 (left-top), 2013 (left-bottom) and 2020 (right) Related Searches, seen in Figure 14, is a common element on SERP pages from a very early age and offers suggestions for related searches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', queries that are in some way related to the current query and may be good candidates for follow-on queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These suggestions can be helpful to support exploration or provide query statements that express information needs in different ways [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Usually, these suggestions are generated based on search log data, either picking queries that frequently follow the current query [28] or clustering queries based on results’ clicking [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Each link takes the user to the respective SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This element appeared for the first time in 2008, lasting until now, except for 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Its positioning is always at the bottom of the results container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The content was diversified regarding how many suggestions would appear and its layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Each suggestion of search is a hyperlinked title pointing to its respective SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Initially, it was organized in a matrix of columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In 2011, Google reduced this schema to two columns, which could be displayed in just one column for suggestions with longer text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Until mid-2020, suggestions were blue and, until 2014, underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Google applied a search icon to each entry in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Later, a new version changed the graphics, making each entry a button with a solid gray background, a search icon, and a title in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This latter change contributed to a recent increase in the element’s average area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 14: Related Searches from 2008 (left - top), 2017 (left bottom) and 2020 (right) 5 AGGREGATED ANALYSIS This section analyzes SERP’s evolution from an aggregated per- spective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1 Elements’ lifetime As shown, SERP have always had a large variety of elements, each with its evolution and active periods, as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Ac- cording to the framework proposed by Wilson [63], input features were analyzed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2, with the search box being the main one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' To support control, besides the input features, we also iden- tified the related searches feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' All the other analyzed features are informational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It is important to note that, given the nature of this study, it was not possible to analyze dynamic features such as interactive querying or personalizable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Black cells in Table 2 identify the years we detected elements in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' For all years in-between black cells, we have con- ducted a manual search in other sources to avoid false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In these cases, we have searched specific websites10 for evidence of elements in such years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' If found, we paint the cell grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It is noticeable how SERP features have emerged in the last decade, contributing to a matrix full of element possibilities in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Almost every SERP element includes well-known design patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' A visual timeline with screenshots of these elements’ presence in SERP is available online11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2 Design pattern application Table 3 maps each element with the patterns proposed by Tidwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' [57], along with the start date of that appliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Older SERP elements make later use of design patterns for individual improve- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In contrast, some contemporary elements may have arisen 10https://searchengineland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com, https://googlesystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='blogspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com and Internet Archive 11https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/timeline/2010 ImagestorParisweather paris france weather forecast climate temperature eiffel tower average rainfall accuwea Images for"complex event processing" < <国Topstories News results for"american idol" - View today\'s top stories \'ldol Winner Hicks Sues Former Producer - Washington Post - 9 hours ago LIVE LIVE LIVE \'ldolFinalist OK After Vegas Robbery - FOX News - 14 hours ago Life after \'ldol\' - MLive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com - 19 hours ago NIA LIVECOVERAGE CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com News forfacebook Che New ork Cimes OCBSNEWS Facebook Surpasses $30_Ahead of Q4 Earnings Electionresults2020: Live Trump vs Biden Pennsylvania2020 Wall Street Journal (blog) - 1 hour ago - 239 related articles LivenewsonTrump Tracker: Presidential election results Shares of Facebook Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' breached the $30 mark on Wednesday, the first Biden and electoral Election time in six months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' votes CBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='ca As Facebook shares hit $30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' ETFs with holdingS_gain MarketWatch (blog) - 2 hours ago - 2 related articles 10 mins ago 24mins ago 19 mins ago Facebook rolling out Timeline changes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' including redesign ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Chicago Tribune - 8 hours ago - 28 related articles 》 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='< ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='7 ' metadata={'source': 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the navigator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='how did prince henry the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Searches related to amazon diversity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='school ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='navigator die ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='amazon diversity officer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='amazon black employee network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='prince henry the navigator facts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='prince henry the navigator fun ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='amazon diversity 2016 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='prince henry the navigator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='amazon supplier diversity registration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='amazonworkforcephonenumber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='accomplishments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='timelineThe Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='CHIIR’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' March 19–23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' USA Table 2: Presence of SERP elements from 2000 to 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In black the years in which the element appeared in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In grey are the years in which the element’s existence is documented elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Visual Identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Search Statistics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Navigation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='User Inputs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Regular Results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Enriched Results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Textual Ads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Shopping Ads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Knowledge Panel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Featured Snippets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Direct Answer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Local Pack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Image Pack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Video Pack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Top Stories ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Carousel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Carousel Grid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='People Also Ask ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Related Searches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Twitter Pack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Recipe Cards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Category Hierarchy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Covid-19 Left Panel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='from the need to apply a design pattern solution whose traces are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='evident from the element’s beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The website12 lists design patterns with images and elements that use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='3 Highlights We identify the most relevant changes along the upper part of a two-decade timeline of SERP in Figures 15 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These changes correspond to the entry of new elements or significant changes in input and navigation options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This timeline of changes is also available online13, enhanced with images and the option to filter the entries for navigation changes or element additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' As stated before, the first stage of the analysis used a visual selection from a monthly sample from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figures 15 and 16 also display, in their lower part, how Google’s SERP interfaces evolved in terms of design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In these timelines, we include the main versions of the SERP interface based on significant changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Apart from how the overall interfaces visually evolved, similar timelines about visual identity, search statistics, and navigation are available on the website14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Google’s initial interfaces had few depth levels in their HTML DOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The first interface design, traced in 2000, differentiated the sponsored results with a colored background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' A second search 12https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/patterns 13https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/timeline 14https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/design query bar existed at the bottom of the page, and the user could change the number of results presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Google removed this bar in the following interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The one traced from 2000 to 2004 revealed a right block of results, exclusive to sponsored ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It marked the appearance of the first bar with tabs directing the user to other types of content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', images and news).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The fourth interface design, traced from 2010 to 2012, was not left-oriented but varied in a spaced manner depending on the screen’s width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' It introduced a sidebar on the left, containing tabs to manage the results, but some of these tabs were duplicated due to the navbar’s tabs bar mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Significant aesthetic changes occurred in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The fifth interface design relates to the launch of the Knowledge Graph, with the right column being divided between it and sponsored results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Some modifications were found earlier in the dataset during those years, the sixth interface design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' However, a design close to the current one began at the end of 2018, the seventh interface design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' As noted in some elements’ graphics, this interface focused on modernizing its lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='4 User interface area We calculated the area of all screenshots in the dataset to analyze its evolution over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 17 shows the development of interface area per month (dots) and per year (line), measured in pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Each entry in the chart corresponds to the average area per month for all CHIIR’23, March 19–23, 2023, Austin, TX, USA Bruno Oliveira and Carla Teixeira Lopes Table 3: Design Patterns and time of their appliance to SERP elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Cells without a dash after the year represent single years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Organizing Navigation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Layout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Lists ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Streams ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='and Feeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Bread- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='crumbs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Grid of ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Twitter Pack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2017- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2017- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Recipe Cards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2020- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2020- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Category Hierarchy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2000-2004 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Covid-19 Left Panel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Right sidebar with sponsored results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='Top Stories introduced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='SERP tabs bar introduced (Images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Groups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Directory) Tabs bar placed above the query bar Image Pack introduced Left sidebar introduced Local Pack introduced Related Searches introduced Tabs bar placed on navbar News tab added Videos and Maps tabs added Shopping and Gmail tabs added 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 Figure 15: Highlights of SERP overall evolution (top) and Interfaces’ visual evolution (bottom) from 2000 to 2010 captures in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Months without values are months without captures in the dataset, as indicated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Results show an increase close to exponential due to the appearance of SERP fea- tures that have added extra content to SERP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' making them Googlem GoogleThe Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP CHIIR’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' March 19–23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' USA Shopping ads introduced Bottom query bar removed Carousel introduced Left sidebar removed Knowledge panel introduced Ads loose their colored background Tabs bar placed under search query Direct Answers introduced Feature Snippets introduced Covid-19 left panel 2017 2016 2015 2014 2013 2012 2011 2018 2019 2020 Figure 16: Highlights of SERP overall evolution (top) and Interfaces’ visual evolution (bottom) from 2011 to 2020 more extensive over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We can also notice this evolution in the animated overlaying of elements on the website15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' px (103) Figure 17: Interface area evolution, measured in squared pixel units 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='5 Files’ size and number We made a similar approach to study the variation in file size re- garding the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Figure 18 represents the SERP captures’ file size evolution and the number of files in its folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' For each capture, we summed the size of each associated file and averaged it per month and, consequently, year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We did not consider embedded files in the HTML but files linked through the associated files folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Size results accompany the development of the interface area, as seen in Figure 17, expressing a steep rise in the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This increase cannot be related to a surge in the associated files, as later values share similar values with the first years of SERP existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Nevertheless, results suggest that SERP sought to reduce the number of related files, achieving this aim during the first decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This number started to rise again because interfaces evolved 15Available at https://bedgarone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='io/serpevolution/layout and demanded more images and graphics, which can increase the files needed to load a SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Besides, protocol advancements such as HTTP pipelining or SPDY may have contributed to the increase of these associated assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Bytes (103) Average nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' of fles Figure 18: Source code size (left - line - circumference), Source code + Files folder size (left - dashed - plus sign) and average number of files associated (right) over time 6 DISCUSSION SERP are no longer pages with “10 blue links”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We have shown that the range of SERP elements has been growing continuously but steeper in the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Although this increase might con- tribute to cluttering the page and falling into the “more is less” trap, we expect SERP to continue evolving quickly [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 512].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Interfaces have kept track of web development’s evolution, as shown by the regular adoption of design patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Category Hierarchy stands out as the only feature used in the first years and later discontinued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The decrease in web directories’ popularity may have been the reason for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Although organic results have been keeping a relatively stable format, SERP have become more diversified over time, providing increasingly sophisticated navigational aids to enhance results in- terpretation, relevance assessment, and user satisfaction [8, 21, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Goole m Google cnn sgin at Archi YouTube NNG Z> Gogle G CNN ND CNN The 10 Best Paris Tou CNN(@CN)ITVCHIIR’23, March 19–23, 2023, Austin, TX, USA Bruno Oliveira and Carla Teixeira Lopes These features complement organic and sponsored results and at- tempt to infer users’ needs and quickly satisfy them even if not ex- plicitly mentioned in their search queries [8] following a “universal search” vision [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' As shown, most SERP elements are informa- tional, providing information about results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The growth in SERP elements almost exponentially increased interface area since modern pages need more vertical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These pages are getting heavier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Surprisingly, given the number of current features that use non-textual content, the average number of files is not much more significant than in 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Notwithstanding, we noticed a higher dispersion in the average number of files in more recent years, as seen in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Aggregating results from heterogeneous sources - verticals - and presenting them in a single interface – aggregated search – has become standard practice [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We could notice this in the Image Pack, Video Pack, Local Pack, and Top Stories features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Other features like the Featured Snippet assemble information in different formats extracted from one source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Since 2020, information on the Web has dramatically increased in quantity and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Videos are nowadays much more popular than they were at the beginning of the century, and content from new platforms such as location technology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', Google Maps) or social media (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', Twitter) has emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This evolution naturally affected the need for the change of the SERP elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The fact that graphical information is pro- cessed before the textual information [23] might also explain the recurrent appearance of images and videos in features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' On the other hand, the evolution of SERP is strongly informed and influenced by users’ search behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Users scarcely look at results other than the first ones [26] and tend to reformulate the query if they cannot find promising results at the top of the list [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' They rarely look at results, such as videos or news, in their respective SERP ‘tabs’ [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This behavior may also motivate search engines to include aggre- gations of other types of results [23], not for the diversity, because most users will not reach them outside the first results page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The SERP are also affected by the interests of the search engine providers who provide users not only with relevant results but also with results of their interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This reality gained prominence when the European Commission concluded that Google abused its market dominance by the way it presented sponsored results [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Although not a focus of analysis of this work, it is frequent to see Google showing results from its maps service, YouTube results in the video container, shopping results from its shopping ads service, and blurring the lines between organic and sponsored results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These decisions have a higher impact on users with less search engine knowledge, who are more likely to trust and use Google [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' SERP features often allow the user to interact with the contents of a web page directly from the SERP [21, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' This cannibalizes clicks [9] and might mean that users get satisfied without clicking on search results, which was defined by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' [33] as “good abandonment”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Studies have found that features that provide direct answers improve user engagement on SERP, reduce user effort, and promote user satisfaction [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Besides contributing to user satisfaction, these features also encourage user engagement and, thus, revenue [21, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Our results reinforce the idea that evaluation measures solely based on the list of “10 blue links” must be rethought based on the SERP we have today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The standard practices of aggregating results from heterogeneous verticals and including features that provide direct answers on SERP have implications for how users interact with search systems and, therefore, on their evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The cannibalization of clicks requests evaluations that consider other types of interactions with the SERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Challenges emerge in the way users’ feedback is explored, either explicitly from user studies or implicitly from weblogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Work has already been conducted to rethink evaluation in the context of aggregated search pages [66] and good abandonment scenarios [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 7 CONCLUSIONS AND FUTURE WORK Using Google as a case study, we studied how SERP user interfaces evolved over two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' While existing research has relied on the actual states of these interfaces, we have updated and improved the analysis with an evolution perspective, addressing old and new elements, their positioning, size, and patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We extracted and provide a dataset with 5,000+ SERP captures, including HTML versions and screenshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We showed that SERP are becoming more diverse in terms of elements, aggregating content from different verticals and including more features that provide direct answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' These changes affect user behavior that, more often, abandon the page satisfied, the so-called “good abandonment”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In the future, we want to analyze other web search engines’ SERP and compare results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' We would also like to explore the evolution of SERP in mobile environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Here, we would like to know if the increase in the SERP area found in this work results in a more significant differentiation of user interfaces between desktop and mobile environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' As stated previously, features that require interaction with the SERP were not analyzed here because our page captures from the Internet Archive don’t allow such interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Given the importance of such features, we would like to explore them in current SERP versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Studies that analyze SERP charac- teristics by types of queries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=', informational, navigational) and user studies comparing old and contemporary SERP would also be interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' ACKNOWLEDGMENTS The Master in Informatics and Computing Engineering and the Department of Informatics Engineering of the Faculty of Engineer- ing of the University of Porto 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://googleblog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='blogspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/2007/05/universal-search-best-answer-is-still.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Accessed: 2022-10-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' [40] Kate Moran and Cami Goray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' The Anatomy of a Search-Results Page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/web/20220511210638/https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='nngroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/ articles/anatomy-search-results-page/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' [41] Kate Moran and Cami Goray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Good Abandonment on Search Results Pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='archive.' metadata={'source': 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Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (Vancouver, BC, Canada) (CHI ’11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 1245–1254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1145/1978942.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/library/view/designing-interfaces-3rd/9781492051954/ [58] Anders Toxboe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Design patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' http://ui-patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' [59] Kristen Vaughn.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1145/3449078 [61] Ryen White and Resa Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Exploratory Search: Beyond the Query-Response Paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Morgan & Claypool Publishers, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' [62] Ryen W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Interactions with Search Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Cambridge University Press, Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://doi.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Providing Direct Answers in Search Results: A Study of User Behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In Proceedings of the 29th ACM International Conference on Information and Knowledge Management (Virtual Event, Ireland) (CIKM ’20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 1635–1644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1145/ 3340531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='3412017 [65] Ann Wylie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' What’s the best length of a word online?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' org/web/20221012095143/https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='wyliecomm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='com/2021/11/whats-the- best-length-of-a-word-online/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' [66] Ke Zhou, Ronan Cummins, Mounia Lalmas, and Joemon M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Jose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Eval- uating Aggregated Search Pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' In Proceedings of the 35th International ACM SIGIR Conference on Research and Development in Information Retrieval (Portland, Oregon, USA) (SIGIR ’12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 115–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='1145/2348283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} +page_content='2348302' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfkx2Z/content/2301.08613v1.pdf'} diff --git a/3NAzT4oBgHgl3EQfuf1J/vector_store/index.faiss b/3NAzT4oBgHgl3EQfuf1J/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4f1f92cc3db55f1fa2e938c364c1b3affc7345ac --- /dev/null +++ b/3NAzT4oBgHgl3EQfuf1J/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8a6597ef136284d377fe198bbc2e06e21be850720f7e3c7068ab6555e1f58acb +size 2228269 diff --git a/3dAzT4oBgHgl3EQfD_oT/content/tmp_files/2301.00984v1.pdf.txt b/3dAzT4oBgHgl3EQfD_oT/content/tmp_files/2301.00984v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e50ea58fcbb592a2f0226333dec9a5711b12f20 --- /dev/null +++ b/3dAzT4oBgHgl3EQfD_oT/content/tmp_files/2301.00984v1.pdf.txt @@ -0,0 +1,1795 @@ +Protein-Ligand Complex Generator & Drug Screening via +Tiered Tensor Transform +Jonathan P. Mailoa,1†* Zhaofeng Ye,1† Jiezhong Qiu,1 Chang-Yu Hsieh,1 and Shengyu +Zhang2* + +1) Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, China +2) Tencent Quantum Laboratory, Tencent, Hong Kong SAR, China + +† These authors contributed equally to this work +* corresponding author: jpmailoa@alum.mit.edu, shengyzhang@tencent.com + + +Protein-Ligand Complex Generator & Drug Screening via +Tiered Tensor Transform + +Abstract +Accurate determination of a small molecule candidate (ligand) binding pose in its target protein +pocket is important for computer-aided drug discovery. Typical rigid-body docking methods ignore the +pocket flexibility of protein, while the more accurate pose generation using molecular dynamics is +hindered by slow protein dynamics. We develop a tiered tensor transform (3T) algorithm to rapidly +generate diverse protein-ligand complex conformations for both pose and affinity estimation in drug +screening, requiring neither machine learning training nor lengthy dynamics computation, while +maintaining both coarse-grain-like coordinated protein dynamics and atomistic-level details of the +complex pocket. The 3T conformation structures we generate are closer to experimental co-crystal +structures than those generated by docking software, and more importantly achieve significantly +higher accuracy in active ligand classification than traditional ensemble docking using hundreds of +experimental protein conformations. 3T structure transformation is decoupled from the system +physics, making future usage in other computational scientific domains possible. + + +Introduction +Structure generation and optimization is an essential topic in the field of life and physical +sciences. The applications of generative model algorithm in these fields are diverse, ranging from +precipitation nowcasting1 and airfoil aerodynamics optimization2–4 down to the optimization of optical +nanostructures5–7, electrode microstructures,8 microfluidic devices,9,10 and material design.11–13 +Structure generation capability is even more prevalent and important for drug discovery +applications,14–20 where protein and drug candidate small molecule (ligand) structures are of interest +in the determination of suitable target-specific drug molecules. Recent development in the deep +learning community has enabled generative model algorithms to more accurately predict single +protein structures,21,22 with a notable recent example being the success of AlphaFold 2 in the CASP14 +competition.23 Other notable recent examples are similarly focused on either the generation of small +molecules with desired properties24,25 or on the high-efficiency sampling for protein structures (RiD, +RIP, L-RIP, etc).26–30 +Targeting a specific pocket of a protein with a ligand poses an additional challenge because the +compound is composed of multiple interacting structures. Finding the binding pose of a flexible ligand +molecule when it is docked onto a flexible protein receptor pocket is a daunting task due to the degree +of flexibilities built onto this two-system complex. The most common method (with lower +computational cost) is to freeze the target protein receptor pocket and use a docking software such +as AutoDock Vina,31 Smina,32 and Glide33 to attach the ligand molecule onto the rigid protein pocket34 +and generate multiple candidate ligand docking poses. Unfortunately it is known that ligands docked +onto a rigid protein pocket are not representative of how ligands look like in a real protein-ligand +complex and are difficult to use for active ligand classification,35 partly because the protein structure +is flexible and can undergo intrinsic or induced conformational changes.36,37 The state-of-the-art +solution to this problem is to perform ensemble docking, where multiple structures of the same +protein pocket (usually 102 – 103 structures) are either obtained experimentally or generated through + +long molecular dynamics simulations (MD with millisecond-long simulations, which correspond to +~1012 MD time steps)35,38 and other generative methods;39 the ligands of interest are then docked to +all these protein structures to generate multiple conformations of protein-ligand complexes. It is +worth noting that the protein structure generation is done with no regard to the specific ligand +existence in the protein pocket, as it remains difficult to simultaneously modify the protein and ligand +structures due to the combined degree of flexibilities in the protein-ligand complex. To the best of the +authors’ knowledge, MD simulation of the entire protein-ligand complex structure remains the only +reliable way to sample and generate diverse complex conformations of entire protein-ligand +pockets.40,41 Unfortunately, it is computationally expensive to do so due to the slow dynamics of the +protein, and this kind of process is usually reserved for the last step of a drug screening workflow such +as the free energy perturbation (FEP) approach, where 2-4 ligands can be scored per day on a 4-GPU +server,42,43 although there is an attempt to do this over larger scale for the initial screening steps.44 +In this work we propose the tiered tensor transform (3T) algorithm, a general framework to +generate diverse physical or biological conformation structures of a multi-scale complex system for +optimization purposes. We demonstrate the usage of this algorithm on the problem of protein-ligand +pocket complex conformation generation by simultaneously generating multiple conformations of the +entire protein-ligand complex. 3T requires one example of the structure to optimize as its initial +starting point (Figure 1a), which is then segmented into local groups in a hierarchical manner into +several smaller micro-groups and larger macro-groups as appropriate. In the context of physics or life +sciences, connectivity-based segmentation is the most appropriate choice. We segment protein-ligand +complexes based on rotatable bonds, protein residues, and protein secondary structures +(Supplementary Figure 1). We then apply multiple structure tensor transformations on these micro- +groups and macro-groups in a hierarchical manner, enabling both macro-scale and micro-scale +optimizations which more effectively escape local energy minima and sample multiple and diverse +protein-ligand pocket complex conformations. These hierarchical tensor transformation parameters +are updated based on the cost function gradient (force field energy, see Figure 1b). While our + +approach is neither a machine learning (ML) approach nor an MD approach, it extensively uses tools +commonly found in deep learning (PyTorch45) in its structure transformation part and MD (atomistic +force field) in its structure evaluation.46–49 This modularity makes it possible to apply 3T-like algorithm +for other types of complex multi-scale structure generation and optimization purposes if a +differentiable domain-specific structure evaluation cost function is available (especially helpful in +physics-informed deep learning work).50 3T method enables full flexibility for all of the protein +backbones and sidechain groups during pocket structure generation (Figure 1c), unlike semi-flexible +protein-ligand docking methods available in state-of-the-art tools such as smina, rDock, GOLD, FLAP, +GRID, and ICM which only allow for select protein sidechain dihedral rotation.32,51–55 These semi- +flexible tools’ complexity varies, ranging from only allowing rotations of –OH and –NH3 groups of the +sidechain54 to going through most available sidechain rotations exhaustively.32 The full protein +flexibility of 3T method, while theoretically should be much more computationally expensive than the +semi-flexible methods, is possible because 3T quickly eliminates energetically unfavourable protein- +ligand conformations through a coarse-grain-MD-like hierarchical structure transformations. We +demonstrate that 3T can successfully generate structures which are closer to the actual experimental +protein-ligand co-crystal compared to the initial structures obtained from cross-docking pose on rigid +protein structures in three representative protein targets, i.e. cyclin-dependent kinase 2 (CDK2), heat +stock protein 90 (HSP90) and coagulation factor X (FXa). This 3T structure generation can be +performed with more than 80 × lower computation cost vs comparable MD simulation. More +importantly, we further demonstrate that ten of these 3T protein-ligand complex conformations are +superior when used to identify active and decoy ligands of a protein target in DUD-E and DEKOIS 2.0 +datasets when compared to state-of-the-art ensemble docking procedure performed on hundreds of +experimentally obtained rigid protein host conformations,56,57 in part because we also have access to +the 3T pocket energy landscape near the binding pose local energy minimum structures. + + + + +Figure 1 | Flowchart of 3T structure generation and analyses. a) Initial protein-ligand pocket volume definition. b) +3T energetic kick and optimization in the tiered tensor transform parameter hyperspace. c) Protein-ligand pocket structures +generated by 3T (only one ligand pose shown for clarity), compared to the initial docked structure and the corresponding +experimental protein-ligand co-crystal structure. + + +Grey +Magenta +Atomistic force field +Green +potential energy +→ 3T energetic kidk +landscape +parametersResults & Discussion +Tiered Tensor Transform Algorithm +In principle, 3T is a generative algorithm which works by transforming an existing structure +through multiple scales of tensor transformations, restrained by physics-based cost function to keep +the generated multi-scale transformed structures physically realistic, and functionally relevant or +optimized. There are three major components in 3T: hierarchical structure segmentation, hierarchical +tensor transformation assignments, and differentiable structure evaluation cost function. +Hierarchical structure segmentation is necessary because we would like to enable local +transformation operations to be performed on our initial structure. It is known that protein-ligand +complexes have a large number of high-dimensional local energy traps, making it more practical to +perform ligand docking on rigid protein structures34 or at most semi-flexible protein structures.53,54 +Performing structure optimization directly on the atomic coordinates of the protein and ligand atoms +will only produce relatively small atomic movements. Structure segmentation eliminates some of this +problem by grouping locally connected atoms into separate groups which move in a coordinated +manner. This grouping is like what is done in coarse-grained molecular dynamics,58 but 3T maintains +its full atomistic level detail. The grouping is also done hierarchically in order to enable different levels +of coordinated movements, including the protein backbone. For our protein-ligand pocket complex +generation, we segment the atoms onto 2 segmentation levels: micro-groups and macro-groups. +Atoms in a pocket (within a cutoff radius ������������������������������������������������������������������������������������ = 20Å from the protein-ligand complex centre +������������������������������������������������������������������������������������) are considered movable, which are then segmented into separate micro-groups: (1) for +protein atoms, the micro-groups are segmented based on their amino acid residue and on +backbone/sidechain distinction; (2) for ligand atoms, the micro-groups are segmented based on their +rotatable bonds. These micro-groups are then grouped further into separate macro-groups: (1) +protein micro-groups in each flexible loop secondary structure (Methods) receive separate macro- + +group assignment while the remaining micro-groups (helixes and sheets) are not assigned into any +macro-group; (2) all ligand micro-groups are assigned into one ligand macro-group. This completes +our two-level 3T hierarchical segmentation (Supplementary Figure 1). In principle, more hierarchical +segmentation levels can be used depending on the physical nature of the system under consideration. +Afterward, we assign separate hierarchical local structural tensor transformations on each +micro and macro-group (Figure 2, and a more visual version in Supplementary Figure 2). Atoms in the +same micro-group i are transformed using rotation or translation tensor transformation parameters +������������������������,������������ and ������������������������,������������ respectively. ������������������������,������������ and ������������������������,������������ each has 3 scalar parameters corresponding to rotation angle +around and translation along the x, y, and z axis originated on the micro-group centre. These tensor +transformations give movable atoms additional coordinated rotation and translation degrees of +freedom in addition to the individual atom translations during 3T optimization. We also apply a special +axis rotation transformation ������������������������,������������ (represented by 1 scalar parameter) on micro-groups which only has +one rotatable bond (anchor), allowing these micro-groups to freely rotate around the anchor. Larger- +scale 3T coordinated movement is also enabled by employing coordinated rotation and translation on +each macro-group j, using the transformation parameters ������������������������,������������ and ������������������������,������������ respectively (containing 3 +scalar parameters each like their micro-group counterparts). + + + + +Figure 2 | Schematic of the 3T scheme. Protein-ligand complex atoms are separated into fixed (grey circles) and movable +atoms (red/blue circles). The movable atoms undergo several layers of multi-scale hierarchical tensor transformations +(individual atom translation, sidechain micro-group rotation around rotatable bond axis, micro-group Cartesian axis rotation +and translation, and macro-group Cartesian axis rotation and translation). Micro-group axis rotation (dashed boxes) around +the rotatable bond is only available for some micro-groups such as protein sidechains and ligand edge fragments. Instead of +directly optimizing the final atomic coordinates using atomistic force field, we optimize the 3T tensor parameters and the +initial movable atom coordinates during the PyTorch cost function gradient backpropagation. Computationally this will look +like a typical deep learning training procedure. See Supplementary Figure 2 for a better visual of the hierarchical structure +transformations. + + + +Protein-Ligand Complex Energy +PyTorch Atomistic Force Field Mode +Final Atom +Coordinates +Ligand +Pocket +Fixed +Translation +Translation +Centre Rotation +Centre Rotation +CR +Ligand +Secondary Structure +ss +Macro-group +Translation +Translation +T +Centre Rotation +CR +Centre Rotation +CR +CR +Axis Rotation +AR +Axis Rotation +AR +AR +Micro-group +Fragment +Frag +Amino Acid +AA +AA +Translation +T +Translation +T +T +Initial Atom +c +Coordinates +N +Ligand +ReceptorAfter these hierarchical segmentation and transformation assignments, the 3T optimization +procedure itself is relatively straightforward using a differentiable structure evaluation cost function. +We start with the initial fixed and movable atom coordinates ������������⃗������������ and ������������⃗������������,������������������������������������������������, pass ������������⃗������������,������������������������������������������������ through the +multiple stages of tensor transformations governed by the transformation parameter ������������‘s above, and +calculate the final movable atom coordinates ������������⃗������������,������������������������������������������������������������. We implement both tensor transformation and +atomistic force field model in PyTorch (Methods) to calculate the total system energy based on ������������⃗������������ and +������������⃗������������,������������������������������������������������������������. Using this force field energy as our primary cost function, backward propagation updates the +3T model’s adjustable parameters ������������⃗������������,������������������������������������������������ and ������������‘s. Simply put, we have converted the structure +generation cost function calculation from the original (Equation 1) to a 3T version (Equation 2): +������������������������,������������������������������������������������������������������������������������������������ = �������������������������������������������������⃗������������ , ������������⃗������������,������������������������������������������������� + + + + +(1) +������������������������,3������������ = ������������������������������������ �3�������������������������⃗������������ , ������������⃗������������,������������������������������������������������, ������������������������,������������, ������������������������,������������, ������������������������,������������, ������������������������,������������, ������������������������,�������������� + +(2) +where ������������������������ is the cost function of the protein-ligand complex, ������������������������������������ is the atomistic force field energy +function, and 3������������ is the hierarchical tensor transformation function illustrated in Figure 2. +During the optimization, we first start by optimizing the ligand within rigid protein pocket for +������������������������������������������������������������ = 200 steps. We then apply an energetic kick in the system to start our protein-ligand complex +pocket conformation generation by initializing small random values on the micro-group ������������‘s. This +energetic kick distorts the structure to a high-energy state, which then gets minimized by 3T over +������������������������������������������������������������ = 2000 backward propagation steps. Different random number seed will generate different 3T +energetic kick and final structures (Methods). Computation-wise, this minimization process looks like +a standard deep learning training (with no classification label or regression target in its cost function). +While the number of optimizable parameters increases when 3T structural transformation is applied, +it is significantly easier to escape local energy minimum and faster to reach lower-energy equilibrium +because we have additional coordinated micro- and macro- degrees of freedom. + + +Evaluation of Generated Protein-Ligand Pocket Structures +For the conformation structure generation quality assessment, we compare these 3T structures +with available experimental co-crystal structures for specific protein-ligand complexes. In this work, +we generate 3T protein-ligand complex conformations for three different proteins: CDK2, HSP90, and +FXa. CDK2 pocket is a flexible deep hydrophobic cavity, HSP90 pocket is surrounded with two long +alpha helixes which are known to take several major conformation changes upon different ligand +binding,59 while FXa active site is a flexible shallow hydrophobic groove. For each of these proteins, +we extract one protein structure example from the Protein Data Bank (PDB): 1fin, 1uyg, and 1ezq +respectively (see Figure 3a). +We first perform individual cross-docking structural analysis on 3T-generated CDK2 +conformations as an example. The ligand from 4ez3 PDB co-crystal structure is cross-docked onto our +1fin protein using smina32 to obtain the initial structure. We calculate the root mean squared +displacement (RMSD,39 see Methods) of this cross-docked ligand when compared to the original 1fin +co-crystal ligand, producing ������������������������������������������������������������������������������������������������ = 3.89 Å. The larger this value is, the farther the predicted +cross-docked structure is from the real co-crystal. We then transform this initial structure through 3T +(see Methods), producing new protein-ligand complex conformation. It can be seen from Figure 3b +that the 3T ligand conformation matches the original 1fin co-crystal ligand conformation better than +the initial structure, with smaller ������������������������������������������������3������������ = 1.86 Å and protein binding sites which are more correctly +attached to the ligand compared to the initial smina structure with the rigid 1fin protein. We +correspondingly have RMSD improvement Δ������������������������������������������������ = ������������������������������������������������������������������������������������������������ − ������������������������������������������������3������������ = 2.03 Å . This +improvement is induced by protein backbone conformation change (red circle), which becomes closer +to the actual 4ezq co-crystal protein and pushes on the ligand slightly, and can be seen in more detail +in Supplementary Figure 3a. +We further analyse our 3T CDK2 protein conformations compared to what will be found across +known co-crystal structures as well as MD simulations. 373 co-crystal CDK2 protein structures are + +extracted from the PDB website. 500 protein structures are extracted every 1 ns interval from an MD +simulation of the 1fin protein-ligand structure in water (see Methods). Finally, 90 protein structures +are extracted from 3T-generated structures originating from 1fin smina re-docked initial structures +(new conformations can be generated by simply changing the random number seed of 3T energetic +kick). Principal component analysis (PCA) of the protein backbone is commonly used to assess protein +conformation diversity,60 and is shown in Figure 3c (see Methods), showing that the co-crystal +structures span a much more diverse range of protein backbone conformations due to the diverse set +of co-crystal ligand chemistry geometries found in nature. The principal components (PC) of ligand- +specific conformations from the MD occupies a fraction of the co-crystal PC sub-space, and crucially +the ligand-specific PC from 3T-generated conformations occupy mostly the MD conformations’ PC +sub-space outside of the irrelevant general co-crystal PC sub-space. This shows that 3T protein +conformations are ligand-induced and more specific compared to general protein-ligand co-crystal +structures. 3T final structures rely on energy minimization scheme, so it does not occupy the entire PC +space of the 1fin MD conformations which is done at the room-temperature. Similar individual ligand +pose and protein microstate examples for HSP90 and FXa pocket conformations are available in +Supplementary Figure 3b-e. + + + + +Figure 3 | Protein-ligand complex conformation generation using 3T. a) Example of ligand-dependent pocket conformations +for three protein structures generated in this work, with CDK2 being the most flexible and HSP90 being the most rigid among +the three. Ligand geometries are hidden for clarity. b) Overlaid visual comparison of cross-docking on CDK2 protein structure +(ligand structure from 4ez3 PDB cross-docked on protein structure from 1fin PDB) using 3T and standard rigid protein docking, +compared to the ground truth 4ez3 co-crystal structure. The red circle indicates the protein backbone structure which is +correctly transformed by 3T and has become significantly more similar to the co-crystal, which is then responsible for pushing +the cross-docked 3T ligand closer to experimental co-crystal ligand pose compared to the initial cross-docked structure. +Individual structures (grey: initial, magenta: 3T, green: co-crystal) are available for clarity purposes, showing that 3T structure +has a smaller RMSD and a better match (ligand pose and protein binding sites) with the experimental co-crystal. c) CDK2 +protein backbone micro-state comparison using PCA for 3T-generated structures (re-docking of 1fin PDB) compared to 1fin +pocket structures generated using 500 ns protein-ligand complex MD and to all known experimental CDK2 co-crystal +structures. The MD only occupies a fraction of the entire CDK2 PC sub-space because there are several major ligand- +dependent protein conformations available for CDK2. 3T correctly occupies only the sub-space corresponding to that of 1fin +MD and does not occupy the remaining subspace which are not physically accessible by 1fin protein-ligand complex. The +experimental 1fin co-crystal structure is shown as the black star. If a semi-flexible cross-docking ensemble is performed using +one initial co-crystal structure, the PC will only show up as a single dot here because the protein backbone cannot move, +unlike the 3T and MD methods. + + + +Co-crystal +MD +3T +1fin +CDK2 +CDK2 +HSP90 +FX +Inita +-cnysta +CDK2 +Intal +3T +CDK2 +CDK2 +4ez3 co-crystalAfter individual ligand structure validation analysis above, we now perform large-scale 3T cross- +docking structure generations to assess the structure improvement statistics over large number of +ligands. For each ligand from the known co-crystal structures we perform a molecular cross-docking +onto their respective protein target structures using smina32 to obtain one initial structure for 3T +generation (Methods). For each initial protein-ligand complex, we generate 10 conformations using +3T. We assess these 10 generated conformations using the scoring function of smina and choose 3 +structures with the lowest docking score for experimental comparison. Similar to the previous section, +we quantitatively show the improvement enabled by our 3T scheme by calculating the ������������������������������������������������������������������������������������������������, +������������������������������������������������3������������, and Δ������������������������������������������������ of these protein-ligand structures. The workflow for this Δ������������������������������������������������ calculation is +shown in Figure 4a. The more ligands with positive Δ������������������������������������������������, the more effective 3T is in generating +protein-ligand complex conformations closer to that of the experimental co-crystal. The probability +distribution function of Δ������������������������������������������������ for the best out of the 3 ligand poses we have previously chosen for +the CDK2 dataset is shown in Figure 4b and Table 1. As can be seen in Table 1, on average 3T is able +to generate new conformations with Δ������������������������������������������������ = 0.54 ± 0.67 Å for the CDK2 dataset, and 83% of the +initial cross-docked structures are improved (positive Δ������������������������������������������������). We further ensure 3T structure +generation quality consistency by repeating the procedure for both the second and third-lowest cross- +docking score initial structures produced by smina (CDK2 dataset), showing that the procedure +consistently generates more realistic ligand docking pose for ≥80% of the ligands compared to its initial +structure (Table 1). + + + + +Figure 4 | 3T-generated cross-docking conformation workflow and analyses. a) Workflow of the Δ������������������������������������������������ calculation +process, starting with initial smina cross-docking, followed by 3T structure transformation and RMSD analysis, generating +������������������������������������������������������������������������������������������������, ������������������������������������������������3������������, and Δ������������������������������������������������ = ������������������������������������������������������������������������������������������������ − ������������������������������������������������3������������. b) Distribution of ligand Δ������������������������������������������������ for generated CDK2 protein +pockets, with Δ������������������������������������������������ > 0 indicating ligand pose improvement over docking software ligand pose. c) Scatterplot of +������������������������������������������������������������������������������������������������ split based on the sign of Δ������������������������������������������������, with diagonal line indicating the physical limit of Δ������������������������������������������������. d) Bar plot showing +the average of Δ������������������������������������������������ for all ligand poses with Δ������������������������������������������������ > 0, indicating that 3T increasingly generates better poses if the +initial pose is less optimal (large ������������������������������������������������������������������������������������������������). e) Bar plot showing the average of Δ������������������������������������������������ for ligand poses with Δ������������������������������������������������ ≤ 0, +indicating that more negative Δ������������������������������������������������ becomes more likely when the initial pose is already very close to the experimental +co-crystal structure (small ������������������������������������������������������������������������������������������������). Bar plot binning is done every 1Å interval of ������������������������������������������������������������������������������������������������. + + + +3T ligand cross-docking onto reference co-crystaf protein +1 Initial +10Pocket +Scoring +3 Min-Score +3T +Pose +Conformations +Function +Conformations +3T ligand RMsD evafuation vs initiaf cross-docking pose +Experimental +RMSD3T +RMSDinit +4RMSD +Co-Crystal +Initial Pose 1 +Initial Pose 1 +Initial Pose 2 +Initial Pose 2 +Initial Pose 3 +Initial Pose3 +CDK2 +CDK2 +CDK2 +CDK2Protein +Initial +Rank +Improved +Ligands +Δ������������������������������������������������ +�Å� +CDK2 +1st pose +255/308 +83% +0.54 ± 0.67 +2nd pose +252/308 +82% +0.53 ± 0.60 +3rd pose +247/308 +80% +0.50 ± 0.62 +HSP90 +1st pose +157/223 +70% +0.41 ± 0.66 +2nd pose +150/223 +67% +0.37 ± 0.59 +3rd pose +159/223 +71% +0.33 ± 0.53 +FXa +1st pose +62/106 +58% +0.16 ± 0.60 +2nd pose +65/106 +61% +0.19 ± 0.60 +3rd pose +78/106 +74% +0.34 ± 0.58 +Table 1 | Statistics of ligand cross-docking pose improvement for 3T poses compared to the original smina cross-docking +pose references. This improvement is measured using Δ������������������������������������������������ with respect to the experimental co-crystal structures. The +statistics are shown for the smina initial cross-docking poses with the 1st, 2nd, and 3rd lowest docking scores, and the fraction +shows the number of ligands with positive Δ������������������������������������������������. For all proteins (CDK2, HSP90, and FXa) and initial poses (ranked 1st, 2nd, +and 3rd), ������������������������������������������������3������������ is smaller than ������������������������������������������������������������������������������������������������ on average, and the improvement is statistically significant (p-value < 0.005 for +FXa’s 1st pose and p-value < 0.001 otherwise, one-sided paired samples t-test). See Supplementary Table 1 and Methods. + + + +We analyse the generated poses to determine why 3T fails to produce better protein-ligand +conformations (Δ������������������������������������������������ ≤ 0 Å) for the remaining 17% of the CDK2 ligands. This can be done by first +plotting the ligands’ Δ������������������������������������������������ vs ������������������������������������������������������������������������������������������������ (Figure 4c). We can further bin these data points into a bar +plot (Figure 4e) which shows that ligands with Δ������������������������������������������������ ≤ 0 Å tends to have worse Δ������������������������������������������������ (worse 3T +poses) if they have small ������������������������������������������������������������������������������������������������ (good initial cross-docked poses). Correspondingly in Figure 4d we +show that ligands with Δ������������������������������������������������ > 0 Å tends to have better Δ������������������������������������������������ (better 3T poses) if they have +larger ������������������������������������������������������������������������������������������������ (bad initial cross-docked poses). This indicates that 3T becomes less capable of +producing more realistic structure than the initial structure if the initial structure itself is already very +similar to the co-crystal (see also Supplementary Figure 3 and Supplementary Table 1 for the +complete breakdown and statistical analysis across protein-ligand complexes and initial poses, +including for HSP90 and FXa proteins). In these cases, the 3T energetic kick during the optimization +processes place the ligands farther from their global minimums, which are then harder to reach back +during the 3T cost function minimization processes. To improve the structure generation further, we +need to use multiple energetic kick strength levels and include more conformations in the candidate +evaluation process (we only include 3 out of the 10 generated conformations for each ligand in this +study). +We subsequently extend our 3T structural generation and quality analysis to protein-ligand +complex pocket conformations of HSP90 and FXa proteins (Supplementary Figure 4). We use the same +CDK2 3T hyperparameters ( ������������������������������������������������������������������������������������ & energetic kick strength) to generate HSP90 and FXa +conformations and study the impact of the protein property difference on the generated protein- +ligand pocket structures. HSP90 is of strong interest to us, as its protein pocket is full of alpha helixes +and beta sheets, quite different compared to the pocket of CDK2 protein which is full of flexible loop +secondary structures. For both HSP90 and FXa proteins, we improve the smina-docked initial +structures for 70% and 58% of the ligands, respectively. While it is clear that larger Δ������������������������������������������������ will be +obtained when the host protein is more flexible (CDK2 has the most positive Δ������������������������������������������������ overall), it is not +immediately obvious if the HSP90’s smaller Δ������������������������������������������������ = 0.41 ± 0.66 Å (Table 1) is the result of its more + +rigid secondary structures or not. Hence, we attempt to stiffen the HSP90 protein further by freezing +the HSP90 protein pocket atoms and only allow docked ligands to undergo 3T energetic kick and +optimization process. In this scenario the Δ������������������������������������������������ for the 1st initial smina docking poses immediately +fell further to 0.20 ± 0.41 Å, showing that protein flexibility during the protein-ligand complex pocket +generation process is essential for more accurate docking pose generation (Supplementary Table 1). +Unfortunately there is an insufficient number of experimental co-crystal structures available for +FXa, where only 106 co-crystals are available (vs 308 and 223 for CDK2 and HSP90, see Table 1). While +we obtain consistent statistics for CDK2 and HSP90 ligands, the statistics is less consistent for the FXa +ligands. The initial docking poses produced by smina are particularly good for FXa ligands’ 1st docking +poses (〈������������������������������������������������������������������������������������������������〉 = 5.00 Å for the entire dataset), which explains the 3T’s lower success rate of 58% +and smaller Δ������������������������������������������������ when attempting to generate more realistic FXa protein-ligand pocket structures +(Supplementary Table 1). We note that 3T’s ability to improve cross-docked ligand pose ������������������������������������������������ is not +affected by how similar the ligands are to the protein host’s co-crystallized (Supplementary Figure 5). + + +Applicability in Active Ligand Classification +We further evaluate the practical utility of our 3T complex conformations for active ligand +classification, compared to complex pocket structures obtained using conventional methods. It has +recently been shown that docking potential drug candidate ligand molecules onto a single rigid protein +pocket is insufficient for the purpose of active ligand classification.35,38 In fact, the ligands need to be +docked onto hundreds of distinct rigid conformations of the target protein pocket.35,38 Simple docking +score evaluation is insufficient and an ML model needs to be built on top of the ensemble docking +scores to obtain a decent active ligand classifier.38 These varieties of protein conformation structures +are difficult or expensive to obtain, and hence ligand (A) is often docked onto non-matching rigid +protein structure (B) taken from a different experimental protein-ligand (B-C) complex, or onto rigid +protein structure (D) generated from lengthy MD of the protein pocket in a solvent. On the other hand, +the 3T structure generation enables us to generate ligand-dependent protein-ligand complex pocket +conformations explicitly tailored to each protein-ligand pair. +We demonstrate this versatility by performing ML-assisted “ensemble docking” similar to that +performed by Ricci-Lopez et al.38 In the prior work, ligand docking was performed onto different +number of rigid protein structures depending on the dataset (CDK2: 402, HSP90: 64, FXa: 136). In this +work, we simply generate ten 3T conformation structures for each ligand docked onto one rigid +protein of CDK2, HSP90, and FXa each. We adopt the identical procedure of 30×4-fold cross validation +(30×4cv) and gradient boosting trees (GBT) classifier algorithm which was used in prior work to ensure +that we only compare the conformation feature quality and not the classification method being +used.38 We also note that 3T generates not only the protein-ligand complex pocket conformations, +but also the potential energy landscape surrounding the local energy minimum during its structure +optimization procedure which can be used as additional features. We hypothesize that it is not simply +the shape of the protein-ligand pocket structure (e.g. docking score) which determines how likely it is +for a ligand to bind onto a target protein pocket, but also how accessible such protein-ligand pocket + +energy minimums are (energy barrier landscape surrounding the local energy minimum), as can be +seen in Supplementary Figure 6. This feature extraction procedure and the subsequent 30×4cv +classification process are shown in Figure 5a, where for each of the ten 3T conformations we generate +for each protein-ligand complex, we extract not only the docking scores but also the protein-ligand +binding formation energy Δ������������ = ������������������������������������������������������������������������������������������������,3������������ − ������������������������������������������������������������������������������������,3������������ − ������������������������������������������������������������������������������������������������,������������������������������������������������ throughout the 3T +optimization process (Methods). Due to the large protein size, CDK2 protein-ligand pocket +conformations are re-generated with ������������������������������������������������������������������������������������ = 25Å for this classification work while ������������������������������������������������������������������������������������ = 20Å +is kept for both HSP90 and FXa (Supplementary Table 2). + + + + +Figure 5 | Active ligand classification using 3T-generated conformations. a) 3T conformation feature extraction process +(pocket cross-docking scores and formation energies Δ������������) and subsequent 30× 4-fold cross validation (with GBT classifier). +b-d) The ������������������������������������������������������������ classification metric is shown for different number of CDK2, HSP90 and FXa pocket conformations +respectively. Similarly, the AUC-ROC metric is shown in e-g). The three proteins differ in structure flexibility, with CDK2 and +FXa being dominated by flexible loops and HSP90 being dominated by alpha helixes and beta sheets. We see that the features +from ten 3T ligand-dependent pocket conformations generated from one experimental protein conformation are equivalent +or better than features from significantly larger number of rigid experimental X-ray diffraction protein conformations (rigid +XRD) or simulated MD conformations (rigid MD). + + + +Raw +Dock +Score +37 +00 +Relaxed +Ligand +口口 +Score +30x +37 +4-fofd cross +A +Fomation +Confornation +vaiaton +Energies +A +Scores +C +CDK2 +CDK2 +ISP90Protein +Active +Ligands +Metric +3T +Classifier +Rigid MD +Classifier +Rigid XRD +Classifier +������������������������������������������������������������,3������������ vs +������������������������������������������������������������,������������������������−������������������������������������ + +CDK2 +442/3764 +(������������������������ = 0.117) +������������������������������������������������������������ +0.771 +± 0.030 +0.608 +± 0.033 +0.624 +± 0.039 +(10 + 2) +vs +402 +AUC-ROC +0.935 +± 0.014 +0.892 +± 0.017 +0.904 +± 0.015 +HSP90 +298/2452 +(������������������������ = 0.122) +������������������������������������������������������������ +0.851 +± 0.035 +0.640 +± 0.042 +0.505 +± 0.046 +(10 + 2) +vs +64 +AUC-ROC +0.949 +± 0.018 +0.903 +± 0.019 +0.836 +± 0.024 +FXa +298/7191 +(������������������������ = 0.040) +������������������������������������������������������������ +0.584 +± 0.043 +0.554 +± 0.046 +0.452 +± 0.044 +(10 + 2) +vs +136 +AUC-ROC +0.913 +± 0.018 +0.902 +± 0.021 +0.855 +± 0.021 +Table 2 | 3T active ligand classification metrics using the GBT classifier across 3 different protein hosts. Generally, the more +������������������������������������������������������������ used for a standard ensemble cross-docking classifier, the better the classifier will be. We show that 3T conformation +classifiers (10 + 2 = 1 initial + 1 with relaxed ligand + 10 energetic kick conformations) consistently outperform rigid protein +conformation classifiers across the 3 different protein hosts even though the standard conformation classifiers use +significantly more rigid experimental protein conformation structures. + + + + +To enable fair comparison with existing work and direct comparison between different dataset +sample distributions, we use the metric area under the curve – receiver operating characteristics +(AUC-ROC) and normalized enrichment factor ������������������������������������������������ = ������������������������ ������������������������������������(������������������������, ������������) +⁄ + where ������������ is the total number +of ligands in the dataset, ������������ is the top fraction of the ranked ligands to be selected (set to ������������ = ������������������������ = +������������ ������������ +⁄ +), ������������ is the total number of true active ligands, and ������������������������ is the total number of the chosen ������������������������ ligands +which are true active ligands.38,61 As we would like to investigate how useful our protein-ligand +complex pocket conformations are compared to experimentally obtained protein conformations, we +re-calculate ������������������������������������������������������������ of the prior work38 for the three proteins for different number of rigid +experimental X-ray diffraction protein conformation hosts (rigid XRD).38 In addition, we also perform +long MD of the proteins in water, extract the structures as shared rigid protein hosts for all of the +ligand dockings, and calculate the corresponding classification metric for this MD-based reference +(rigid MD) similar to another prior work.35 We note that these MD structures are not ligand-dependent +because performing individual MD for each explicit protein-ligand pair (holo-MD) in this work will be +computationally prohibitive. The AUC-ROC result supporting our hypothesis is shown in Figure 5b, +where we show that an ‘ensemble-docking’ CDK2 active ligand classifier built using ten 3T protein- +ligand complex conformations significantly outperforms an ensemble-docking CDK2 active ligand +classifiers built using either 402 rigid protein conformation hosts (both rigid MD and rigid XRD). We +similarly outperform 64 HSP90 and 136 FXa rigid protein conformation classifiers using our respective +ten 3T conformations (Figure 5c-d, Table 2). These classifiers’ ������������������������������������������������������������ metrics are consistent with AUC- +ROC metrics (Figure 5e-g) showing 3T conformations significantly outperforming their rigid +conformation counterparts, further demonstrating the classification utility of our 3T conformations. +The performance improvement is especially big for HSP90, which is the least flexible protein pocket +structure among the three. We also show that constraining or eliminating protein structures’ flexibility +during the 3T conformation generation will significantly degrade 3T classifier performance +(Supplementary Figure 6, Supplementary Table 2). + +While it may seem non-intuitive that a classifier built with the number of conformation +structures ������������������������������������������������������������,3������������ = 10 which originates from ������������������������������������������������������������ = 1 protein host conformation structure achieves +similar or better classification results than a classifier built using a large number of experimental +protein host conformation structures (������������������������������������������������������������ = 64–402), we note that the 3T structures are ligand- +dependent and their features contain more information (Methods). We further note that while 3T +conformations are unique and random for each ligand (making classifier feature usage more difficult +to justify), we mitigate this problem by ensuring that the same 3T random seed is used across different +ligands. This means the ligands share almost identical initial protein structural distortion during the 3T +energetic kick process, before the structures get relaxed into their final protein-ligand complex +conformation geometries. Finally, we note that 3T classifiers with ������������������������������������������������������������,3������������ = 4 is in fact enough to +outperform both rigid MD and rigid XRD-based classifiers (Supplementary Figure 7). +In addition to being significantly more accurate and taking significantly less experimental +resources than conventional approaches, we show that 3T also takes significantly less computation +resources than holo-MD or exhaustive smina semi-flexible docking approach32 (Table 3 for CDK2, +Supplementary Table 3 for HSP90 and FXA), although this prototype version of 3T is still slower than +the lightweight semi-flexible docking approach such as rDock which only allows limited –OH and –NH3 +group rotation on the sidechains.54 If the protein-ligand complexes are generated using holo-MD, 3T +structure generation is computationally cheaper than holo-MD by more than 80× (aggressive MD +assumption, see Methods). This holo-MD approach for each ligand is computationally intractable, and +one way to reduce this cost is by performing a protein pocket MD and sharing the compute cost across +all ligands of interest prior to rigid-protein docking (rigid MD). Under this rigid MD docking scenario, +it is difficult to achieve better active ligand classification performance compared to the rigid protein +docking using experimental structures especially when there are multiple major protein +conformations such as CDK2 (Figure 3c). + + + +Protein Rigidity +Structure Generation Method +XRD Experiment +MD Cost / Ligand +Docking Cost / Ligand +Rigid +Rigid XRD +406× +0 +12.9 CPU-hr +Rigid MD* +1× +0.058 GPU-hr +12.9 CPU-hr +Semi-flexible +Smina (flexible sidechain) +406× +0 +710.1 CPU-hr +rDock (flexible OH, NH3) +406× +0 +6.3 CPU-hr +Fully-flexible +Holo-MD* +1× +217.2 GPU-hr +0 +3T +1× +0 +2.5 GPU-hr / 22.7 CPU-hr +* 69k atoms for the CDK2 protein in water  GROMACS GPU speed = 110.5 ns/day +Table 3 | Experimental and computational resource estimation for various docking-based active ligand classification tasks +on the CDK2 dataset, based on the protein-ligand complex structure generation method being used. The six methods are +categorized based on the rigidity of generated protein structure (rigid: no protein conformation change during ligand cross- +docking, semi-flexible: some protein sidechain rotation is allowed during cross-docking, fully-flexible: all protein backbone +and sidechain atoms can freely move during cross-docking). The three methods in bold are the ones for which we do +classification performance comparison in Figure 5. For rigid and semi-flexible methods, each ligand is cross-docked onto all +the available protein host conformations, obtained from either XRD experiment or MD. The XRD experiment column refers +to the number of X-ray diffraction co-crystal structures which were used in previous work (also in Figure 5).38 For rigid MD, +holo-MD, and 3T, only one such co-crystal structure is needed as the initial structure. The holo-MD-based method requires +additional MD simulations to generate the pocket structures (one holo-MD for each protein-ligand pair, which is +computationally unfeasible). Rigid MD-based method can bypass this computation cost requirement by extracting just the +protein structures generated from an MD simulation and sharing it across all the ligands to reduce the MD cost, followed by +standard rigid-protein cross-dockings, in exchange for losing the ligand-dependent-protein aspect of the holo-MD method. +For MD-based methods, we took the aggressive assumption that 1µs MD is enough to generate sufficiently diverse protein +structures (see Methods). For 3T, all structure generation cost (single rigid protein docking plus 10-conformation generation) +is categorized as ‘docking cost’. The docking computation cost estimates are averaged from three randomly chosen ligands, +except for 3T CPU-hr estimates which are averaged from 16 randomly chosen ligands. + + + +Conclusions +In summary, we demonstrate a novel algorithm tiered tensor transform (3T) to generate +realistic complex multi-scale structures such as protein-ligand complex conformation. The structure +generation works by using a combination of one example initial structure, a differentiable structure +evaluation cost function, a hierarchical multi-scale tensor transformation sequence, and a random +energetic kick for initial structural distortion. Using the 3T algorithm, we can generate unique protein- +ligand complex conformations for a given protein target and a ligand molecule drug candidate. We +demonstrate that these generated pocket structures match experimental co-crystal structures better +for 58–83% of ligand molecules across three different target protein hosts when compared to those +generated by docking software which attaches ligands onto rigid protein target hosts. More +importantly, we demonstrate that these 3T conformations are useful for active ligand classification +purposes. Features from ten 3T conformations significantly surpass features from hundreds of rigid +protein conformations and can be generated with more than 80× lower computation cost vs +comparable MD simulations on a GPU. Due to 3T’s modularity, adaptation onto other fields in physical +sciences such as optical nanostructure or microfluidic structure generations/optimizations should be +straightforward if a relatively low-cost structure evaluation cost function is available. + + +Code Availability +The 3T structure generation code, as well as the resulting generated structures and features +necessary for constructing Figure 4 and Figure 5 will be made publicly available in Tencent Quantum +Laboratory Github upon publication. + + +References +1. +Ravuri, S. et al. Skilful precipitation nowcasting using deep generative models of radar. +Nature 597, 672–677 (2021). +2. +Li, J., Zhang, M., Martins, J. R. R. A. & Shu, C. Efficient aerodynamic shape optimization with +deep-learning-based geometric filtering. AIAA J. 58, 4243–4259 (2020). +3. +Chen, W. & Ramamurthy, A. 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Practical model selection for prospective virtual screening. J. Chem. Inf. Model. +59, 282–293 (2019). + + + + + +Methods +Ligand Structure Collection +The 3D ligand structures for the “COCRY” dataset come from Protein Data Bank1. For these +downloaded pdb files, all the water and solvent molecules were removed. Then, all these co-crystal +structures are aligned to a reference protein structure (1fin in CDK2, 1uyg in HSP90 and 1ezq in FXA). +The natural ligands were then extracted to get the “COCRY” dataset. The small molecules other than +the “COCRY” dataset come from several sources (DUD-E, DEKOIS 2.0 and CSAR) as described in the +work of Ricci-Lopez et al.1–4 For the DUD-E and DEKOIS 2.0 datasets, the 3D structures have already +been generated, which could be used for docking directly. For CSAR dataset, the 3D structures were +generated using OpenBabel from the SMILES.5 +Input Structure Preparation +A single protein (pdb) conformation structure is obtained from the Protein Data Bank (1fin for +CDK2, 1uyg for HSP90 and 1ezq for FXA).1 The ligand molecules are then docked onto the rigid protein +using +smina6 +with +parameters +of +“—scoring=vinardo +–factor=100 +–num_modes=5 +– +exhaustiveness=16”. The docked ligand is then extracted onto a standalone mol2 file. Secondary +structure, residue name and rotatable bond information are extracted from the pdb and mol2 files +using PyMol (v2.5), OpenBabel (v3.1.1) and RDKit (v2020.09.1.0).5,7,8 The protein pdb is assigned +CHARMM force field9 using the GROMACS software10 and converted into protein gro file. Similarly, +CHARMM-style force field is assigned onto the ligand mol2 file using SwissParam webserver,11 which +is then converted into the GROMACS format using charmm2gromacs-pvm functionality.10 They are +then combined onto one protein-ligand complex gro file (with complete CHARMM force field) and +further converted into the LAMMPS input data format12 using a custom version of the InterMol +software13 with some bug fixes. This LAMMPS input data format can be directly loaded onto our 3T +PyTorch model for eventual atomistic force field energy (structure evaluation cost function) +calculation. Secondary structure from the pdb files are assigned with PyMol. Here, we simply use three + +types of markers, i.e. helix, sheet and loop. Rotatable bond information is extracted from mol2 files +using OpenBabel and RDKit. ������������������������������������������������������������������������������������ is calculated using the centre of mass of a batch of aligned +cocrystal ligands. +3T PyTorch Model Development and Structure Generation +The 3T structure generation algorithm is implemented using the autograd functionality of +PyTorch,14 and computationally will look identical to a standard PyTorch deep learning model, except +that there is no machine learning or training data involved in the process. The 3T model is split onto +two parts, with the first being the hierarchical tensor transformation module where structural +transformation happens and the second being the structure – force field energy calculation module. +LAMMPS force field styles which are generated by InterMol are re-implemented in the 3T PyTorch +model to enable native ‘training-like’ PyTorch structure generation. Adam optimizer15 and multi-step +learning rate scheduler are used. In the beginning, 200 optimizer steps are used to relax the ligand +structure in the pocket. Then the entire movable protein-ligand pocket (within ������������������������������������������������������������������������������������) experiences +3T energetic kick, followed by 2000 3T optimizer steps. We use uniform random distribution +[−1.5, +1.5] Å for the micro-group ������������������������ translation kicks and [−0.15, +0.15] radian for the ������������������������ rotation +kicks. Some of the protein micro-groups such as phenylalanine, histidine, and tryptophan sidechains +can be very rigid, which might introduce a very deep local structure energy minimum. Because of that, +when we detect that there is enough space for these structures (no atomic clashes within 1 Å), we +apply additional 180-degree rotation on ������������������������,������������ with 50% probability during the 3T energetic kick step to +enable more diverse protein-ligand complex pocket conformation generation. The generated +conformations are outputted as xyz or cif files, and the cost function (energy landscape) throughout +optimization was recorded. This process was repeated 10 times using 10 different (but consistent +across ligands) random number seeds during the energetic kick, to generate ten 3T conformations. +The PyToch components of 3T are executed on single NVIDIA T4 GPUs in the Tencent Cloud platform. + + +PCA for Protein Structures +Three groups of structures are processed for the analyses: (1) co-crystal protein structures, (2) +protein conformations extracted from long MD simulations, and (3) 3T-generated protein +conformations. First, all structures are aligned to the reference structures (e.g. 1fin for CDK2). The co- +crystal protein conformations are taken from all available PDB’s for the given protein, the MD +conformations are 500 structures sampled every 1ns from a holo-MD simulation of the protein-ligand +structures, while the 3T conformations are 90 structures generated from the 1fin smina re-docked +initial structures. Then, the (x,y,z) coordinates of the protein backbone alpha carbon atoms in the +pockets are extracted as features. Next, the scikit-learn PCA models (n_component=2) are fitted using +the co-crystal data and then used to transform the MD and 3T data. Finally, the principle components +PC1 and PC2 are plotted to show the protein conformation distributions. +������������������������������������������������������������ Calculation +The ten 3T conformations are scored using smina scoring function. Three pocket structures with +the lowest docking scores are chosen as our best candidates, which are then aligned to the +corresponding experimental co-crystals using PyMol based on the pocket atoms of the proteins, and +the ligand RMSD is calculated with spyrmsd packages.16 The best RMSD of the three aligned pocket +structures is then compared to the initial smina pose’ RMSD (similarly aligned to the experimental co- +crystal). We also test how significant is the hypothesis that 〈������������������������������������������������������������������������������������������������ − ������������������������������������������������3������������〉 = 〈Δ������������������������������������������������〉 > 0 +on average, calculating the p-value using the scipy’s stats package. +Active Ligand Classification +The recorded 3T formation energies (2200 steps for each conformation) are down sampled by +100 and scaled down by 1000 (to better match the unit of the docking scores), producing 22 energy +features per conformation. The docking scores associated with initial structure, ligand-relaxed +structure, and the 10 conformations are also included as features (12 features), resulting in a total of + +232 features per ligand. These features are directly used as GBT-based 30×4cv active ligand classifier +input, using the same Jupyter notebook available from previous work for AUC-ROC and ������������������������������������������������������������ +calculations.1 There is no change on the classification algorithm setup to ensure we have fair +conformation feature comparison instead of classification algorithm comparison. +Molecular Dynamics Setup and Semi-Flexible Docking Computation Resource Estimation +Two types of machines are used for this comparison. For GPU machine, we use one NVIDIA T4 +card. For CPU machine, we use 16 cores on a 48-core AMD EPYC 7K62 processor. Semi-flexible Smina +docking (rigid protein backbone, rotatable sidechains) is done using parameters “--scoring=vinardo -- +factor=100 --num_modes=3 -exhaustiveness=16 --flexdist=4 --flexdist_ligand=ref_ligand.sdf”. +For +rDock, the parameter “RECEPTOR_FLEX=4” is used in the PRMFILE and “-n=64” is used for docking. For +the MD computation, the GROMACS protein-ligand structure above is solvated in water and the +system is charge-neutralized and minimized before subsequent NVT and NPT equilibrations. +GROMACS production MD speed (NPT, 2.0 fs time step at temperature of 300 K) is then measured. It +is estimated that 1 µs to 1 ms MD time is needed to obtain enough protein structural diversity, and +we have taken the aggressive assumption that 1 µs MD is enough. The required GPU computation cost +is then calculated accordingly.17 + + + +Method References +1. +Ricci-Lopez, J., Aguila, S. A., Gilson, M. K. & Brizuela, C. A. Improving structure-based virtual +screening with ensemble docking and machine learning. J. Chem. Inf. Model. 61, 5362–5376 +(2021). +2. +Mysinger, M. M., Carchia, M., Irwin, J. J. & Shoichet, B. K. Directory of useful decoys, +enhanced (DUD-E): Better ligands and decoys for better benchmarking. Journal of Medicinal +Chemistry vol. 55 6582–6594 (2012). +3. +Bauer, M. R., Ibrahim, T. M., Vogel, S. M. & Boeckler, F. M. Evaluation and optimization of +virtual screening workflows with DEKOIS 2.0 - A public library of challenging docking +benchmark sets. J. Chem. Inf. Model. 53, 1447–1462 (2013). +4. +Dunbar, J. B. et al. CSAR data set release 2012: Ligands, affinities, complexes, and docking +decoys. J. Chem. Inf. Model. 53, 1842–1852 (2013). +5. +O’Boyle, N. M. et al. Open Babel: An open chemical toolbox - 1758-2946-3-33.pdf. J. +Cheminform. 3, 33 (2011). +6. +Koes, D. R., Baumgartner, M. P. & Camacho, C. J. Lessons learned in empirical scoring with +smina from the CSAR 2011 benchmarking exercise. J. Chem. Inf. Model. 53, 1893–1904 +(2013). +7. +The PyMOL Molecular Graphics System, Schrödinger, LLC. https://pymol.org/2/ (2021). +8. +RDKit: Open-Source cheminformatics. http://www.rdkit.org/ (2020). +9. +Vanommeslaeghe, K. et al. CHARMM General Force Field (CGenFF): A force field for drug-like +molecules compatible with the CHARMM all-atom additive biological force fields. J. Comput. +Chem. 31, 671–690 (2010). +10. +Berendsen, H. J. C., van der Spoel, D. & van Drunen, R. GROMACS: A message-passing parallel +molecular dynamics implementation. Comput. Phys. Commun. 91, 43–56 (1995). +11. +Zoete, V., Cuendet, M. A., Grosdidier, A. & Michielin, O. SwissParam: A fast force field +generation tool for small organic molecules. J. Comput. Chem. 32, 2359–2368 (2012). +12. +Thompson, A. P. et al. LAMMPS - a flexible simulation tool for particle-based materials +modeling at the atomic, meso, and continuum scales. Comput. Phys. Commun. 271, 108171 +(2022). +13. +Shirts, M. R. et al. Lessons learned from comparing molecular dynamics engines on the +SAMPL5 dataset. J. Comput. Aided. Mol. Des. 31, 147–161 (2017). +14. +Paszke, A. et al. PyTorch: An imperative style, high-performance deep learning library. in +Advances in Neural Information Processing Systems vol. 33 (2019). +15. +Kingma, D. P. & Ba, J. L. Adam: A method for stochastic optimization. in Proceedings of the +3rd International Conference on Learning Representations 1–15 (2015). +16. +Meli, R. & Biggin, P. C. Spyrmsd: Symmetry-corrected RMSD calculations in Python. J. +Cheminform. 12, 49 (2020). +17. +Evangelista Falcon, W., Ellingson, S. R., Smith, J. C. & Baudry, J. Ensemble docking in drug +discovery: How many protein configurations from molecular dynamics simulations are +needed to reproduce known ligand binding? J. Phys. Chem. B 123, 5189–5195 (2019). + +Acknowledgements +The authors thank M. Shao from Tencent Quantum Lab for technical support on the Tencent +Cloud platform. This work is fully conducted within Tencent Quantum Laboratory using the Tencent +Cloud platform. + +Author Contributions +J.P.M. is responsible for 3T algorithm development, structure generation and feature extraction. +Z.Y. and J.P.M. are responsible for initial protein-ligand complex structure preparation and input data +pre-processing. Z.Y. is responsible for rigid protein docking, RMSD calculation and GBT classification. +J.P.M and Z.Y. are responsible for MD structure generation and microstate analysis. Z.Y. and J.Q. are +responsible for large-scale rigid protein docking on apo-MD structures. J.P.M. and Z.Y. performs the +data and computation cost analysis. C.-Y.H. and S.Z. provide feedback and guide the research. All +authors contribute into the manuscript preparation. + +Competing interests +The authors declare no competing interests. + +Supplementary Information +The online version contains supplementary material available at … + + + +Materials & Correspondence +Correspondence regarding this manuscript and material requests should be addressed to +Jonathan Mailoa at jpmailoa@alum.mit.edu or Shengyu Zhang at shengyzhang@tencent.com. + +Supplementary Information – +Protein-Ligand Complex Generator & Drug Screening via +Tiered Tensor Transform +Jonathan P. Mailoa,1†* Zhaofeng Ye,1† Jiezhong Qiu,1 Chang-Yu Hsieh,1 and Shengyu +Zhang2* + +1) Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, China +2) Tencent Quantum Laboratory, Tencent, Hong Kong SAR, China + +† These authors contributed equally to this work +* corresponding author: jpmailoa@alum.mit.edu, shengyzhang@tencent.com + + + + +Supplementary Figure 1 | 3T protein-ligand pocket hierarchical structure segmentation. a) protein atom micro-groups +based on amino acid backbone and sidechain, b) ligand atom micro-groups based on ligand rotatable bonds, c) protein atom +macro-groups based on residue secondary structure, and d) ligand atom macro-group including all ligand atoms. + + + +HgN+ +COO +Supplementary Figure 2 | Visual description of 3T hierarchical structure transformations. a) Individual atom translation, b) +individual micro-group sidechain rotations, c) individual micro-group centre rotations, d) individual micro-group translation, +e) individual macro-group centre rotation, and f) individual macro-group translation. The parameters for these +transformations are optimized by PyTorch force field-based cost function optimizers. For macro-group centre rotation, it is +performed by rotating each micro-group within macro-group to its micro-group centre in a coordinated manner. + + + +R +H3N+ +COO +H3N+ +COO +H.N+ +COO +H3N+ +C00 +H3N+ +COO +H3N+ +COO +Supplementary Figure 3 | Physical validation of 3T structures. a) CDK2 ligand pose and protein surface mesh comparison +for ligand from 4ez3 PDB cross-docked onto protein from 1fin, showing that the 3T transformation allows the protein pocket +structure to contract and push the ligand to the right, similar to the actual 4ez3 co-crystal structure. The original 4ez3-1fin +cross-docked structure has a rigid protein pocket which is more open, resulting in the wrong ligand pose. b-c) Individual +example of overlaid structure comparison between cross-docked protein-ligand structure generated by 3T (magenta) vs +initial cross-docked structure generated by smina (grey), as well as experimental co-crystal structure reference (green) for +HSP90 (ligand of 2wi6 PDB cross-docked on protein of 1uyg PDB) and FXa proteins (ligand of 3qke PDB cross-docked on +protein of 1ezq PDB). d-e) Protein backbone structure PCA comparison for all available experimental co-crystal PDB’s (green), +structures generated using a single holo-MD simulation (grey), and re-docked structures generated using 3T (magenta) for +HSP90 and FXa proteins. The holo-MD and 3T re-docking are done using just HSP90 1uyg PDB and FXa 1ezq PDB, respectively. +The PCA of the corresponding experimental co-crystal structures are highlighted in dark grey. For HSP90, there are several +distinct protein-ligand pocket conformations available in nature (green). Holo-MD and 3T only occupy the PC subspace that +belongs to the actual 1uyg PDB experimental co-crystal subspace and nowhere else. For FXa, it seems that there is only a +single conformation space which is being shared by the co-crystals, holo-MD, and 3T. We note that we have much more +limited availability of experimental FXa protein PDB co-crystals compared to CDK2 and HSP90 proteins. + + +CDK2 +Initial +3T +Co-crystal +HSP90 +Grey +Grey +Magenta +Magenta +Green +Green +MD +Co-crystal +Co-crystal +MD +3T +3T +luyg +lezq +0 +-6 +ET5 +-10 +HSP90 +FXa +1uyg on 1uyg +1ezg on 1ezq +Supplementary Figure 4 | 3T-generated cross-docking conformation workflow and analyses. a–d) Distribution of ligand +Δ������������������������������������������������ for generated CDK2 pockets (red, ������������������������������������������������������������������������������������ = 25Å), HSP90 with both flexible and rigid pockets (purple and green), +and FXa pockets (grey). Δ������������������������������������������������ > 0 indicating ligand pose improvement over docking software ligand pose. e–h) Scatterplot +of ������������������������������������������������������������������������������������������������ split based on the sign of Δ������������������������������������������������, with diagonal line indicating the physical limit of Δ������������������������������������������������. i–l) Bar plot +showing the average of Δ������������������������������������������������ for all ligand poses where Δ������������������������������������������������ > 0, indicating that 3T increasingly generates better +poses if the initial pose is less optimal. m–p) Bar plot showing the average of Δ������������������������������������������������ for all ligand poses where Δ������������������������������������������������ ≤ +0, indicating that more negative Δ������������������������������������������������ becomes more likely when the initial pose is already very close to the experimental +co-crystal structure. When the HSP90 protein pocket is made perfectly rigid (3T transformations are only applied to the +ligands, and not applied to the protein structure), Δ������������������������������������������������ is reduced and the trends observed in other cases with flexible 3T +pocket generation are no longer apparent. Bar plot binning is done every 1Å. + + + +Initial Pose 1 +Initial Pose1 +Initial Pose 1 +Initial Pose 1 +Initial Pose 2 +Initial Pose 2 +Initial Pose 2 +Initial Pose 2 + Initial Pose 3 +Initial Pose 3 +Initial Pose 3 +Initial Pose 3 +CDK2 +HSP90 +HSP90 +FXa +(25A) +(rigid) +F +Initial Pose 1 +Initial Pose 1 +Initial Pose 1 +Initial Pose 1 +Initial Pose 2 +Initial Pose 2 +Initial Pose 2 +Initial Pose 2 +Initial Pose 3 +Initial Pose 3 +Initial Pose 3 +Initial Pose 3 +CDK2 +HSP90 +HSP90 +FXa +(25A) +(rigid) +CDK2 +HSP90 +HSP90 +FXa +(25A) +(rigid) +CDK2 +HSP90 +HSP90 +FXa +(25A) +m +n +(rigid)Protein +Initial +Rank +Improved +Ligands +Δ������������������������������������������������ +�Å� +〈������������������������������������������������������������������������������������������������,������������������������������������〉 +�Å� +〈������������������������������������������������������������������������������������������������,Δ������������������������������������������������≤0〉 +�Å� +〈������������������������������������������������������������������������������������������������,Δ������������������������������������������������>0〉 +�Å� +p-value +CDK2 +(20 Å) +1st pose +255/308 +83% +0.54 ± 0.67 +6.42 ± 3.55 +4.86 ± 3.38 +6.75 ± 3.49 +3e-35 +2nd pose +252/308 +82% +0.53 ± 0.60 +6.75 ± 3.43 +5.64 ± 3.02 +7.00 ± 3.47 +9e-41 +3rd pose +247/308 +80% +0.50 ± 0.62 +6.74 ± 3.49 +5.60 ± 3.59 +7.01 ± 3.41 +9e-35 +CDK2 +(25 Å) +1st pose +246/308 +80% +0.42 ± 0.60 +6.42 ± 3.55 +5.98 ± 4.51 +6.53 ± 3.25 +2e-28 +2nd pose +235/308 +76% +0.43 ± 0.62 +6.75 ± 3.44 +5.76 ± 2.68 +7.05 ± 3.58 +9e-28 +3rd pose +234/308 +76% +0.42 ± 0.58 +6.76 ± 3.57 +6.32 ± 4.66 +6.90 ± 3.14 +3e-30 +HSP90 +1st pose +157/223 +70% +0.41 ± 0.66 +6.29 ± 3.05 +4.69 ± 2.86 +6.95 ± 2.88 +2e-17 +2nd pose +150/223 +67% +0.37 ± 0.59 +6.62 ± 2.57 +5.39 ± 2.47 +7.15 ± 2.42 +1e-17 +3rd pose +159/223 +71% +0.33 ± 0.53 +6.60 ± 2.64 +4.83 ± 2.47 +7.17 ± 2.43 +3e-17 +HSP90 +(rigid) +1st pose +157/223 +70% +0.20 ± 0.41 +6.29 ± 3.05 +6.02 ± 2.70 +6.40 ± 3.18 +6e-12 +2nd pose +150/223 +67% +0.20 ± 0.46 +6.62 ± 2.57 +6.37 ± 2.44 +6.72 ± 2.61 +3e-10 +3rd pose +133/223 +60% +0.18 ± 0.46 +6.60 ± 2.64 +6.21 ± 2.85 +6.82 ± 2.48 +4e-8 +FXa +1st pose +62/106 +58% +0.16 ± 0.60 +5.00 ± 3.54 +3.29 ± 3.21 +6.21 ± 3.26 +5e-3 +2nd pose +65/106 +61% +0.19 ± 0.60 +5.72 ± 3.44 +4.08 ± 3.32 +6.76 ± 3.09 +6e-4 +3rd pose +78/106 +74% +0.34 ± 0.58 +6.67 ± 3.02 +6.61 ± 2.85 +6.69 ± 3.07 +2e-8 +Supplementary Table 1 | Statistics of ligand cross-docking pose improvement for 3T poses compared to the original smina +cross-docking pose references. In addition to the information available in Table 1 of the main text, we have also included +the mean and standard distribution of initial structure error ������������������������������������������������������������������������������������������������ for 3T processes which produce positive and negative +Δ������������������������������������������������, as well as those for the cases where HSP90 protein pocket atoms are frozen during 3T process. It can be seen that +for the case where 3T performs badly (lower fraction of initial cross-docked ligand poses are improved), it is because the +initial structures themselves are already quite close to experimental co-crystal structures (small ������������������������������������������������������������������������������������������������). For all proteins +(CDK2, HSP90, and FXa) and initial poses (ranked 1st, 2nd, and 3rd), ������������������������������������������������3������������ is smaller than ������������������������������������������������������������������������������������������������ on average, and the +improvement is statistically significant (p-value < 0.005 for FXa’s 1st pose and p-value < 0.001 otherwise, one-sided paired +samples t-test). See main text Methods. + + + + +Supplementary Figure 5 | Impact of ligand similarity to reference co-crystal PDB ligand on 3T ability to improve docking +pose. The scatterplot of Δ������������������������������������������������ vs the Morgan fingerprint similarity between the cross-docked ligands and the ligand which +is part of the reference co-crystal for a) CDK2 (������������������������������������������������������������������������������������ = 25Å, reference host protein: 1fin PDB), b) HSP90 (reference host +protein: 1fin PDB), and c) FXa (reference host protein: 1ezq PDB). The low Pearson correlation coefficient r indicates that +3T’s ability to improve the cross-docked ligand RMSD is independent from such ligand’s similarity to the reference co-crystal +ligand of the protein conformation host. + + + +Initial Pose 1 +Initial Pose 1 +Initial Pose 1 +Initial Pose 2 +Initial Pose 2 +Initial Pose 2 +Initial Pose 3 +Initial Pose 3 +Initial Pose 3 += -0.060 +=-0.028 +0.091 +1. +CDK2 +HSP90 +Fxa +(20A) +Supplementary Figure 6 | Impact of reducing or eliminating 3T protein flexibility on active ligand classification +performance. a–b) The ������������������������������������������������������������ active ligand classification metric for CDK2 pockets (������������������������������������������������������������������������������������ = 20Å) and HSP90 pockets +(pockets intentionally made to be perfectly rigid and non-transformable by 3T), and c-d) AUC-ROC active ligand classification +metric for the same proteins. 30× 4-fold cross validation (with GBT classifier) is used for the classification statistics. It can be +seen that 3T on small CDK2 pocket is insufficient to build good classifiers and a larger radius of pocket flexibility needs to be +allowed. Similarly, it can also be seen that performing 3T just on the ligands (with perfectly rigid HSP90 protein pocket +structure) will significantly degrade the active ligand classifier performance. + + + +7 +CDK2 +06dsH +(20A) +(ngid) +[+ +: +K +HSP90 +OT +(rigid)Protein +Active +Ligands + +3T +Classifier +Rigid MD +Classifier +Rigid XRD +Classifier +������������������������������������������������������������,3������������ +vs +������������������������������������������������������������,������������������������−������������������������������������ +CDK2 +(25 Å) +442/3764 +(������������������������ = 0.117) +������������������������������������������������������������ +0.771 +± 0.030 +0.608 +± 0.033 +0.624 +± 0.039 +(10 + 2) +vs +402 +AUC-ROC +0.935 +± 0.014 +0.892 +± 0.017 +0.904 +± 0.015 +CDK2 +(20 Å) +442/3764 +(������������������������ = 0.117) +������������������������������������������������������������ +0.469 +± 0.037 +0.608 +± 0.033 +0.624 +± 0.039 +(10 + 2) +vs +402 +AUC-ROC +0.828 +± 0.020 +0.892 +± 0.017 +0.904 +± 0.015 +HSP90 +298/2452 +(������������������������ = 0.122) +������������������������������������������������������������ +0.851 +± 0.035 +0.640 +± 0.042 +0.505 +± 0.046 +(10 + 2) +vs +64 +AUC-ROC +0.949 +± 0.018 +0.903 +± 0.019 +0.836 +± 0.024 +HSP90 +(rigid) +298/2452 +(������������������������ = 0.122) +������������������������������������������������������������ +0.524 +± 0.042 +0.640 +± 0.042 +0.505 +± 0.046 +(10 + 2) +vs +64 +AUC-ROC +0.830 +± 0.024 +0.903 +± 0.019 +0.836 +± 0.024 +FXa +298/7191 +(������������������������ = 0.040) +������������������������������������������������������������ +0.584 +± 0.043 +0.554 +± 0.046 +0.452 +± 0.044 +(10 + 2) +vs +136 +AUC-ROC +0.913 +± 0.018 +0.902 +± 0.021 +0.855 +± 0.021 +Supplementary Table 2 | 3T active ligand classification metrics using the GBT classifier across 3 different protein hosts. In +addition to the information available in Table 2 of the main text, we have also included the classification statistics when +������������������������������������������������������������������������������������ = 20Å pocket is used for CDK2 and when perfectly rigid HSP90 protein pocket is used for 3T protein-ligand pocket +conformation generations. It can be seen that when the flexible protein pocket generation ability of 3T is reduced or removed, +its active ligand classification utility will be diminished. + + + + +Supplementary Figure 7 | The impact of incorporating 3T energy landscape features vs excluding such features during +classification for different number of 3T-generated conformations. a–c) The ������������������������������������������������������������ active ligand classification metric for +CDK2 pockets (������������������������������������������������������������������������������������ = 25Å), HSP90 pockets (������������������������������������������������������������������������������������ = 20Å, non-rigid), and FXa pockets (������������������������������������������������������������������������������������ = 20Å). d–f) AUC-ROC +active ligand classification metric for the same proteins as in a–c). 30× 4-fold cross validation (with GBT classifier) is used for +the classification statistics. It can very clearly be seen that incorporating 3T energy landscape features is very helpful for +identifying active ligands from the decoys. ������������������������������������������������������������ = 0 means that we do not generate any fully-flexible protein-ligand pocket +structure through 3T energetic kick, and only utilize the initial smina cross-docked structure and the structure obtained after +relaxing the ligand while maintaining rigid protein pocket (see Figure 5). + + + +[] +CDK2 +(25A) +d +e +CDK2 +(25A)Protein +Protein +Rigidity +Method +XRD +Experiment +MD Cost / Ligand +Docking Cost / Ligand +CDK2 +Rigid +Rigid XRD +406× +0 +12.9 CPU-hr +Rigid MD* +1× +0.058 GPU-hr +12.9 CPU-hr +Semi- +flexible +Smina (flexible sidechain) +406× +0 +710.1 CPU-hr +rDock (flexible OH, NH3) +406× +0 +6.3 CPU-hr +Fully- +flexible +Holo-MD* +1× +217.2 GPU-hr +0 +3T +1× +0 +2.5 GPU-hr / 22.7 CPU-hr +* 69k atoms for the CDK2 protein --> GROMACS GPU speed = 110.5 ns/day +HSP90 +Rigid +Rigid XRD +64× +0 +4.1 CPU-hr +Rigid MD* +1× +0.051 GPU-hr +4.1 CPU-hr +Semi- +flexible +Smina (flexible sidechain) +64× +0 +178.3 CPU-hr +rDock (flexible OH, NH3) +64× +0 +1.7 CPU-hr +Fully- +flexible +Holo-MD* +1× +124.3 GPU-hr +0 +3T +1× +0 +2.1 GPU-hr / 10.1 CPU-hr +* 37k atoms for the HSP90 protein --> GROMACS GPU speed = 193.1 ns/day +FXa +Rigid +Rigid XRD +136× +0 +6.5 CPU-hr +Rigid MD* +1× +0.028 GPU-hr +6.5 CPU-hr +Semi- +flexible +Smina (flexible sidechain) +136× +0 +170.5 CPU-hr +rDock (flexible OH, NH3) +136× +0 +2.5 CPU-hr +Fully- +flexible +Holo-MD* +1× +201.7 GPU-hr +0 +3T +1× +0 +1.8 GPU-hr / 10.7 CPU-hr +* 60k atoms for the FXa protein --> GROMACS GPU speed = 119.0 ns/day +Supplementary Table 3 | Experimental and computational resource estimation for various docking-based active ligand +classification tasks on the CDK2, HSP90, and FXa dataset based on the protein-ligand complex structure generation +method being used. Active ligand Experimental and computational resource estimation for various docking-based active +ligand classification tasks on the CDK2, HSP90, and FXa dataset, based on the protein-ligand complex structure generation +method being used. The six methods are categorized based on the rigidity of generated protein structure (rigid: no protein +conformation change during ligand cross-docking, semi-flexible: some protein sidechain rotation is allowed during cross- +docking, fully-flexible: all protein backbone and sidechain atoms can freely move during cross-docking). The three methods +in bold are the ones for which we do classification performance comparison in main text Figure 5. The XRD experiment +column refers to the number of X-ray diffraction co-crystal structures which are needed to enable active ligand classification +task based on such methods (the same number of experimental structures being used in main text Figure 5). For rigid MD, +holo-MD, and 3T, only one such experimental co-crystal protein structure is needed as the initial structure. However, the +holo-MD and rigid MD methods will require additional MD simulation cost for the subsequent protein-ligand complex +structure generations. For 3T, all structure generation cost (single rigid protein docking plus 10-conformation generation) is +categorized as ‘docking cost’. The docking computation cost estimates are averaged from three randomly chosen ligands, +except for 3T CPU-hr estimates which are averaged from 16 randomly chosen ligands. + diff --git a/3dAzT4oBgHgl3EQfD_oT/content/tmp_files/load_file.txt b/3dAzT4oBgHgl3EQfD_oT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1e9c6eb6d874176aeef74e83514e7257667a646 --- /dev/null +++ b/3dAzT4oBgHgl3EQfD_oT/content/tmp_files/load_file.txt @@ -0,0 +1,1610 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf,len=1609 +page_content='Protein-Ligand Complex Generator & Drug Screening via Tiered Tensor Transform Jonathan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Mailoa,1†* Zhaofeng Ye,1† Jiezhong Qiu,1 Chang-Yu Hsieh,1 and Shengyu Zhang2* 1) Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, China 2) Tencent Quantum Laboratory, Tencent, Hong Kong SAR, China † These authors contributed equally to this work corresponding author: jpmailoa@alum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='edu, shengyzhang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='com Protein-Ligand Complex Generator & Drug Screening via Tiered Tensor Transform Abstract Accurate determination of a small molecule candidate (ligand) binding pose in its target protein pocket is important for computer-aided drug discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Typical rigid-body docking methods ignore the pocket flexibility of protein, while the more accurate pose generation using molecular dynamics is hindered by slow protein dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We develop a tiered tensor transform (3T) algorithm to rapidly generate diverse protein-ligand complex conformations for both pose and affinity estimation in drug screening, requiring neither machine learning training nor lengthy dynamics computation, while maintaining both coarse-grain-like coordinated protein dynamics and atomistic-level details of the complex pocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The 3T conformation structures we generate are closer to experimental co-crystal structures than those generated by docking software, and more importantly achieve significantly higher accuracy in active ligand classification than traditional ensemble docking using hundreds of experimental protein conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 3T structure transformation is decoupled from the system physics, making future usage in other computational scientific domains possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Introduction Structure generation and optimization is an essential topic in the field of life and physical sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The applications of generative model algorithm in these fields are diverse, ranging from precipitation nowcasting1 and airfoil aerodynamics optimization2–4 down to the optimization of optical nanostructures5–7, electrode microstructures,8 microfluidic devices,9,10 and material design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='11–13 Structure generation capability is even more prevalent and important for drug discovery applications,14–20 where protein and drug candidate small molecule (ligand) structures are of interest in the determination of suitable target-specific drug molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Recent development in the deep learning community has enabled generative model algorithms to more accurately predict single protein structures,21,22 with a notable recent example being the success of AlphaFold 2 in the CASP14 competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='23 Other notable recent examples are similarly focused on either the generation of small molecules with desired properties24,25 or on the high-efficiency sampling for protein structures (RiD, RIP, L-RIP, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='26–30 Targeting a specific pocket of a protein with a ligand poses an additional challenge because the compound is composed of multiple interacting structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Finding the binding pose of a flexible ligand molecule when it is docked onto a flexible protein receptor pocket is a daunting task due to the degree of flexibilities built onto this two-system complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The most common method (with lower computational cost) is to freeze the target protein receptor pocket and use a docking software such as AutoDock Vina,31 Smina,32 and Glide33 to attach the ligand molecule onto the rigid protein pocket34 and generate multiple candidate ligand docking poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Unfortunately it is known that ligands docked onto a rigid protein pocket are not representative of how ligands look like in a real protein-ligand complex and are difficult to use for active ligand classification,35 partly because the protein structure is flexible and can undergo intrinsic or induced conformational changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='36,37 The state-of-the-art solution to this problem is to perform ensemble docking, where multiple structures of the same protein pocket (usually 102 – 103 structures) are either obtained experimentally or generated through long molecular dynamics simulations (MD with millisecond-long simulations, which correspond to ~1012 MD time steps)35,38 and other generative methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='39 the ligands of interest are then docked to all these protein structures to generate multiple conformations of protein-ligand complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' It is worth noting that the protein structure generation is done with no regard to the specific ligand existence in the protein pocket, as it remains difficult to simultaneously modify the protein and ligand structures due to the combined degree of flexibilities in the protein-ligand complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' To the best of the authors’ knowledge, MD simulation of the entire protein-ligand complex structure remains the only reliable way to sample and generate diverse complex conformations of entire protein-ligand pockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='40,41 Unfortunately, it is computationally expensive to do so due to the slow dynamics of the protein, and this kind of process is usually reserved for the last step of a drug screening workflow such as the free energy perturbation (FEP) approach, where 2-4 ligands can be scored per day on a 4-GPU server,42,43 although there is an attempt to do this over larger scale for the initial screening steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='44 In this work we propose the tiered tensor transform (3T) algorithm, a general framework to generate diverse physical or biological conformation structures of a multi-scale complex system for optimization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We demonstrate the usage of this algorithm on the problem of protein-ligand pocket complex conformation generation by simultaneously generating multiple conformations of the entire protein-ligand complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 3T requires one example of the structure to optimize as its initial starting point (Figure 1a), which is then segmented into local groups in a hierarchical manner into several smaller micro-groups and larger macro-groups as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In the context of physics or life sciences, connectivity-based segmentation is the most appropriate choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We segment protein-ligand complexes based on rotatable bonds, protein residues, and protein secondary structures (Supplementary Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We then apply multiple structure tensor transformations on these micro- groups and macro-groups in a hierarchical manner, enabling both macro-scale and micro-scale optimizations which more effectively escape local energy minima and sample multiple and diverse protein-ligand pocket complex conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' These hierarchical tensor transformation parameters are updated based on the cost function gradient (force field energy, see Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' While our approach is neither a machine learning (ML) approach nor an MD approach, it extensively uses tools commonly found in deep learning (PyTorch45) in its structure transformation part and MD (atomistic force field) in its structure evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='46–49 This modularity makes it possible to apply 3T-like algorithm for other types of complex multi-scale structure generation and optimization purposes if a differentiable domain-specific structure evaluation cost function is available (especially helpful in physics-informed deep learning work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='50 3T method enables full flexibility for all of the protein backbones and sidechain groups during pocket structure generation (Figure 1c), unlike semi-flexible protein-ligand docking methods available in state-of-the-art tools such as smina, rDock, GOLD, FLAP, GRID, and ICM which only allow for select protein sidechain dihedral rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='32,51–55 These semi- flexible tools’ complexity varies, ranging from only allowing rotations of –OH and –NH3 groups of the sidechain54 to going through most available sidechain rotations exhaustively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='32 The full protein flexibility of 3T method, while theoretically should be much more computationally expensive than the semi-flexible methods, is possible because 3T quickly eliminates energetically unfavourable protein- ligand conformations through a coarse-grain-MD-like hierarchical structure transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We demonstrate that 3T can successfully generate structures which are closer to the actual experimental protein-ligand co-crystal compared to the initial structures obtained from cross-docking pose on rigid protein structures in three representative protein targets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' cyclin-dependent kinase 2 (CDK2), heat stock protein 90 (HSP90) and coagulation factor X (FXa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This 3T structure generation can be performed with more than 80 × lower computation cost vs comparable MD simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' More importantly, we further demonstrate that ten of these 3T protein-ligand complex conformations are superior when used to identify active and decoy ligands of a protein target in DUD-E and DEKOIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='0 datasets when compared to state-of-the-art ensemble docking procedure performed on hundreds of experimentally obtained rigid protein host conformations,56,57 in part because we also have access to the 3T pocket energy landscape near the binding pose local energy minimum structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Figure 1 | Flowchart of 3T structure generation and analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a) Initial protein-ligand pocket volume definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' b) 3T energetic kick and optimization in the tiered tensor transform parameter hyperspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' c) Protein-ligand pocket structures generated by 3T (only one ligand pose shown for clarity), compared to the initial docked structure and the corresponding experimental protein-ligand co-crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Grey Magenta Atomistic force field Green potential energy → 3T energetic kidk landscape parametersResults & Discussion Tiered Tensor Transform Algorithm In principle, 3T is a generative algorithm which works by transforming an existing structure through multiple scales of tensor transformations, restrained by physics-based cost function to keep the generated multi-scale transformed structures physically realistic, and functionally relevant or optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' There are three major components in 3T: hierarchical structure segmentation, hierarchical tensor transformation assignments, and differentiable structure evaluation cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Hierarchical structure segmentation is necessary because we would like to enable local transformation operations to be performed on our initial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' It is known that protein-ligand complexes have a large number of high-dimensional local energy traps, making it more practical to perform ligand docking on rigid protein structures34 or at most semi-flexible protein structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='53,54 Performing structure optimization directly on the atomic coordinates of the protein and ligand atoms will only produce relatively small atomic movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Structure segmentation eliminates some of this problem by grouping locally connected atoms into separate groups which move in a coordinated manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This grouping is like what is done in coarse-grained molecular dynamics,58 but 3T maintains its full atomistic level detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The grouping is also done hierarchically in order to enable different levels of coordinated movements, including the protein backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For our protein-ligand pocket complex generation, we segment the atoms onto 2 segmentation levels: micro-groups and macro-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Atoms in a pocket (within a cutoff radius ������������������������������������������������������������������������������������ = 20Å from the protein-ligand complex centre ������������������������������������������������������������������������������������) are considered movable, which are then segmented into separate micro-groups: (1) for protein atoms, the micro-groups are segmented based on their amino acid residue and on backbone/sidechain distinction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' (2) for ligand atoms, the micro-groups are segmented based on their rotatable bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' These micro-groups are then grouped further into separate macro-groups: (1) protein micro-groups in each flexible loop secondary structure (Methods) receive separate macro- group assignment while the remaining micro-groups (helixes and sheets) are not assigned into any macro-group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' (2) all ligand micro-groups are assigned into one ligand macro-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This completes our two-level 3T hierarchical segmentation (Supplementary Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In principle, more hierarchical segmentation levels can be used depending on the physical nature of the system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Afterward, we assign separate hierarchical local structural tensor transformations on each micro and macro-group (Figure 2, and a more visual version in Supplementary Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Atoms in the same micro-group i are transformed using rotation or translation tensor transformation parameters ������������������������,������������ and ������������������������,������������ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������,������������ and ������������������������,������������ each has 3 scalar parameters corresponding to rotation angle around and translation along the x, y, and z axis originated on the micro-group centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' These tensor transformations give movable atoms additional coordinated rotation and translation degrees of freedom in addition to the individual atom translations during 3T optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We also apply a special axis rotation transformation ������������������������,������������ (represented by 1 scalar parameter) on micro-groups which only has one rotatable bond (anchor), allowing these micro-groups to freely rotate around the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Larger- scale 3T coordinated movement is also enabled by employing coordinated rotation and translation on each macro-group j, using the transformation parameters ������������������������,������������ and ������������������������,������������ respectively (containing 3 scalar parameters each like their micro-group counterparts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Figure 2 | Schematic of the 3T scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Protein-ligand complex atoms are separated into fixed (grey circles) and movable atoms (red/blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The movable atoms undergo several layers of multi-scale hierarchical tensor transformations (individual atom translation, sidechain micro-group rotation around rotatable bond axis, micro-group Cartesian axis rotation and translation, and macro-group Cartesian axis rotation and translation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Micro-group axis rotation (dashed boxes) around the rotatable bond is only available for some micro-groups such as protein sidechains and ligand edge fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Instead of directly optimizing the final atomic coordinates using atomistic force field, we optimize the 3T tensor parameters and the initial movable atom coordinates during the PyTorch cost function gradient backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Computationally this will look like a typical deep learning training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' See Supplementary Figure 2 for a better visual of the hierarchical structure transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Protein-Ligand Complex Energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='PyTorch Atomistic Force Field Mode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Final Atom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Coordinates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Ligand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Pocket ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Fixed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Centre Rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Centre Rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Ligand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Secondary Structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='ss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Macro-group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Centre Rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Centre Rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Axis Rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='AR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Axis Rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='AR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='AR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Micro-group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Fragment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Frag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Amino Acid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='AA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='AA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Atom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Coordinates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Ligand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='ReceptorAfter these hierarchical segmentation and transformation assignments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' the 3T optimization procedure itself is relatively straightforward using a differentiable structure evaluation cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We start with the initial fixed and movable atom coordinates ������������⃗������������ and ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' pass ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������������������������������������������ through the multiple stages of tensor transformations governed by the transformation parameter ������������‘s above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' and calculate the final movable atom coordinates ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������������������������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We implement both tensor transformation and atomistic force field model in PyTorch (Methods) to calculate the total system energy based on ������������⃗������������ and ������������⃗������������,������������������������������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Using this force field energy as our primary cost function, backward propagation updates the 3T model’s adjustable parameters ������������⃗������������,������������������������������������������������ and ������������‘s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Simply put,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' we have converted the structure generation cost function calculation from the original (Equation 1) to a 3T version (Equation 2): ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������������������������������������������������������������������������������������������ = �������������������������������������������������⃗������������ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������������������������������������������� (1) ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3������������ = ������������������������������������ �3�������������������������⃗������������ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='�������������� (2) where ������������������������ is the cost function of the protein-ligand complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������������������ is the atomistic force field energy function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' and 3������������ is the hierarchical tensor transformation function illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' During the optimization, we first start by optimizing the ligand within rigid protein pocket for ������������������������������������������������������������ = 200 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We then apply an energetic kick in the system to start our protein-ligand complex pocket conformation generation by initializing small random values on the micro-group ������������‘s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This energetic kick distorts the structure to a high-energy state, which then gets minimized by 3T over ������������������������������������������������������������ = 2000 backward propagation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Different random number seed will generate different 3T energetic kick and final structures (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Computation-wise, this minimization process looks like a standard deep learning training (with no classification label or regression target in its cost function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' While the number of optimizable parameters increases when 3T structural transformation is applied, it is significantly easier to escape local energy minimum and faster to reach lower-energy equilibrium because we have additional coordinated micro- and macro- degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Evaluation of Generated Protein-Ligand Pocket Structures For the conformation structure generation quality assessment, we compare these 3T structures with available experimental co-crystal structures for specific protein-ligand complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In this work, we generate 3T protein-ligand complex conformations for three different proteins: CDK2, HSP90, and FXa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' CDK2 pocket is a flexible deep hydrophobic cavity, HSP90 pocket is surrounded with two long alpha helixes which are known to take several major conformation changes upon different ligand binding,59 while FXa active site is a flexible shallow hydrophobic groove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For each of these proteins, we extract one protein structure example from the Protein Data Bank (PDB): 1fin, 1uyg, and 1ezq respectively (see Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We first perform individual cross-docking structural analysis on 3T-generated CDK2 conformations as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The ligand from 4ez3 PDB co-crystal structure is cross-docked onto our 1fin protein using smina32 to obtain the initial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We calculate the root mean squared displacement (RMSD,39 see Methods) of this cross-docked ligand when compared to the original 1fin co-crystal ligand, producing ������������������������������������������������������������������������������������������������ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='89 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The larger this value is, the farther the predicted cross-docked structure is from the real co-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We then transform this initial structure through 3T (see Methods), producing new protein-ligand complex conformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' It can be seen from Figure 3b that the 3T ligand conformation matches the original 1fin co-crystal ligand conformation better than the initial structure, with smaller ������������������������������������������������3������������ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='86 Å and protein binding sites which are more correctly attached to the ligand compared to the initial smina structure with the rigid 1fin protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We correspondingly have RMSD improvement Δ������������������������������������������������ = ������������������������������������������������������������������������������������������������ − ������������������������������������������������3������������ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='03 Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This improvement is induced by protein backbone conformation change (red circle), which becomes closer to the actual 4ezq co-crystal protein and pushes on the ligand slightly, and can be seen in more detail in Supplementary Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We further analyse our 3T CDK2 protein conformations compared to what will be found across known co-crystal structures as well as MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 373 co-crystal CDK2 protein structures are extracted from the PDB website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 500 protein structures are extracted every 1 ns interval from an MD simulation of the 1fin protein-ligand structure in water (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Finally, 90 protein structures are extracted from 3T-generated structures originating from 1fin smina re-docked initial structures (new conformations can be generated by simply changing the random number seed of 3T energetic kick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Principal component analysis (PCA) of the protein backbone is commonly used to assess protein conformation diversity,60 and is shown in Figure 3c (see Methods), showing that the co-crystal structures span a much more diverse range of protein backbone conformations due to the diverse set of co-crystal ligand chemistry geometries found in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The principal components (PC) of ligand- specific conformations from the MD occupies a fraction of the co-crystal PC sub-space, and crucially the ligand-specific PC from 3T-generated conformations occupy mostly the MD conformations’ PC sub-space outside of the irrelevant general co-crystal PC sub-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This shows that 3T protein conformations are ligand-induced and more specific compared to general protein-ligand co-crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 3T final structures rely on energy minimization scheme, so it does not occupy the entire PC space of the 1fin MD conformations which is done at the room-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Similar individual ligand pose and protein microstate examples for HSP90 and FXa pocket conformations are available in Supplementary Figure 3b-e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Figure 3 | Protein-ligand complex conformation generation using 3T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a) Example of ligand-dependent pocket conformations for three protein structures generated in this work, with CDK2 being the most flexible and HSP90 being the most rigid among the three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Ligand geometries are hidden for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' b) Overlaid visual comparison of cross-docking on CDK2 protein structure (ligand structure from 4ez3 PDB cross-docked on protein structure from 1fin PDB) using 3T and standard rigid protein docking, compared to the ground truth 4ez3 co-crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The red circle indicates the protein backbone structure which is correctly transformed by 3T and has become significantly more similar to the co-crystal, which is then responsible for pushing the cross-docked 3T ligand closer to experimental co-crystal ligand pose compared to the initial cross-docked structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Individual structures (grey: initial, magenta: 3T, green: co-crystal) are available for clarity purposes, showing that 3T structure has a smaller RMSD and a better match (ligand pose and protein binding sites) with the experimental co-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' c) CDK2 protein backbone micro-state comparison using PCA for 3T-generated structures (re-docking of 1fin PDB) compared to 1fin pocket structures generated using 500 ns protein-ligand complex MD and to all known experimental CDK2 co-crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The MD only occupies a fraction of the entire CDK2 PC sub-space because there are several major ligand- dependent protein conformations available for CDK2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 3T correctly occupies only the sub-space corresponding to that of 1fin MD and does not occupy the remaining subspace which are not physically accessible by 1fin protein-ligand complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The experimental 1fin co-crystal structure is shown as the black star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' If a semi-flexible cross-docking ensemble is performed using one initial co-crystal structure, the PC will only show up as a single dot here because the protein backbone cannot move, unlike the 3T and MD methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Co-crystal MD 3T 1fin CDK2 CDK2 HSP90 FX Inita cnysta CDK2 Intal 3T CDK2 CDK2 4ez3 co-crystalAfter individual ligand structure validation analysis above, we now perform large-scale 3T cross- docking structure generations to assess the structure improvement statistics over large number of ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For each ligand from the known co-crystal structures we perform a molecular cross-docking onto their respective protein target structures using smina32 to obtain one initial structure for 3T generation (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For each initial protein-ligand complex, we generate 10 conformations using 3T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We assess these 10 generated conformations using the scoring function of smina and choose 3 structures with the lowest docking score for experimental comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Similar to the previous section, we quantitatively show the improvement enabled by our 3T scheme by calculating the ������������������������������������������������������������������������������������������������, ������������������������������������������������3������������, and Δ������������������������������������������������ of these protein-ligand structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The workflow for this Δ������������������������������������������������ calculation is shown in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The more ligands with positive Δ������������������������������������������������, the more effective 3T is in generating protein-ligand complex conformations closer to that of the experimental co-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The probability distribution function of Δ������������������������������������������������ for the best out of the 3 ligand poses we have previously chosen for the CDK2 dataset is shown in Figure 4b and Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' As can be seen in Table 1, on average 3T is able to generate new conformations with Δ������������������������������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='67 Å for the CDK2 dataset, and 83% of the initial cross-docked structures are improved (positive Δ������������������������������������������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We further ensure 3T structure generation quality consistency by repeating the procedure for both the second and third-lowest cross- docking score initial structures produced by smina (CDK2 dataset), showing that the procedure consistently generates more realistic ligand docking pose for ≥80% of the ligands compared to its initial structure (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Figure 4 | 3T-generated cross-docking conformation workflow and analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a) Workflow of the Δ������������������������������������������������ calculation process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' starting with initial smina cross-docking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' followed by 3T structure transformation and RMSD analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' generating ������������������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������������������������������3������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' and Δ������������������������������������������������ = ������������������������������������������������������������������������������������������������ − ������������������������������������������������3������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' b) Distribution of ligand Δ������������������������������������������������ for generated CDK2 protein pockets, with Δ������������������������������������������������ > 0 indicating ligand pose improvement over docking software ligand pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' c) Scatterplot of ������������������������������������������������������������������������������������������������ split based on the sign of Δ������������������������������������������������, with diagonal line indicating the physical limit of Δ������������������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' d) Bar plot showing the average of Δ������������������������������������������������ for all ligand poses with Δ������������������������������������������������ > 0, indicating that 3T increasingly generates better poses if the initial pose is less optimal (large ������������������������������������������������������������������������������������������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' e) Bar plot showing the average of Δ������������������������������������������������ for ligand poses with Δ������������������������������������������������ ≤ 0, indicating that more negative Δ������������������������������������������������ becomes more likely when the initial pose is already very close to the experimental co-crystal structure (small ������������������������������������������������������������������������������������������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Bar plot binning is done every 1Å interval of ������������������������������������������������������������������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3T ligand cross-docking onto reference co-crystaf protein ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 Initial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='10Pocket ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Scoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3 Min-Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Pose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Conformations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Conformations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3T ligand RMsD evafuation vs initiaf cross-docking pose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Experimental ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='RMSD3T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='RMSDinit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='4RMSD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Co-Crystal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CDK2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CDK2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CDK2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CDK2Protein ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Rank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Improved ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Ligands ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Δ������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='�Å� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CDK2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1st pose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='255/308 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='83% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='67 2nd pose 252/308 82% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 3rd pose 247/308 80% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='62 HSP90 1st pose 157/223 70% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='66 2nd pose 150/223 67% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='59 3rd pose 159/223 71% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='53 FXa 1st pose 62/106 58% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 2nd pose 65/106 61% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 3rd pose 78/106 74% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='58 Table 1 | Statistics of ligand cross-docking pose improvement for 3T poses compared to the original smina cross-docking pose references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This improvement is measured using Δ������������������������������������������������ with respect to the experimental co-crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The statistics are shown for the smina initial cross-docking poses with the 1st, 2nd, and 3rd lowest docking scores, and the fraction shows the number of ligands with positive Δ������������������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For all proteins (CDK2, HSP90, and FXa) and initial poses (ranked 1st, 2nd, and 3rd), ������������������������������������������������3������������ is smaller than ������������������������������������������������������������������������������������������������ on average, and the improvement is statistically significant (p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='005 for FXa’s 1st pose and p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='001 otherwise, one-sided paired samples t-test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' See Supplementary Table 1 and Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We analyse the generated poses to determine why 3T fails to produce better protein-ligand conformations (Δ������������������������������������������������ ≤ 0 Å) for the remaining 17% of the CDK2 ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This can be done by first plotting the ligands’ Δ������������������������������������������������ vs ������������������������������������������������������������������������������������������������ (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We can further bin these data points into a bar plot (Figure 4e) which shows that ligands with Δ������������������������������������������������ ≤ 0 Å tends to have worse Δ������������������������������������������������ (worse 3T poses) if they have small ������������������������������������������������������������������������������������������������ (good initial cross-docked poses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Correspondingly in Figure 4d we show that ligands with Δ������������������������������������������������ > 0 Å tends to have better Δ������������������������������������������������ (better 3T poses) if they have larger ������������������������������������������������������������������������������������������������ (bad initial cross-docked poses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This indicates that 3T becomes less capable of producing more realistic structure than the initial structure if the initial structure itself is already very similar to the co-crystal (see also Supplementary Figure 3 and Supplementary Table 1 for the complete breakdown and statistical analysis across protein-ligand complexes and initial poses, including for HSP90 and FXa proteins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In these cases, the 3T energetic kick during the optimization processes place the ligands farther from their global minimums, which are then harder to reach back during the 3T cost function minimization processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' To improve the structure generation further, we need to use multiple energetic kick strength levels and include more conformations in the candidate evaluation process (we only include 3 out of the 10 generated conformations for each ligand in this study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We subsequently extend our 3T structural generation and quality analysis to protein-ligand complex pocket conformations of HSP90 and FXa proteins (Supplementary Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We use the same CDK2 3T hyperparameters ( ������������������������������������������������������������������������������������ & energetic kick strength) to generate HSP90 and FXa conformations and study the impact of the protein property difference on the generated protein- ligand pocket structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' HSP90 is of strong interest to us, as its protein pocket is full of alpha helixes and beta sheets, quite different compared to the pocket of CDK2 protein which is full of flexible loop secondary structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For both HSP90 and FXa proteins, we improve the smina-docked initial structures for 70% and 58% of the ligands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' While it is clear that larger Δ������������������������������������������������ will be obtained when the host protein is more flexible (CDK2 has the most positive Δ������������������������������������������������ overall), it is not immediately obvious if the HSP90’s smaller Δ������������������������������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='66 Å (Table 1) is the result of its more rigid secondary structures or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Hence, we attempt to stiffen the HSP90 protein further by freezing the HSP90 protein pocket atoms and only allow docked ligands to undergo 3T energetic kick and optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In this scenario the Δ������������������������������������������������ for the 1st initial smina docking poses immediately fell further to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='41 Å, showing that protein flexibility during the protein-ligand complex pocket generation process is essential for more accurate docking pose generation (Supplementary Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Unfortunately there is an insufficient number of experimental co-crystal structures available for FXa, where only 106 co-crystals are available (vs 308 and 223 for CDK2 and HSP90, see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' While we obtain consistent statistics for CDK2 and HSP90 ligands, the statistics is less consistent for the FXa ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The initial docking poses produced by smina are particularly good for FXa ligands’ 1st docking poses (〈������������������������������������������������������������������������������������������������〉 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='00 Å for the entire dataset), which explains the 3T’s lower success rate of 58% and smaller Δ������������������������������������������������ when attempting to generate more realistic FXa protein-ligand pocket structures (Supplementary Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We note that 3T’s ability to improve cross-docked ligand pose ������������������������������������������������ is not affected by how similar the ligands are to the protein host’s co-crystallized (Supplementary Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Applicability in Active Ligand Classification We further evaluate the practical utility of our 3T complex conformations for active ligand classification, compared to complex pocket structures obtained using conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' It has recently been shown that docking potential drug candidate ligand molecules onto a single rigid protein pocket is insufficient for the purpose of active ligand classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='35,38 In fact, the ligands need to be docked onto hundreds of distinct rigid conformations of the target protein pocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='35,38 Simple docking score evaluation is insufficient and an ML model needs to be built on top of the ensemble docking scores to obtain a decent active ligand classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='38 These varieties of protein conformation structures are difficult or expensive to obtain, and hence ligand (A) is often docked onto non-matching rigid protein structure (B) taken from a different experimental protein-ligand (B-C) complex, or onto rigid protein structure (D) generated from lengthy MD of the protein pocket in a solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' On the other hand, the 3T structure generation enables us to generate ligand-dependent protein-ligand complex pocket conformations explicitly tailored to each protein-ligand pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We demonstrate this versatility by performing ML-assisted “ensemble docking” similar to that performed by Ricci-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='38 In the prior work, ligand docking was performed onto different number of rigid protein structures depending on the dataset (CDK2: 402, HSP90: 64, FXa: 136).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In this work, we simply generate ten 3T conformation structures for each ligand docked onto one rigid protein of CDK2, HSP90, and FXa each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We adopt the identical procedure of 30×4-fold cross validation (30×4cv) and gradient boosting trees (GBT) classifier algorithm which was used in prior work to ensure that we only compare the conformation feature quality and not the classification method being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='38 We also note that 3T generates not only the protein-ligand complex pocket conformations, but also the potential energy landscape surrounding the local energy minimum during its structure optimization procedure which can be used as additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We hypothesize that it is not simply the shape of the protein-ligand pocket structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' docking score) which determines how likely it is for a ligand to bind onto a target protein pocket, but also how accessible such protein-ligand pocket energy minimums are (energy barrier landscape surrounding the local energy minimum), as can be seen in Supplementary Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This feature extraction procedure and the subsequent 30×4cv classification process are shown in Figure 5a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' where for each of the ten 3T conformations we generate for each protein-ligand complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' we extract not only the docking scores but also the protein-ligand binding formation energy Δ������������ = ������������������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3������������ − ������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3������������ − ������������������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������������������������������������������ throughout the 3T optimization process (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Due to the large protein size, CDK2 protein-ligand pocket conformations are re-generated with ������������������������������������������������������������������������������������ = 25Å for this classification work while ������������������������������������������������������������������������������������ = 20Å is kept for both HSP90 and FXa (Supplementary Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Figure 5 | Active ligand classification using 3T-generated conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a) 3T conformation feature extraction process (pocket cross-docking scores and formation energies Δ������������) and subsequent 30× 4-fold cross validation (with GBT classifier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' b-d) The ������������������������������������������������������������ classification metric is shown for different number of CDK2, HSP90 and FXa pocket conformations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Similarly, the AUC-ROC metric is shown in e-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The three proteins differ in structure flexibility, with CDK2 and FXa being dominated by flexible loops and HSP90 being dominated by alpha helixes and beta sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We see that the features from ten 3T ligand-dependent pocket conformations generated from one experimental protein conformation are equivalent or better than features from significantly larger number of rigid experimental X-ray diffraction protein conformations (rigid XRD) or simulated MD conformations (rigid MD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Raw Dock Score 37 00 Relaxed Ligand 口口 Score 30x 37 4-fofd cross A Fomation Confornation vaiaton Energies A Scores C CDK2 CDK2 ISP90Protein Active Ligands Metric 3T Classifier Rigid MD Classifier Rigid XRD Classifier ������������������������������������������������������������,3������������ vs ������������������������������������������������������������,������������������������−������������������������������������ CDK2 442/3764 (������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='117) ������������������������������������������������������������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='771 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='608 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='624 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='039 (10 + 2) vs 402 AUC-ROC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='935 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='892 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='904 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='015 HSP90 298/2452 (������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='122) ������������������������������������������������������������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='851 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='640 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='505 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='046 (10 + 2) vs 64 AUC-ROC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='949 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='903 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='836 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='024 FXa 298/7191 (������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='040) ������������������������������������������������������������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='584 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='452 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='044 (10 + 2) vs 136 AUC-ROC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='913 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='902 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='855 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='021 Table 2 | 3T active ligand classification metrics using the GBT classifier across 3 different protein hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Generally, the more ������������������������������������������������������������ used for a standard ensemble cross-docking classifier, the better the classifier will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We show that 3T conformation classifiers (10 + 2 = 1 initial + 1 with relaxed ligand + 10 energetic kick conformations) consistently outperform rigid protein conformation classifiers across the 3 different protein hosts even though the standard conformation classifiers use significantly more rigid experimental protein conformation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' To enable fair comparison with existing work and direct comparison between different dataset sample distributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' we use the metric area under the curve – receiver operating characteristics (AUC-ROC) and normalized enrichment factor ������������������������������������������������ = ������������������������ ������������������������������������(������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������) ⁄ where ������������ is the total number of ligands in the dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������ is the top fraction of the ranked ligands to be selected (set to ������������ = ������������������������ = ������������ ������������ ⁄ ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������ is the total number of true active ligands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' and ������������������������ is the total number of the chosen ������������������������ ligands which are true active ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='38,61 As we would like to investigate how useful our protein-ligand complex pocket conformations are compared to experimentally obtained protein conformations, we re-calculate ������������������������������������������������������������ of the prior work38 for the three proteins for different number of rigid experimental X-ray diffraction protein conformation hosts (rigid XRD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='38 In addition, we also perform long MD of the proteins in water, extract the structures as shared rigid protein hosts for all of the ligand dockings, and calculate the corresponding classification metric for this MD-based reference (rigid MD) similar to another prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='35 We note that these MD structures are not ligand-dependent because performing individual MD for each explicit protein-ligand pair (holo-MD) in this work will be computationally prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The AUC-ROC result supporting our hypothesis is shown in Figure 5b, where we show that an ‘ensemble-docking’ CDK2 active ligand classifier built using ten 3T protein- ligand complex conformations significantly outperforms an ensemble-docking CDK2 active ligand classifiers built using either 402 rigid protein conformation hosts (both rigid MD and rigid XRD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We similarly outperform 64 HSP90 and 136 FXa rigid protein conformation classifiers using our respective ten 3T conformations (Figure 5c-d, Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' These classifiers’ ������������������������������������������������������������ metrics are consistent with AUC- ROC metrics (Figure 5e-g) showing 3T conformations significantly outperforming their rigid conformation counterparts, further demonstrating the classification utility of our 3T conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The performance improvement is especially big for HSP90, which is the least flexible protein pocket structure among the three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We also show that constraining or eliminating protein structures’ flexibility during the 3T conformation generation will significantly degrade 3T classifier performance (Supplementary Figure 6, Supplementary Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' While it may seem non-intuitive that a classifier built with the number of conformation structures ������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3������������ = 10 which originates from ������������������������������������������������������������ = 1 protein host conformation structure achieves similar or better classification results than a classifier built using a large number of experimental protein host conformation structures (������������������������������������������������������������ = 64–402),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' we note that the 3T structures are ligand- dependent and their features contain more information (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We further note that while 3T conformations are unique and random for each ligand (making classifier feature usage more difficult to justify), we mitigate this problem by ensuring that the same 3T random seed is used across different ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This means the ligands share almost identical initial protein structural distortion during the 3T energetic kick process, before the structures get relaxed into their final protein-ligand complex conformation geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Finally, we note that 3T classifiers with ������������������������������������������������������������,3������������ = 4 is in fact enough to outperform both rigid MD and rigid XRD-based classifiers (Supplementary Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In addition to being significantly more accurate and taking significantly less experimental resources than conventional approaches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' we show that 3T also takes significantly less computation resources than holo-MD or exhaustive smina semi-flexible docking approach32 (Table 3 for CDK2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Supplementary Table 3 for HSP90 and FXA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' although this prototype version of 3T is still slower than the lightweight semi-flexible docking approach such as rDock which only allows limited –OH and –NH3 group rotation on the sidechains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='54 If the protein-ligand complexes are generated using holo-MD, 3T structure generation is computationally cheaper than holo-MD by more than 80× (aggressive MD assumption, see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This holo-MD approach for each ligand is computationally intractable, and one way to reduce this cost is by performing a protein pocket MD and sharing the compute cost across all ligands of interest prior to rigid-protein docking (rigid MD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Under this rigid MD docking scenario, it is difficult to achieve better active ligand classification performance compared to the rigid protein docking using experimental structures especially when there are multiple major protein conformations such as CDK2 (Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Protein Rigidity Structure Generation Method XRD Experiment MD Cost / Ligand Docking Cost / Ligand Rigid Rigid XRD 406× 0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='9 CPU-hr Rigid MD* 1× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='058 GPU-hr 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='9 CPU-hr Semi-flexible Smina (flexible sidechain) 406× 0 710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 CPU-hr rDock (flexible OH, NH3) 406× 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3 CPU-hr Fully-flexible Holo-MD* 1× 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='2 GPU-hr 0 3T 1× 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5 GPU-hr / 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='7 CPU-hr 69k atoms for the CDK2 protein in water \uf0e0 GROMACS GPU speed = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5 ns/day Table 3 | Experimental and computational resource estimation for various docking-based active ligand classification tasks on the CDK2 dataset, based on the protein-ligand complex structure generation method being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The six methods are categorized based on the rigidity of generated protein structure (rigid: no protein conformation change during ligand cross- docking, semi-flexible: some protein sidechain rotation is allowed during cross-docking, fully-flexible: all protein backbone and sidechain atoms can freely move during cross-docking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The three methods in bold are the ones for which we do classification performance comparison in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For rigid and semi-flexible methods, each ligand is cross-docked onto all the available protein host conformations, obtained from either XRD experiment or MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The XRD experiment column refers to the number of X-ray diffraction co-crystal structures which were used in previous work (also in Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='38 For rigid MD, holo-MD, and 3T, only one such co-crystal structure is needed as the initial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The holo-MD-based method requires additional MD simulations to generate the pocket structures (one holo-MD for each protein-ligand pair, which is computationally unfeasible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Rigid MD-based method can bypass this computation cost requirement by extracting just the protein structures generated from an MD simulation and sharing it across all the ligands to reduce the MD cost, followed by standard rigid-protein cross-dockings, in exchange for losing the ligand-dependent-protein aspect of the holo-MD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For MD-based methods, we took the aggressive assumption that 1µs MD is enough to generate sufficiently diverse protein structures (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For 3T, all structure generation cost (single rigid protein docking plus 10-conformation generation) is categorized as ‘docking cost’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The docking computation cost estimates are averaged from three randomly chosen ligands, except for 3T CPU-hr estimates which are averaged from 16 randomly chosen ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Conclusions In summary, we demonstrate a novel algorithm tiered tensor transform (3T) to generate realistic complex multi-scale structures such as protein-ligand complex conformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The structure generation works by using a combination of one example initial structure, a differentiable structure evaluation cost function, a hierarchical multi-scale tensor transformation sequence, and a random energetic kick for initial structural distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Using the 3T algorithm, we can generate unique protein- ligand complex conformations for a given protein target and a ligand molecule drug candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We demonstrate that these generated pocket structures match experimental co-crystal structures better for 58–83% of ligand molecules across three different target protein hosts when compared to those generated by docking software which attaches ligands onto rigid protein target hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' More importantly, we demonstrate that these 3T conformations are useful for active ligand classification purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Features from ten 3T conformations significantly surpass features from hundreds of rigid protein conformations and can be generated with more than 80× lower computation cost vs comparable MD simulations on a GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Due to 3T’s modularity, adaptation onto other fields in physical sciences such as optical nanostructure or microfluidic structure generations/optimizations should be straightforward if a relatively low-cost structure evaluation cost function is available.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Amaral, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Protein conformational flexibility modulates kinetics and thermodynamics of drug binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 8, 2276 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' David, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' & Jacobs, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Principal component analysis: A method fo determining the essential dynamics of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 1084 (Humana Press, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Practical model selection for prospective virtual screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 59, 282–293 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Methods Ligand Structure Collection The 3D ligand structures for the “COCRY” dataset come from Protein Data Bank1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For these downloaded pdb files, all the water and solvent molecules were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Then, all these co-crystal structures are aligned to a reference protein structure (1fin in CDK2, 1uyg in HSP90 and 1ezq in FXA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The natural ligands were then extracted to get the “COCRY” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The small molecules other than the “COCRY” dataset come from several sources (DUD-E, DEKOIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='0 and CSAR) as described in the work of Ricci-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1–4 For the DUD-E and DEKOIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='0 datasets, the 3D structures have already been generated, which could be used for docking directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For CSAR dataset, the 3D structures were generated using OpenBabel from the SMILES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5 Input Structure Preparation A single protein (pdb) conformation structure is obtained from the Protein Data Bank (1fin for CDK2, 1uyg for HSP90 and 1ezq for FXA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 The ligand molecules are then docked onto the rigid protein using smina6 with parameters of “—scoring=vinardo –factor=100 –num_modes=5 – exhaustiveness=16”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The docked ligand is then extracted onto a standalone mol2 file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Secondary structure, residue name and rotatable bond information are extracted from the pdb and mol2 files using PyMol (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5), OpenBabel (v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1) and RDKit (v2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5,7,8 The protein pdb is assigned CHARMM force field9 using the GROMACS software10 and converted into protein gro file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Similarly, CHARMM-style force field is assigned onto the ligand mol2 file using SwissParam webserver,11 which is then converted into the GROMACS format using charmm2gromacs-pvm functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='10 They are then combined onto one protein-ligand complex gro file (with complete CHARMM force field) and further converted into the LAMMPS input data format12 using a custom version of the InterMol software13 with some bug fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This LAMMPS input data format can be directly loaded onto our 3T PyTorch model for eventual atomistic force field energy (structure evaluation cost function) calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Secondary structure from the pdb files are assigned with PyMol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Here, we simply use three types of markers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' helix, sheet and loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Rotatable bond information is extracted from mol2 files using OpenBabel and RDKit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������������������������������������������������������������������ is calculated using the centre of mass of a batch of aligned cocrystal ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 3T PyTorch Model Development and Structure Generation The 3T structure generation algorithm is implemented using the autograd functionality of PyTorch,14 and computationally will look identical to a standard PyTorch deep learning model, except that there is no machine learning or training data involved in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The 3T model is split onto two parts, with the first being the hierarchical tensor transformation module where structural transformation happens and the second being the structure – force field energy calculation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' LAMMPS force field styles which are generated by InterMol are re-implemented in the 3T PyTorch model to enable native ‘training-like’ PyTorch structure generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Adam optimizer15 and multi-step learning rate scheduler are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In the beginning, 200 optimizer steps are used to relax the ligand structure in the pocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Then the entire movable protein-ligand pocket (within ������������������������������������������������������������������������������������) experiences 3T energetic kick, followed by 2000 3T optimizer steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We use uniform random distribution [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5, +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5] Å for the micro-group ������������������������ translation kicks and [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='15, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='15] radian for the ������������������������ rotation kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Some of the protein micro-groups such as phenylalanine, histidine, and tryptophan sidechains can be very rigid, which might introduce a very deep local structure energy minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Because of that, when we detect that there is enough space for these structures (no atomic clashes within 1 Å), we apply additional 180-degree rotation on ������������������������,������������ with 50% probability during the 3T energetic kick step to enable more diverse protein-ligand complex pocket conformation generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The generated conformations are outputted as xyz or cif files, and the cost function (energy landscape) throughout optimization was recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' This process was repeated 10 times using 10 different (but consistent across ligands) random number seeds during the energetic kick, to generate ten 3T conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The PyToch components of 3T are executed on single NVIDIA T4 GPUs in the Tencent Cloud platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' PCA for Protein Structures Three groups of structures are processed for the analyses: (1) co-crystal protein structures, (2) protein conformations extracted from long MD simulations, and (3) 3T-generated protein conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' First, all structures are aligned to the reference structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 1fin for CDK2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The co- crystal protein conformations are taken from all available PDB’s for the given protein, the MD conformations are 500 structures sampled every 1ns from a holo-MD simulation of the protein-ligand structures, while the 3T conformations are 90 structures generated from the 1fin smina re-docked initial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Then, the (x,y,z) coordinates of the protein backbone alpha carbon atoms in the pockets are extracted as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Next, the scikit-learn PCA models (n_component=2) are fitted using the co-crystal data and then used to transform the MD and 3T data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Finally, the principle components PC1 and PC2 are plotted to show the protein conformation distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������������������������������������������ Calculation The ten 3T conformations are scored using smina scoring function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Three pocket structures with the lowest docking scores are chosen as our best candidates, which are then aligned to the corresponding experimental co-crystals using PyMol based on the pocket atoms of the proteins, and the ligand RMSD is calculated with spyrmsd packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='16 The best RMSD of the three aligned pocket structures is then compared to the initial smina pose’ RMSD (similarly aligned to the experimental co- crystal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We also test how significant is the hypothesis that 〈������������������������������������������������������������������������������������������������ − ������������������������������������������������3������������〉 = 〈Δ������������������������������������������������〉 > 0 on average, calculating the p-value using the scipy’s stats package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Active Ligand Classification The recorded 3T formation energies (2200 steps for each conformation) are down sampled by 100 and scaled down by 1000 (to better match the unit of the docking scores), producing 22 energy features per conformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The docking scores associated with initial structure, ligand-relaxed structure, and the 10 conformations are also included as features (12 features), resulting in a total of 232 features per ligand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' These features are directly used as GBT-based 30×4cv active ligand classifier input, using the same Jupyter notebook available from previous work for AUC-ROC and ������������������������������������������������������������ calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 There is no change on the classification algorithm setup to ensure we have fair conformation feature comparison instead of classification algorithm comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Molecular Dynamics Setup and Semi-Flexible Docking Computation Resource Estimation Two types of machines are used for this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For GPU machine, we use one NVIDIA T4 card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For CPU machine, we use 16 cores on a 48-core AMD EPYC 7K62 processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Semi-flexible Smina docking (rigid protein backbone, rotatable sidechains) is done using parameters “--scoring=vinardo -- factor=100 --num_modes=3 -exhaustiveness=16 --flexdist=4 --flexdist_ligand=ref_ligand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='sdf”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For rDock, the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' is responsible for 3T algorithm development, structure generation and feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' are responsible for initial protein-ligand complex structure preparation and input data pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' is responsible for rigid protein docking, RMSD calculation and GBT classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='M and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' are responsible for MD structure generation and microstate analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' are responsible for large-scale rigid protein docking on apo-MD structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' performs the data and computation cost analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' provide feedback and guide the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' All authors contribute into the manuscript preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Competing interests The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Supplementary Information The online version contains supplementary material available at … Materials & Correspondence Correspondence regarding this manuscript and material requests should be addressed to Jonathan Mailoa at jpmailoa@alum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='edu or Shengyu Zhang at shengyzhang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Supplementary Information – Protein-Ligand Complex Generator & Drug Screening via Tiered Tensor Transform Jonathan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Mailoa,1†* Zhaofeng Ye,1† Jiezhong Qiu,1 Chang-Yu Hsieh,1 and Shengyu Zhang2* 1) Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong, China 2) Tencent Quantum Laboratory, Tencent, Hong Kong SAR, China † These authors contributed equally to this work corresponding author: jpmailoa@alum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='edu, shengyzhang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='com Supplementary Figure 1 | 3T protein-ligand pocket hierarchical structure segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a) protein atom micro-groups based on amino acid backbone and sidechain, b) ligand atom micro-groups based on ligand rotatable bonds, c) protein atom macro-groups based on residue secondary structure, and d) ligand atom macro-group including all ligand atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' HgN+ COO Supplementary Figure 2 | Visual description of 3T hierarchical structure transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a) Individual atom translation, b) individual micro-group sidechain rotations, c) individual micro-group centre rotations, d) individual micro-group translation, e) individual macro-group centre rotation, and f) individual macro-group translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The parameters for these transformations are optimized by PyTorch force field-based cost function optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For macro-group centre rotation, it is performed by rotating each micro-group within macro-group to its micro-group centre in a coordinated manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' R H3N+ COO H3N+ COO H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='N+ COO H3N+ C00 H3N+ COO H3N+ COO Supplementary Figure 3 | Physical validation of 3T structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a) CDK2 ligand pose and protein surface mesh comparison for ligand from 4ez3 PDB cross-docked onto protein from 1fin, showing that the 3T transformation allows the protein pocket structure to contract and push the ligand to the right, similar to the actual 4ez3 co-crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The original 4ez3-1fin cross-docked structure has a rigid protein pocket which is more open, resulting in the wrong ligand pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' b-c) Individual example of overlaid structure comparison between cross-docked protein-ligand structure generated by 3T (magenta) vs initial cross-docked structure generated by smina (grey), as well as experimental co-crystal structure reference (green) for HSP90 (ligand of 2wi6 PDB cross-docked on protein of 1uyg PDB) and FXa proteins (ligand of 3qke PDB cross-docked on protein of 1ezq PDB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' d-e) Protein backbone structure PCA comparison for all available experimental co-crystal PDB’s (green), structures generated using a single holo-MD simulation (grey), and re-docked structures generated using 3T (magenta) for HSP90 and FXa proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The holo-MD and 3T re-docking are done using just HSP90 1uyg PDB and FXa 1ezq PDB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The PCA of the corresponding experimental co-crystal structures are highlighted in dark grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For HSP90, there are several distinct protein-ligand pocket conformations available in nature (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Holo-MD and 3T only occupy the PC subspace that belongs to the actual 1uyg PDB experimental co-crystal subspace and nowhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For FXa, it seems that there is only a single conformation space which is being shared by the co-crystals, holo-MD, and 3T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' We note that we have much more limited availability of experimental FXa protein PDB co-crystals compared to CDK2 and HSP90 proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' CDK2 Initial 3T Co-crystal HSP90 Grey Grey Magenta Magenta Green Green MD Co-crystal Co-crystal MD 3T 3T luyg lezq 0 6 ET5 10 HSP90 FXa 1uyg on 1uyg 1ezg on 1ezq Supplementary Figure 4 | 3T-generated cross-docking conformation workflow and analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a–d) Distribution of ligand Δ������������������������������������������������ for generated CDK2 pockets (red, ������������������������������������������������������������������������������������ = 25Å), HSP90 with both flexible and rigid pockets (purple and green), and FXa pockets (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Δ������������������������������������������������ > 0 indicating ligand pose improvement over docking software ligand pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' e–h) Scatterplot of ������������������������������������������������������������������������������������������������ split based on the sign of Δ������������������������������������������������, with diagonal line indicating the physical limit of Δ������������������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' i–l) Bar plot showing the average of Δ������������������������������������������������ for all ligand poses where Δ������������������������������������������������ > 0, indicating that 3T increasingly generates better poses if the initial pose is less optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' m–p) Bar plot showing the average of Δ������������������������������������������������ for all ligand poses where Δ������������������������������������������������ ≤ 0, indicating that more negative Δ������������������������������������������������ becomes more likely when the initial pose is already very close to the experimental co-crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' When the HSP90 protein pocket is made perfectly rigid (3T transformations are only applied to the ligands, and not applied to the protein structure), Δ������������������������������������������������ is reduced and the trends observed in other cases with flexible 3T pocket generation are no longer apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Bar plot binning is done every 1Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 3 ' metadata={'source': 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+page_content='Initial Pose 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Initial Pose 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='CDK2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Improved ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Ligands ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Δ������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='�Å� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='〈������������������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='������������������������������������〉 �Å� 〈������������������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Δ������������������������������������������������≤0〉 �Å� 〈������������������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='Δ������������������������������������������������>0〉 �Å� p-value CDK2 (20 Å) 1st pose 255/308 83% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='42 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='86 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='75 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='49 3e-35 2nd pose 252/308 82% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='75 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='64 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='00 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='47 9e-41 3rd pose 247/308 80% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='74 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='49 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='59 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='01 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='41 9e-35 CDK2 (25 Å) 1st pose 246/308 80% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='42 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='55 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='98 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='51 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='53 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='25 2e-28 2nd pose 235/308 76% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='75 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='44 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='76 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='68 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='05 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='58 9e-28 3rd pose 234/308 76% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='58 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='76 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='57 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='32 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='66 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='90 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='14 3e-30 HSP90 1st pose 157/223 70% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='66 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='29 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='69 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='86 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='95 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='88 2e-17 2nd pose 150/223 67% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='59 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='62 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='57 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='39 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='47 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='15 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='42 1e-17 3rd pose 159/223 71% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='53 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='83 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='47 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='17 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='43 3e-17 HSP90 (rigid) 1st pose 157/223 70% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='29 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='02 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='70 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='40 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='18 6e-12 2nd pose 150/223 67% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='62 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='57 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='37 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='44 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='72 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='61 3e-10 3rd pose 133/223 60% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='64 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='21 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='85 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='82 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='48 4e-8 FXa 1st pose 62/106 58% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='00 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='54 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='29 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='21 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='26 5e-3 2nd pose 65/106 61% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='72 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='08 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='76 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='09 6e-4 3rd pose 78/106 74% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='58 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='67 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='61 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='85 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='69 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='07 2e-8 Supplementary Table 1 | Statistics of ligand cross-docking pose improvement for 3T poses compared to the original smina cross-docking pose references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In addition to the information available in Table 1 of the main text, we have also included the mean and standard distribution of initial structure error ������������������������������������������������������������������������������������������������ for 3T processes which produce positive and negative Δ������������������������������������������������, as well as those for the cases where HSP90 protein pocket atoms are frozen during 3T process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' It can be seen that for the case where 3T performs badly (lower fraction of initial cross-docked ligand poses are improved), it is because the initial structures themselves are already quite close to experimental co-crystal structures (small ������������������������������������������������������������������������������������������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For all proteins (CDK2, HSP90, and FXa) and initial poses (ranked 1st, 2nd, and 3rd), ������������������������������������������������3������������ is smaller than ������������������������������������������������������������������������������������������������ on average, and the improvement is statistically significant (p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='005 for FXa’s 1st pose and p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='001 otherwise, one-sided paired samples t-test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' See main text Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Supplementary Figure 5 | Impact of ligand similarity to reference co-crystal PDB ligand on 3T ability to improve docking pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The scatterplot of Δ������������������������������������������������ vs the Morgan fingerprint similarity between the cross-docked ligands and the ligand which is part of the reference co-crystal for a) CDK2 (������������������������������������������������������������������������������������ = 25Å, reference host protein: 1fin PDB), b) HSP90 (reference host protein: 1fin PDB), and c) FXa (reference host protein: 1ezq PDB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The low Pearson correlation coefficient r indicates that 3T’s ability to improve the cross-docked ligand RMSD is independent from such ligand’s similarity to the reference co-crystal ligand of the protein conformation host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Initial Pose 1 Initial Pose 1 Initial Pose 1 Initial Pose 2 Initial Pose 2 Initial Pose 2 Initial Pose 3 Initial Pose 3 Initial Pose 3 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='060 =-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='091 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' CDK2 HSP90 Fxa (20A) Supplementary Figure 6 | Impact of reducing or eliminating 3T protein flexibility on active ligand classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a–b) The ������������������������������������������������������������ active ligand classification metric for CDK2 pockets (������������������������������������������������������������������������������������ = 20Å) and HSP90 pockets (pockets intentionally made to be perfectly rigid and non-transformable by 3T), and c-d) AUC-ROC active ligand classification metric for the same proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 30× 4-fold cross validation (with GBT classifier) is used for the classification statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' It can be seen that 3T on small CDK2 pocket is insufficient to build good classifiers and a larger radius of pocket flexibility needs to be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Similarly, it can also be seen that performing 3T just on the ligands (with perfectly rigid HSP90 protein pocket structure) will significantly degrade the active ligand classifier performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 7 CDK2 06dsH (20A) (ngid) [+ : K HSP90 OT (rigid)Protein Active Ligands 3T Classifier Rigid MD Classifier Rigid XRD Classifier ������������������������������������������������������������,3������������ vs ������������������������������������������������������������,������������������������−������������������������������������ CDK2 (25 Å) 442/3764 (������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='117) ������������������������������������������������������������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='771 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='608 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='624 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='039 (10 + 2) vs 402 AUC-ROC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='935 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='892 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='904 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='015 CDK2 (20 Å) 442/3764 (������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='117) ������������������������������������������������������������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='469 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='608 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='624 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='039 (10 + 2) vs 402 AUC-ROC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='828 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='892 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='904 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='015 HSP90 298/2452 (������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='122) ������������������������������������������������������������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='851 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='640 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='505 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='046 (10 + 2) vs 64 AUC-ROC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='949 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='903 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='836 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='024 HSP90 (rigid) 298/2452 (������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='122) ������������������������������������������������������������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='524 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='640 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='505 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='046 (10 + 2) vs 64 AUC-ROC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='830 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='903 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='836 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='024 FXa 298/7191 (������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='040) ������������������������������������������������������������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='584 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='452 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='044 (10 + 2) vs 136 AUC-ROC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='913 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='902 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='855 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='021 Supplementary Table 2 | 3T active ligand classification metrics using the GBT classifier across 3 different protein hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' In addition to the information available in Table 2 of the main text, we have also included the classification statistics when ������������������������������������������������������������������������������������ = 20Å pocket is used for CDK2 and when perfectly rigid HSP90 protein pocket is used for 3T protein-ligand pocket conformation generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' It can be seen that when the flexible protein pocket generation ability of 3T is reduced or removed, its active ligand classification utility will be diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Supplementary Figure 7 | The impact of incorporating 3T energy landscape features vs excluding such features during classification for different number of 3T-generated conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' a–c) The ������������������������������������������������������������ active ligand classification metric for CDK2 pockets (������������������������������������������������������������������������������������ = 25Å), HSP90 pockets (������������������������������������������������������������������������������������ = 20Å, non-rigid), and FXa pockets (������������������������������������������������������������������������������������ = 20Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' d–f) AUC-ROC active ligand classification metric for the same proteins as in a–c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' 30× 4-fold cross validation (with GBT classifier) is used for the classification statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' It can very clearly be seen that incorporating 3T energy landscape features is very helpful for identifying active ligands from the decoys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' ������������������������������������������������������������ = 0 means that we do not generate any fully-flexible protein-ligand pocket structure through 3T energetic kick, and only utilize the initial smina cross-docked structure and the structure obtained after relaxing the ligand while maintaining rigid protein pocket (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' [] CDK2 (25A) d e CDK2 (25A)Protein Protein Rigidity Method XRD Experiment MD Cost / Ligand Docking Cost / Ligand CDK2 Rigid Rigid XRD 406× 0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='9 CPU-hr Rigid MD* 1× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='058 GPU-hr 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='9 CPU-hr Semi- flexible Smina (flexible sidechain) 406× 0 710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 CPU-hr rDock (flexible OH, NH3) 406× 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3 CPU-hr Fully- flexible Holo-MD* 1× 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='2 GPU-hr 0 3T 1× 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5 GPU-hr / 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='7 CPU-hr 69k atoms for the CDK2 protein --> GROMACS GPU speed = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5 ns/day HSP90 Rigid Rigid XRD 64× 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 CPU-hr Rigid MD* 1× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='051 GPU-hr 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 CPU-hr Semi- flexible Smina (flexible sidechain) 64× 0 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3 CPU-hr rDock (flexible OH, NH3) 64× 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='7 CPU-hr Fully- flexible Holo-MD* 1× 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='3 GPU-hr 0 3T 1× 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 GPU-hr / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 CPU-hr 37k atoms for the HSP90 protein --> GROMACS GPU speed = 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='1 ns/day FXa Rigid Rigid XRD 136× 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5 CPU-hr Rigid MD* 1× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='028 GPU-hr 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5 CPU-hr Semi- flexible Smina (flexible sidechain) 136× 0 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5 CPU-hr rDock (flexible OH, NH3) 136× 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='5 CPU-hr Fully- flexible Holo-MD* 1× 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='7 GPU-hr 0 3T 1× 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='8 GPU-hr / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='7 CPU-hr 60k atoms for the FXa protein --> GROMACS GPU speed = 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content='0 ns/day Supplementary Table 3 | Experimental and computational resource estimation for various docking-based active ligand classification tasks on the CDK2, HSP90, and FXa dataset based on the protein-ligand complex structure generation method being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' Active ligand Experimental and computational resource estimation for various docking-based active ligand classification tasks on the CDK2, HSP90, and FXa dataset, based on the protein-ligand complex structure generation method being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The six methods are categorized based on the rigidity of generated protein structure (rigid: no protein conformation change during ligand cross-docking, semi-flexible: some protein sidechain rotation is allowed during cross- docking, fully-flexible: all protein backbone and sidechain atoms can freely move during cross-docking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The three methods in bold are the ones for which we do classification performance comparison in main text Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The XRD experiment column refers to the number of X-ray diffraction co-crystal structures which are needed to enable active ligand classification task based on such methods (the same number of experimental structures being used in main text Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For rigid MD, holo-MD, and 3T, only one such experimental co-crystal protein structure is needed as the initial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' However, the holo-MD and rigid MD methods will require additional MD simulation cost for the subsequent protein-ligand complex structure generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' For 3T, all structure generation cost (single rigid protein docking plus 10-conformation generation) is categorized as ‘docking cost’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} +page_content=' The docking computation cost estimates are averaged from three randomly chosen ligands, except for 3T CPU-hr estimates which are averaged from 16 randomly chosen ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfD_oT/content/2301.00984v1.pdf'} diff --git a/3tE0T4oBgHgl3EQfeADx/content/2301.02386v1.pdf b/3tE0T4oBgHgl3EQfeADx/content/2301.02386v1.pdf new file mode 100644 index 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a/5tE1T4oBgHgl3EQfTAOS/content/tmp_files/2301.03073v1.pdf.txt b/5tE1T4oBgHgl3EQfTAOS/content/tmp_files/2301.03073v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7ed414b05800eee0e75255295514a9f83d10041f --- /dev/null +++ b/5tE1T4oBgHgl3EQfTAOS/content/tmp_files/2301.03073v1.pdf.txt @@ -0,0 +1,3373 @@ +MNRAS 000, 000–000 (0000) +Preprint 10 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Detection of He++ ion in the star-forming ring of the +Cartwheel using MUSE data and ionizing mechanisms +Y. D. Mayya⋆1 +ID, A. Plat2 +ID, V. M. A. G´omez-Gonz´alez3 +ID, J. Zaragoza-Cardiel1,4 +ID, +S. Charlot5 +ID and G. Bruzual6 +ID +1Instituto Nacional de Astrof´ısica, ´Optica y Electr´onica, Luis Enrique Erro 1, Tonantzintla 72840, Puebla, Mexico +2Steward Observatory, 933 N. Cherry Avenue, University of Arizona, Tucson, AZ 85721, USA +3Institute for Physics and Astronomy, Universit¨at Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam, Germany +4Consejo Nacional de Ciencia y Tecnolog´ıa, Av. Insurgentes Sur 1582, 03940, Mexico City, Mexico +5Sorbonne Universit´e, CNRS, UMR7095, Institut d’Astrophysique de Paris, F-75014, Paris, France +6Instituto de Radioastronom´ıa y Astrof´ısica, UNAM Campus Morelia, Apartado postal 3-72, 58090 Morelia, Michoac´an, Mexico +10 January 2023 +ABSTRACT +We here report the detection of the nebular He ii λ4686 line in 32 H ii regions in the +metal-poor collisional ring galaxy Cartwheel using the Multi-Unit Spectroscopic Ex- +plorer (MUSE) dataset. The measured I(He ii λ4686)/I(Hβ) ratio varies from 0.004 +to 0.07, with a mean value of 0.010±0.003. Ten of these 32 H ii regions are coinci- +dent with the location of an Ultra Luminous X-ray (ULX) source. We used the flux +ratios of important diagnostic lines and results of photoionization by Simple Stellar +Populations (SSPs) to investigate the likely physical mechanisms responsible for the +ionization of He+. We find that the majority of the regions (27) are consistent with +photoionization by star clusters in their Wolf-Rayet (WR) phase with initial ioniza- +tion parameter −3.5 −1; (3) hard radiation from binary stars (in par- +ticular, X-ray binaries); (4) ionization of He+ by radiative +shocks; and/or (5) ionization of He+ by an active galactic +nucleus (AGN), if and when present. +Over the last decade or so, it has been possible to ob- +tain reliable fluxes of the relatively weak He ii λ4686 line +for a large sample of objects, thanks to dedicated spectro- +scopic surveys such as Sloan Digital Sky Survey (SDSS). +Given that the often-used nebular diagnostic lines (see e.g. +P´erez-Montero 2017) are much brighter than the He ii λ4686 +line, the available dataset has also allowed accurate deter- +mination of nebular metallic abundances. Dataset obtained +from these large surveys are the ones often used to confront +with the predictions of SSP models (e.g. Plat et al. 2019; +Schaerer, Fragos & Izotov 2019). These data typically sam- +ple zones that span several kiloparsecs in size, centered on +the nucleus of the galaxy. Though the diagnostic lines (Bald- +win, Phillips, & Terlevich 1981) allow the rejection of spectra +dominated by an AGN at least at high enough metallicities +(Z >0.008; see Feltre, Charlot, & Gutkin 2016), the pres- +ence of a weak AGN in a spectrum dominated by a starburst +component cannot be ruled out. Over the large spatial scales +sampled in these studies, several of the above mentioned five +physical mechanisms are likely to be at work, thus inhibit- +ing discerning the relative importance of each mechanism. +Data on smaller scales, ideally of selected regions in nearby +galaxies, are required so as to explore the role of each of the +above-listed mechanisms in increasing the He ii λ4686/Hβ +ratio above the canonical values predicted by the SSP mod- +els. +Availability of spectrographs incorporating integral +field units (IFUs) on large telescopes such as Multi Unit +Spectroscopic Explorer (MUSE) on the Very Large Tele- +scope (VLT) (Bacon et al. 2010) and MEGARA on the +Gran Telescopio Canarias (GTC) (Gil de Paz et al. 2018) +is allowing such studies possible in recent years. Kehrig et +al. (2015) and Kehrig et al. (2018) used MUSE data to spa- +tially map the He ii λ4686 line in two of the most metal- +poor galaxies: I Zw 18 and SBS 0335 − 052E, finding that +the observed He+ ionization cannot be explained by the +WR stars present in these galaxies. Recently, Mayya et al. +(2020) used MEGARA to map the central starburst cluster +of NGC 1569, a galaxy with the oxygen abundance similar to +that of the Large Magellanic Cloud (LMC), finding that the +WR stars in the starburst cluster are able to completely ex- +plain the observed ionization. Data on regions where hydro- +gen ionization is dominated by photons from massive stars +of metallicity below that of the LMC are needed to address +the sources of He+ ionization in distant metal-poor galax- +ies. The collisional-ring galaxy Cartwheel provides such a +laboratory, as explained below. +The Cartwheel is considered as the archetype of the +class of collisional ring galaxies (Appleton & Struck-Marcel +1996; Struck 2010). Ring galaxies are characterized by a ring +that harbours a chain of star forming knots. The star for- +mation in the ring is believed to be triggered by a radially +expanding density wave that was formed as a result of a com- +pact galaxy plunging through a massive gas-rich disk galaxy +close to its center and almost perpendicular to it (Lynds +& Toomre 1976). Higdon (1995) found that the Hα emis- +sion in the Cartwheel, a tracer of current star formation, +is distributed along a ring as predicted by the collisional +scenario of the formation of ring galaxies. MUSE dataset is +available on this galaxy at the seeing-limited spatial reso- +lution of ∼0.6 arcsec. This dataset provides optical spectra +covering a rest wavelength range of ∼4600 to 9100 ˚A over +the entire galaxy. On the Hα image constructed using this +dataset, we have identified more than 200 individual H ii +regions in and around the ring. A colour-composite image +formed using this Hα image is shown in Figure 1. At the +distance of the Cartwheel (128 Mpc using the Hubble con- +stant of 71 km s−1 Mpc−1), MUSE spectra are available at +physical scales of ∼370 pc, which is an order of magnitude +better as compared to the typical size scale where He ii λ4686 +is detected in metal-poor galaxies. At the spatial resolu- +tion of the Wide Field and Planetary Camera 2 (WFPC2) +images of the Hubble Space Telescope (HST), which are +the highest resolution images (∼0.2 arcsec=125 pc) avail- +able for this galaxy, we can associate each MUSE-identified +H ii region with a population of super star clusters (SSCs), +which provide the ionization of the H ii regions. Fosbury & +Hawarden (1977) have measured an oxygen abundance of +12+log(O/H)∼8.0, corresponding roughly to Z ∼ 0.003 (see +Table 2 of Gutkin et al. 2016), a value at which the observed +He ii λ4686/Hβ ratio in galaxy samples is higher than that +predicted by most of the SSPs. The wide spectral coverage of +MUSE data allowed the flux measurement of nebular lines +useful to study the ionization state of the H ii regions us- +ing the standard line ratio diagrams (Baldwin, Phillips, & +Terlevich 1981, hereafter BPT). We used this new dataset +to measure an average oxygen abundance for the ring re- +gions of 12+log(O/H)∼8.19±0.15 (Zaragoza-Cardiel et al. +2022), which is marginally higher than the value reported +by Fosbury & Hawarden (1977) for three of the brightest +H ii regions. +An additional aspect that makes the Cartwheel a good +candidate for understanding the He ii ionization problem is +the presence of more than 15 ultra-luminous X-ray sources +(ULXs) in its star-forming ring (Gao et al. 2003; Wolter +& Trinchieri 2004). In fact the Cartwheel is the record +holder for the maximum number of ULX sources in a sin- +gle galaxy (Wolter, Fruscione & Mapelli 2018). The ULX +emission is believed to be originating in high-mass X-ray +binaries (HMXBs), with the compact object most likely an +intermediate-mass black hole (IMBH) (Mapelli et al. 2010; +Wolter, Fruscione & Mapelli 2018). The presence of these +sources allows us to explore the role of HMXBs in the He+ +ionization in regions where massive stars contribute to the +ionization of hydrogen and other ions of ionization potential +much lower than 54.4 eV. +In Section 2, we describe the dataset, extraction of indi- +MNRAS 000, 000–000 (0000) + +HeII in the Cartwheel galaxy +3 +221 +220 +219 +218 +217 +216 +215 +214 +213 +212 +211 +210 +209 +208 +207 +206 +205 +204 +203 +202 +201 +200 +199 +198 +197 +196 +195 +194 +193 +192191 +190 +189 +188187186 +185 +184 +183 +182 +181 +180 +179 +178 +177 +176175 174 +173172 +171 +170 +169 +168 +167 +166 +165 +164 +163 +162 +161 +160 +159 +158 +157 +156 +155154 +153 +152 +151 +150 +149 +148 +147 +146 +145 144143 +142 +141 +140 +139138137 +136 +135 +134 +133 +132 +131 +130 +129 +128 +127 +126 +125 +124 +123 +122 +121 +120 +119 +118 +117 +116 +115 +114 +113 +112 +111 +110 +109 +108 +107 +106 +105 +104 +103 +102 +101 +100 +99 +98 +97 +96 +95 +94 +93 +92 +91 +90 +89 +88 +87 +86 +85 +84 +83 +82 +81 +80 +79 +78 +77 +76 +75 +74 +73 +72 +71 +70 +69 +68 +67 +66 +65 +64 +63 +62 +61 +60 +59 +58 +57 +56 +55 +54 +53 +52 +51 +50 +49 +48 +4746 +45 +44 +43 +42 +41 +40 +39 +38 +3736 +35 +34 33 +32 +31 +30 +29 28 +27 +26 +25 +24 +23 +22 +21 +20 +19 +18 +17 +16 +15 14 +131211109 +8 +7 +6 +5 +4 +3 +2 1 +E +N +Figure 1. Colour-composite HST/WFPC2 and VLT/MUSE image of the Cartwheel galaxy. The RGB image is formed using the +HST/WFPC2 filters (PSF=0.2 arcsec) in F814W, pseudo-green and F450W as red, green and blue components, respectively; the Hα +image is constructed from MUSE data (PSF=0.6 arcsec) as a fourth reddish component. The H ii regions in the ring are identified by +numbers and shown by green circles of 0.6 arcsec radius, which corresponds to 370 pc at the distance of the Cartwheel. +vidual spectrum, and details of measurement of line fluxes. +The analysis of nebular line ratios is described in Section +3. A detailed discussion of results on the He+ ionization in +each star-forming complex is carried out in Section 4. Our +conclusions are given in Section 5. +2 +THE SAMPLE OF HE++ NEBULAE AND +THE CONTROL SAMPLE OF BRIGHT H ii +REGIONS +2.1 +The H ii region parent sample +Given that the ionization potential of hydrogen is four times +lower as compared to the second ionization potential of he- +lium, the He++ nebulae are expected to be a subset of the +ionized nebulae. The outer ring of the Cartwheel is cur- +rently experiencing an intense burst of star formation. Hig- +don (1995) found that the Hα emission, a tracer of current +star formation, is predominantly confined to 29 ionized com- +MNRAS 000, 000–000 (0000) + +4 +Y. D. Mayya et al. +Table 1. Cartwheel H ii regions with He ii λ4686 nebular emission line. +ID +Hβ +He ii +O +and WR stars Cont +H ii Higdon ULX +log f(Hβ) +SNR +EW log M∗ +log Q(He+) I(He ii)/I(Hβ) +NO7V +EWBB +NWNL +NWR +SNR +[erg cm−2 s−1] +[˚A] +M⊙ +ph s−1 +×100 +3σ +C&B +C&B +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +(10) +(11) +(12) +(13) (14) +10 +H1 +— +−15.082 ± 0.009 +109.9 +57.2 +4.84 +49.29 +1.186 ± 0.326 +341 +6.07 +2 +7 17.9 +16 +H3 +G18 +−14.672 ± 0.005 +183.5 +71.7 +5.25 +49.56 +0.845 ± 0.196 +876 +3.28 +7 +19 34.1 +17 +H4 +— +−15.533 ± 0.014 +70.7 +29.6 +4.39 +49.19 +2.666 ± 0.784 +120 +5.07 +1 +2 21.9 +20 +H5 +— +−14.841 ± 0.005 +204.6 +63.7 +5.08 +49.55 +1.232 ± 0.168 +594 +2.77 +4 +13 38.3 +60 +H6 +— +−14.808 ± 0.006 +165.3 154.4 +5.11 +49.43 +0.861 ± 0.160 +640 +7.22 +5 +14 14.7 +75 +H10 +G19/W23 +−14.512 ± 0.003 +287.3 +90.5 +5.41 +49.68 +0.775 ± 0.151 +1267 +3.48 +10 +28 34.0 +81 +H12 +— +−14.366 ± 0.003 +363.8 159.6 +5.55 +50.03 +1.256 ± 0.098 +1773 +3.43 +14 +39 30.2 +84 +H14 +— +−14.364 ± 0.002 +408.3 120.7 +5.56 +49.91 +0.936 ± 0.110 +1781 +2.93 +14 +39 35.4 +85 +H14 +G17/W7 +−14.517 ± 0.003 +303.4 +79.8 +5.40 +49.87 +1.221 ± 0.149 +1252 +2.66 +10 +27 38.3 +86 +H14 +— +−14.994 ± 0.005 +195.7 +77.8 +4.93 +49.12 +0.648 ± 0.195 +417 +4.91 +3 +9 21.7 +90 +H15 +— +−14.339 ± 0.003 +338.4 126.8 +5.58 +49.80 +0.686 ± 0.101 +1887 +3.04 +15 +41 33.2 +91 +H15 +G14 +−14.756 ± 0.006 +170.9 +49.4 +5.16 +49.55 +1.008 ± 0.207 +722 +2.14 +6 +15 46.0 +92 +H15 +— +−14.808 ± 0.005 +194.1 +66.4 +5.11 +49.42 +0.854 ± 0.195 +640 +3.15 +5 +14 31.0 +97 +H17 +— +−14.984 ± 0.007 +136.5 +68.8 +4.94 +49.18 +0.733 ± 0.161 +427 +3.13 +3 +9 35.6 +98 +H17 +— +−14.887 ± 0.006 +163.3 +72.9 +5.03 +49.17 +0.565 ± 0.171 +534 +3.70 +4 +11 27.6 +99 +H17 +G12/W24 +−13.575 ± 0.001 1055.0 282.6 +6.35 +50.55 +0.664 ± 0.052 +10960 +2.78 +92 +242 35.2 +100 H17 +— +−14.721 ± 0.005 +221.9 +69.9 +5.20 +49.42 +0.694 ± 0.159 +783 +2.68 +6 +17 39.2 +105 H19 +— +−14.824 ± 0.006 +171.0 +61.4 +5.10 +49.31 +0.676 ± 0.162 +617 +2.63 +5 +13 42.0 +106 H19 +— +−14.563 ± 0.004 +241.5 +81.3 +5.36 +49.57 +0.683 ± 0.161 +1126 +2.99 +9 +24 34.2 +108 H21 +— +−15.148 ± 0.009 +116.2 +52.5 +4.77 +49.32 +1.461 ± 0.323 +292 +4.66 +2 +6 21.8 +111 H20 +G11/W10 +−14.894 ± 0.007 +152.5 +34.9 +5.03 +49.43 +1.066 ± 0.240 +525 +2.38 +4 +11 47.5 +112 H22 +— +−14.493 ± 0.004 +271.0 219.4 +5.43 +49.37 +0.361 ± 0.107 +1323 +5.50 +11 +29 22.5 +113 H22 +— +−15.201 ± 0.009 +109.4 +46.1 +4.72 +49.02 +0.831 ± 0.213 +259 +4.07 +2 +5 26.4 +116 H23 +— +−14.614 ± 0.004 +259.1 +93.4 +5.31 +49.81 +1.341 ± 0.153 +1001 +3.14 +8 +22 34.1 +118 H23 +G8 +−14.432 ± 0.003 +360.0 123.3 +5.49 +49.71 +0.686 ± 0.126 +1523 +3.24 +12 +33 35.1 +119 H24 +G6/W14 +−14.498 ± 0.006 +170.1 +60.6 +5.42 +49.83 +1.053 ± 0.184 +1308 +2.61 +11 +28 39.1 +120 H24 +— +−14.679 ± 0.007 +152.4 +51.0 +5.24 +49.60 +0.951 ± 0.251 +862 +2.98 +7 +19 36.5 +121 H24 +— +−14.983 ± 0.011 +91.3 +69.8 +4.94 +49.28 +0.908 ± 0.239 +428 +5.69 +3 +9 18.4 +141 H26 +— +−14.700 ± 0.005 +188.6 +37.4 +5.22 +49.56 +0.918 ± 0.223 +821 +2.11 +6 +18 54.6 +144 H26 +G2/W17 +−15.036 ± 0.007 +134.8 +38.0 +4.88 +49.61 +2.223 ± 0.322 +379 +3.01 +3 +8 34.8 +148 — +— +−15.785 ± 0.021 +46.8 +21.8 +4.14 +49.31 +6.283 ± 1.599 +67 +7.90 +0 +1 13.8 +213 H29 +G9/W12 +−14.907 ± 0.006 +164.5 +88.9 +5.01 +49.20 +0.644 ± 0.196 +510 +4.53 +4 +11 24.4 +(1) H ii region identification number used in this work, (2) H ii complex number from Higdon (1995), (3) ULX sources associated to the +H ii region, (4-7) Measured and derived quantities using Hβ nebular line, where stellar mass in column 7 is derived assuming a Lyman +continuum photons rate of 4.94×1046 using C&B model corresponding to an age of 3.4 Myr for the ionizing cluster, (8) He+ ionizing +photon rate determined from the observed He ii λ4686 flux for a Case B H ii region, (9) 100× the intensity of He ii λ4686 line with +respect to that of Hβ, (10) Equivalent number of O7V stars derived assuming a typical Hβ luminosity of 4.76×1036 erg s−1 for an O7V +star from L´opez-S´anchez & Esteban (2010), (11) Upper limit for the equivalent width of the BB from WNL type inferred as the 3σ flux +of a broad (FWHM=20 ˚A) BB divided by the underlying continuum flux, (12-13) The number of WNL and total WR stars expected in +C&B models during the WR phase for the cluster mass in column 7, (14) SNR of the continuum adjacent to the BB bump. +plexes in the outer ring. As a first step, we used the MUSE +datacube to obtain a narrow-band image at the red-shifted +wavelength of the Hα line to locate ionized regions at the +resolution of MUSE (FWHM=0.6 arcsec∼370 pc), which is +∼3 times better as compared to the image used by Hig- +don (1995). In Figure 1, we show an RGB image formed +by combining the HST/WFPC2 filters in F814W, pseudo- +green and F450W as red, green and blue components, re- +spectively; the Hα image constructed from MUSE data is +used as a fourth reddish component. We identified visually +221 prominent Hα-emitting knots in the ring or near to it +on the MUSE image, which are shown by numbers in this +image. At the resolution of the HST/WFPC2 images, most +of the Hα-emitting knots are associated with one or more +compact clusters, which are the most likely ionizing sources +of the nebulae. Thus, the selected regions are star-forming +complexes spread over the entire extracted area. However, +at the spatial resolution offered by MUSE, each complex is +basically a compact unresolved knot. We hence used a uni- +form aperture of 0.6 arcsec radius to extract spectrum of +each region. We note that the Hα surface brightness in the +outer ring is 10 to 100 times brighter than the typical cut-off +surface brightness of 2×10−17 erg cm−2 s−1 arcsec−2 for the +Diffuse Ionized Gas (DIG) (see e.g. Belfiore et al. 2022), and +hence the DIG contribution to the flux ratios of lines in the +extracted spectra can be ignored. +We did not attempt to subtract the disk spectrum from +the extracted spectra as it is non-trivial to locate a re- +gion for extraction of the local disk spectrum in a galaxy +such as the Cartwheel because of its peculiar morphology +and star formation history. Thus the extracted spectra con- +tain contributions from any pre-collisional, as well as from +all post-collisional, populations in the extracted area. From +an analysis of stellar populations in the extracted spectra, +MNRAS 000, 000–000 (0000) + +HeII in the Cartwheel galaxy +5 +10 +16 +17 +20 +60 +75 +81 +84 +85 +86 +90 +91 +92 +97 +98 +99 +100 +105 +106 +108 +111 +112 +113 +116 +118 +119 +120 +121 +141 +144 +148 +213 +Figure 2. The He ii λ4686/Hβ ratio as a function of SNR(Hβ) +for the Cartwheel H ii regions. Detections and the 3-σ upper lim- +its for the He ii λ4686 line are shown by blue circles and red +triangles, respectively. All detections are tagged by their iden- +tification numbers. The inclined dotted line denotes the lowest +He ii λ4686/Hβ ratio that can be detected at 3-σ for H ii regions +having EW(Hβ)=125 ˚A. See text for details. +Zaragoza-Cardiel et al. (2022) concluded that the spectra +lack absorption features characteristics of disk stellar pop- +ulations. This is understandable because the pre-collsional +disk is around a factor of five fainter than the ring (Mar- +cum, Appleton & Higdon 1992; Higdon 1995), and hence +the extracted spectra are dominated by contribution from all +populations formed after the collision around 100 Myr ago +(Renaud et al. 2018). The extracted spectra, however, could +contain contribution from non-ionizing populations formed +over the last 100 Myr. Such a population will not contribute +to the measured emission line fluxes, but will make the ob- +served values of emission-line EWs lower that that expected +for a single burst young population. We will take this into +account while we carry out a detailed comparison of the ob- +served EWs with that predicted from population synthesis +models in Sec.4. +2.2 +The He++ nebular sample +We measured fluxes of all major nebular lines in all the ex- +tracted spectra. We used the Gaussian profile fitting routine +of the splot task in iraf1 for this purpose. The measured +line fluxes are free from the underlying continuum. We fit- +ted each extracted spectrum with a Gaussian profile at the +redshifted wavelengths of a list of nebular lines commonly re- +ported in extragalactic H ii regions, including the He ii λ4686 +1 IRAF is distributed by the National Optical Astronomy Obser- +vatories, which are operated by the Association of Universities for +Research in Astronomy, Inc., under cooperative agreement with +the National Science Foundation. +line. The output quantities of the Gaussian fitting task that +we used here are: line flux, FWHM and EW. We measured +the root mean square (rms) noise at feature-less parts of the +continuum adjacent to the measured line, which we used +to determine the signal-to-noise ratio (SNR) of the mea- +sured line. An emission line is deemed detected if the fitted +FWHM is comparable to that of the Hβ line (2.5 ˚A) and +that the integrated line flux has a SNR⩾3. Using these line +fluxes, we constructed two sub-samples: (1) a sub-sample of +87 bright H ii regions defined as those with SNR>40 in the +Hβ line, and (2) a sub-sample of 32 He ii nebulae defined +as those where the He ii λ4686 nebular line is detected. As +expected, the latter sample is a subset of the former. The +bright H ii region sample is used as a control sample to un- +derstand the physical conditions that favour the detection +of the He ii λ4686 line in ionized regions. +The criterion that the FWHM of a line should be com- +parable to that of the Hβ emission line ensures that the +detected line is of nebular origin. Nevertheless, the spectra +of the He ii nebulae were visually inspected for the possible +presence of an underlying broad He ii λ4686 emission feature, +commonly referred to as the ’blue bump’ (BB), from WR +stars. None of the spectra showed this feature, and hence +any contribution of the WR bump to the measured nebular +He ii λ4686 fluxes can be ignored. +The He ii nebular sample2 is listed in Table 1. In col- +umn 2, we list the cross identification of our sources with +the H ii complexes of Higdon (1995). In other columns, we +give the basic measured quantities of these regions such as +the flux, SNR and EW of the Hβ line (columns 4, 5 and +6, respectively), and the He ii λ4686 intensity normalized to +I(Hβ) and multiplied by 100, along with error (column 9). +The error in measured flux (σl) of each line is calculated +using the expression (Tresse et al. 1999):, +σl = σcD +� +(2Npix + EW +D ), +(1) +where D is the spectral dispersion in ˚A per pixel (1.25 for +MUSE), σc is the mean standard deviation per pixel of the +continuum, which is measured in a line-free part of the con- +tinuum adjacent to each line, Npix is the number of pixels +covered by the line, which is equated to the FWHM of the +fitted Gaussian profile. In all these 32 regions, the Hβ flux +has SNR>50, with a median SNR of 176. The He ii λ4686 +line is detected at SNR=3–15. +Detection of a line depends on the SNR of the spec- +trum. In Figure 2, we show the He ii λ4686/Hβ ratio for all +the H ii regions with SNR(Hβ)⩾40. Detections are shown +in blue solid circles, whereas the upper limits, defined as +3-σ fluxes where σ is calculated using the error equation 1 +with Npix=FWHM(Hβ), are shown by red inverted trian- +gles. As a reference, we show by the dotted line the min- +imum value of He ii λ4686/Hβ ratio that a region should +have so that the He ii λ4686 line is detected, given the SNR +of Hβ in its spectra. The intercept of this line depends +2 Fluxes of all measured lines for the entire sample of H ii re- +gions are presented in companion paper by Zaragoza-Cardiel et +al. (2022), which deals with the nebular elemental abundances. +The ratios of fluxes of lines used in this work are given in a Sup- +plementary Electronic Table. +MNRAS 000, 000–000 (0000) + +6 +Y. D. Mayya et al. +Figure 3. The VLT/MUSE rest-frame spectra in the blue part +of three representative H ii regions #99, #112 and #148 in the +Cartwheel. Each spectrum is shown normalized to the best fit +continuum spectrum and displaced vertically for clarity sake. The +nebular lines in the plotted spectral range are indicated. In the +three spectra, the He ii λ4686 line is detected at SNR=12, 3.3, and +3.6 (from top to bottom), with the bottom two spectra illustrating +detections just above our chosen 3-σ detection limit. In addition, +all detections have line widths comparable to that of the typical +line spread function. We mark by a cross a spurious feature at +λ=4810 ˚A in the bottom-most spectrum, which is narrower than +the typical line spread function. +slightly on EW(Hβ), with the plotted line corresponding +to EW(Hβ)=125 ˚A. The line would shift upwards (down- +wards) by ∼0.005 for regions with EWs a factor of two +lower (higher). Our sample of He ii nebulae has a mean +value of I(He ii λ4686)/I(Hβ)=0.010±0.003, with the low- +est and highest ratios being 0.004 (#112) and 0.07 (#148). +We can expect He ii line detection only in H ii regions +with SNR(Hβ)>100, if their He ii λ4686/Hβ ratio is close or +higher than the mean value of the sample. Conversely, the +detection of He ii λ4686 line would require abnormally high +ratio of He ii λ4686/Hβ in regions with SNR(Hβ)<100, such +as the case in #148 and #17. As expected, the He ii λ4686 +line is detected in all MUSE spectra with SNR(Hβ)⩾200 +(12 regions). For regions having 100100 M⊙) Stars. This illustrates that +the absence of the blue bump does not necessarily imply the +absence of WR stars. At the metallicity of the Cartwheel +H ii regions, WR stars in numbers sufficient to doubly ionize +helium could be present even when the characteristic broad +BB is not detected. This could be the case not only in the +Cartwheel regions, but in general in all metal-poor systems +that show He ii λ4686 nebular line (e.g. Shirazi & Brinch- +mann 2012). +It is interesting to note that we would have been able +to detect the BB when Very Massive Stars in clusters, if +present, go through the WR phase. It can be inferred from +Figure 7, that the Mu=300 M⊙ models provide He+ ion- +izing photons for double the duration as compared to the +MNRAS 000, 000–000 (0000) + +10 +Y. D. Mayya et al. +Figure 8. Observational upper limits (inverted triangles) on the +EW(BB) plotted against the He ii λ4686/Hβ ratio for Cartwheel +H ii regions with He ii λ4686 line detections. The expected values +during the WR phase from the C&B SSP model for two upper cut- +off masses are shown. In the presence of massive stars of 300 M⊙ +the EW(BB) reaches values as high as 10 ˚A for a brief period +around 2.5 Myr, whereas the EW(BB) stays lower than 1.5 ˚A +when Mu=100 M⊙, which is at least a factor of two lower than +the detection threshold as indicated by the 3-σ upper limits. H ii +regions with extreme values of He ii λ4686/Hβ ratio are identified +with their numbers. See text for details. +Mu=100 M⊙ models. This implies that if all the He ii λ4686- +detected H ii regions had an IMF with Mu=300 M⊙, the BB +would have been at the detectable level in 50% of the cases. +The non-detection of the BB in all regions points to the ab- +sence of Mu >100 M⊙ stars in H ii regions of the Cartwheel. +Given that our He ii λ4686-line detection criteria are +tuned to detect nebular (narrow) lines, there exists a possi- +bility that we inadvertently excluded possible WR sources +in spectra where we did not detect the He ii λ4686 narrow +line. In order to verify this possibility, we analyzed the out- +put results of the Gaussian-fitting for all the regions to look +for a broad He ii λ4686 component with SNR⩾3. None of the +spectra showed evidence for it. If stars more massive than +100 M⊙ were common, the BB should have been present at +detectable levels in at least a few of the 80 Hβ-bright regions. +In fact, none of the spectra of our original sample of 221 H ii +regions showed the BB. Such a non-detection reinforces the +inference drawn from the He ii λ4686-detected regions that +the upper mass cut-off of the IMF rarely exceeds 100 M⊙ in +H ii regions. +In summary, WR stars are viable sources of He+ ion- +ization in He ii λ4686-detected regions of the Cartwheel, in +spite of the non-detection of the BB. In rest of this paper, +we use photoionization models to investigate whether the +intensity ratios of bright nebular lines support the scenario +of WR stars as the only source of ionization in majority of +the regions. Four regions with extreme He ii λ4686/Hβ ratio +Table 2. Cartwheel H ii regions nearest to an ULX source. +H ii ID Higdon +Gao +Wolter +He++ +ID +offset +ID +offset +log LX +[arcsec] +[arcsec] erg s−1 +(1) (2) +(3) +(4) +(5) +(6) +(7) +(8) +13 H2 +— +— +W21 +0.61 +38.70 +no +16 H3 +G18 +0.84 +— +— +— +yes +49 — +— +— +W18 +0.29 +no +75 H10 +G19 +0.65 +W23 +0.33 +38.78 +yes +85 H14 +G17 +0.16 +W7 +0.42 +39.70 +yes +88 — +G15 +0.12 +W8 +0.52 +39.18 +no +91 H15 +G14 +0.44 +— +— +— +yes +99 H17 +G12 +0.77 +W24 +2.39 +38.57 +yes +111 H20 +G11 +2.17 +W10 +1.75 +40.44 +yes +117 — +G7 +1.33 +W13 +1.21 +no +118 H23 +G8 +0.24 +— +— +— +yes +119 H24 +G6 +0.56 +W14 +1.28 +39.88 +yes +125 — +G5 +0.65 +W15 +1.42 +39.06 +no +144 H26 +G2 +0.50 +W17 +0.70 +39.92 +yes +146 — +G3 +1.74 +W16 +2.32 +39.83 +no +155 — +G4 +0.41 +— +— +— +no +213 H29 +G9 +0.81 +W12 +0.72 +39.49 +yes +(identified in Figure 8), possibly require alternative sources +of ionization, which are also investigated in the paper. +2.6.2 +Main sequence stars +The bulk of the ionization of hydrogen in H ii regions is pro- +vided by star clusters in their early phase when massive O +stars are in the main sequence (MS). Some of these stars are +hot enough to doubly ionize helium. The emission EW(Hβ) +is maximum during this early phase having values larger +than 500 ˚A. The highest values of He ii λ4686/Hβ during +this phase are 0.002 and 0.001, respectively in C&B and +BPASS models, both with Mu=300 M⊙. It can be inferred +from Figure 2 that we require SNR above 400 to detect +He ii λ4686 line ionized by the MS stars. There are 5 regions +with SNR>330, all of which have He ii λ4686/Hβ>0.006, +i.e. at least a factor of three higher than the MS values. +On the other hand, the region with the lowest value of +He ii λ4686/Hβ is #112, which is around twice the MS value. +Stripped binary stars, which are not taken into account in +the C&B and BPASS models, may have a role in increas- +ing the He ii λ4686/Hβ above the calculated values. The MS +phase is characterized by high EW(Hβ). We analyse all re- +gions in He ii λ4686/Hβ vs EW(Hβ) plane to address this +issue in Sec. 3. +2.6.3 +Ultra-luminous X-ray sources +Pointlike non-nuclear X-ray sources with isotropic bolomet- +ric luminosity in the the 0.5–10 keV band (LX) exceeding +3×1039 erg s−1are referred to as ULX sources. Gao et al. +(2003) and Wolter & Trinchieri (2004) analyzed the Chan- +dra/Acis data of Cartwheel finding 31 and 24 ULX sources, +respectively, in the FoV of the Cartwheel, the majority of +them coinciding with the star-forming ring of the Cartwheel. +The most luminous of these sources (#11 in Gao et al. 2003 +and #10 in Wolter & Trinchieri 2004) has LX > 1041 erg s−1, +thus satisfying the criterion to be called as a hyperluminous +MNRAS 000, 000–000 (0000) + +HeII in the Cartwheel galaxy +11 +X-ray source (HLX). However, a one-to-one correspondence +with an optical knot in the ring was poor in both the stud- +ies, with offsets between the H ii complexes defined by Hig- +don (1995) and the ULX coordinates, generally exceeding +the 1 arcsec beam of the Chandra/Acis observations, even +after correcting for zeropoint offsets in the respective coordi- +nate systems. The reason for these large offsets is that there +is more than one star cluster within the seeing-limited res- +olution of ∼1.7 arcsec (1 kpc) of the Hα image of Higdon +(1995), with the coordinates referring to that of the bright- +est H ii region in the complex, which is not necessarily the +ULX source. The astrometrically calibrated MUSE and HST +dataset that we use in this study offers ∼3 and 8 times better +spatial resolutions, respectively, as compared to the Hα im- +age of Higdon (1995), which allows us to improve upon the +identification of the optical counterpart of the ULX sources. +In Figure 4, we mark the positions of ULX/HLX sources +by red circles overlaid on the HST image. Fourteen of the 17 +X-ray sources coincide with the position of an H ii region to +better than an arcsec, the beam of the X-ray observations. +The H ii region closest to a ULX/HLX source is identified in +Table 2, where we also give the offsets for each source from +the coordinates reported by Gao et al. (2003) and Wolter & +Trinchieri (2004). It is worth noting that given the high den- +sity of H ii regions in the ring, more than one H ii region can +be associated for sources with offsets exceeding an arcsec. +The offsets are systematically smaller for Gao et al. (2003) +coordinates. Source #111, the identified counterpart of the +HLX source (G11) is outside the Chandra/Acis beam, sug- +gesting that the identification is likely to be wrong, and that +the source may be associated to a non-Hα-emitting object. +In order to identify such a candidate, we looked for any stel- +lar knot in the HST images. We find a faint red knot at the +edge of the Chandra/Acis beam, which could be a likely +counterpart of the HLX source (see the region G11 in the +figure). A He++ nebula is present within the beam of the +X-ray observations for ten and seven ULX sources identi- +fied by Gao et al. (2003) and Wolter & Trinchieri (2004), +respectively (see the last column of the table). We analyse +below the possible role of X-rays from the ULX sources in +the ionization of He+. +Schaerer, Fragos & Izotov (2019) found that the ob- +served He ii λ4686/Hβ ratio in metal-poor galaxies can be +explained if the bulk of the He+ ionizing photons is emit- +ted by HMXBs, whose numbers are found to increase with +decreasing metallicity. They obtained an empirical relation +between Q(He+) and the X-ray luminosity, LX, suggesting +an almost constant ratio q = Q(He+)/LX = 2 × 1010 pho- +ton erg−1, with extreme values of q being 1–3×1010 pho- +ton erg−1. Plat et al. (2019) warned that this process is not +efficient at EW(Hβ)>200 ˚A, as these systems are too young +to form compact objects (neutron stars and stellar mass +black holes) necessary for the existence of HMXBs. Only +two Cartwheel He ii-emitting regions have EW(Hβ)>200 ˚A, +and hence ionization of He+ by ULX sources is a possibility +in majority of the He ii-emitting regions with an associated +ULX source. +The presence of an ULX source in 17 of the Cartwheel +H ii regions allows us to calculate the value of q directly for +these regions. For this purpose, we use the Q(He+) for each +region in column 8 of Table 1 and the LX of column 7 of +Table 2, which was taken from Wolter & Trinchieri (2004). +Figure 9. +Analysis of the possibility of ionization of He+ by +the ULX sources associated with the H ii regions of Cartwheel in +He ii λ4686/Hβ ratio vs. q = Q(He+)/LX plane. Circles are detec- +tions, and inverted triangles are 3-σ upper limits on He ii λ4686 +detection. Regions in which The range of values for this ratio pro- +posed by Schaerer, Fragos & Izotov (2019) is shown by the vertical +hatched area. The points are annotated with the optically iden- +tified H ii region numbers associated with each ULX source (see +Table 1). +In Figure 9, we plot the He ii λ4686/Hβ ratio against the q +values for the 11 sources for which we have well-determined +values of LX. The He ii λ4686 line is detected in seven of +these sources, with the remaining four only having an upper +limit for the detection of the He ii λ4686 line. We find a +dispersion of more than 2 orders of magnitudes in the value +of q for the individual H ii regions in the Cartwheel, with +only one of these regions having q in the range found by +Schaerer, Fragos & Izotov (2019). This large variation in +the q value suggests that the X-rays cannot be the unique +source of ionization in these sources. It is likely that not +all ULX sources in the Cartwheel are HMXBs, and instead +the X-ray luminosity may be originating in supernova (SN) +remnants. Wolter, Fruscione & Mapelli (2018) discuss them +as HMXBs, whereas in the original detection papers (Gao +et al. 2003; Wolter & Trinchieri 2004), such a possibility +was firmly established for the only HLX source (#111) in +the Cartwheel. The q-value obtained for this source is more +than an order of magnitude lower than the value proposed by +Schaerer, Fragos & Izotov (2019). We analyse the fluxes of +lines from high ionization levels such as [Ar iv] to address the +role of X-ray ionization in each of the H ii regions associated +with an ULX source. +3 +ANALYSIS OF NEBULAR LINE RATIOS +The wealth of spatial and spectroscopic information con- +tained in the MUSE data of the Cartwheel offers us a great +MNRAS 000, 000–000 (0000) + +12 +Y. D. Mayya et al. +Figure 10. Relation between the He ii λ4686/Hβ ratio and EW(Hβ) for the Cartwheel H ii regions: circles are detections whereas triangles +are 3-σ upper limits. Filled symbols correspond to H ii regions associated to ULX sources. The radius of the circle is proportional to its +log(He ii λ4686/Hβ) ratio. All H ii regions where the He ii λ4686 line is detected as well as those without He ii detection, but with 3-σ +upper limits of He ii λ4686/Hβ<0.06 are identified by their H ii region number. +opportunity to comprehensively address the nature of ion- +izing sources based on the ionization state of the nebulae. +Specifically, the data allow us to study whether the H ii re- +gions containing the He ii λ4686 line show any difference +with respect to our control sample of 87 H ii regions in +the same galaxy, in any of the diagnostic line ratios com- +monly used in ionized nebulae (Baldwin, Phillips, & Ter- +levich 1981). +In this section, we discuss the general trends seen in dif- +ferent line ratio diagrams. We also discuss the possible role +of ULX sources in the ionization of He+. In Sec.4, we com- +pare the observed trends with that expected from different +theoretical scenarios of ionization of He+. +3.1 +He ii λ4686/Hβ, [O iii] λ5007/Hβ and EW(Hβ) +We start our analysis of the diagnostic line ratios by plotting +in Figure 10 the He ii λ4686/Hβ ratio versus EW(Hβ), which +is a standard indicator of age of stellar populations during +their early nebular phase, as illustrated in Figure 5. In this +and all the upcoming figures, we distinguish H ii regions with +and without the detection of the He ii λ4686 nebular line by +circles and inverted triangles, respectively. Filled symbols +show the H ii regions that have an associated ULX. Error +bars are shown only when the errors are significantly larger +than the symbol size. H ii regions with detected He ii λ4686 +line are annotated with their number designation from Ta- +ble 1. Care is taken to avoid superposition of the annota- +tions, but it was not always possible due to the crowding of +points in some of the plots. +Figure 10 shows that the He ii λ4686 line is detected in +seven of the 12 H ii regions with EW(Hβ)>100 ˚A. Among +the high EW(Hβ) regions without He ii λ4686 detection, re- +gion#96 is identified in the plot. This region lies slightly +outside the ring in the bright southern arc (see Figure 4). +The H ii regions with lower emission EW(Hβ) have, in gen- +eral, fainter nebular lines making the detection of the faint +He ii λ4686 line dependent of the SNR of each spectra (see +Figure 2). Three regions (#144, #17 and #148) standout in +the diagram, as they are among the regions with the lowest +EW(Hβ), but having the highest values of He ii λ4686/Hβ +ratio. In fact, these three regions exemplify a tendency for +MNRAS 000, 000–000 (0000) + +HeII in the Cartwheel galaxy +13 +Figure 11. (left) [O iii] λ5007/Hβ ratio vs. EW(Hβ); (right) He ii λ4686/Hβ vs. [O iii] λ5007/Hβ ratio, plotted against each other. +Symbols have the same meaning as in Figure 10. +the upper boundary of the He ii λ4686/Hβ ratio to increase +with decreasing EW(Hβ). +We show the [O iii] λ5007/Hβ ratio against EW(Hβ) on +the left panel of the Figure 11. The [O iii] λ5007/Hβ ratio is a +well-known indicator of the ionization state of an H ii region, +with high ionization regions having a higher value of the ra- +tio. We clarify that the error bars on the ratio are negligibly +small, including for those regions without the He ii λ4686 de- +tection (inverted triangles). Note that the higher EW(Hβ) +regions have higher [O iii] λ5007/Hβ ratio, independent of +whether the He ii λ4686 line is detected or not, with region +#99 (the brightest H ii region) and #148 (H ii region with +the highest He ii λ4686/Hβ ratio) lying at the extreme ends +of the relation shown by the rest of the regions. The region +#111 stands out from the relation for having a too high +ionization for its observed low EW(Hβ). We recall that this +source is the H ii region nearest to the HLX source. However, +its association with the HLX source is doubtful as discussed +in Sec. 2.6.3. Dilution of EW(Hβ) from a non-ionizing clus- +ter inside the aperture used for extracting the spectrum is +the most likely reason for this region to displace from the +observed correlation. This is supported by a visual exami- +nation of the HST images of this region, which reveals two +sources, with the source brighter in the F814W image the +likely non-ionizing cluster. +In +the +right +panel, +we +show +the +He ii λ4686/Hβ +ratio +against +the +[O iii] λ5007/Hβ +ratio. +As +expected, +He ii λ4686-line detection is more frequent in high ion- +ization H ii regions as compared to relatively low ion- +ization regions — it is detected in as much as 75 per +cent +(15 +out +of +21) +of +the +high +ionization +regions +(log([O iii] λ5007/Hβ)>0.60). Surprisingly, low ionization +H ii regions (log([O iii] λ5007/Hβ)<0.40) have a non-zero +detection frequency (10 per cent; two out of 20). The in- +creasing tendency for the upper boundary of He ii λ4686/Hβ +value with decreasing EW(Hβ) is also manifested in the +He ii λ4686/Hβ vs [O iii] λ5007/Hβ plot, with #99 and #148 +marking the endpoints of this tendency. +3.2 +Location of He ii-emitting H ii regions in +diagnostic diagrams +In order to understand the sources of ionization of He+ in +H ii regions of Cartwheel, we show in Figure 12 all our re- +gions in the most commonly used BPT diagrams (Bald- +win, Phillips, & Terlevich 1981). In the first three plots +(top two and the bottom left), we use lines of low ion- +ization potential in the x-axis, whereas the y-axis con- +tains [O iii] λ5007 line, which as discussed before origi- +nates in the high ionization zone. The H ii regions lie +along a sequence wherein the ratios of [N ii] λ6583/Hα, +[S ii] λ6717 + 6731/Hα and [O i] λ6300/Hα, systematically +increase as the [O iii] λ5007/Hβ ratio decreases. The frac- +tion of He ii-emitting regions decreases along the sequence +from top-left to bottom-right. In the bottom right panel, +we show line ratios that maximize the values for high ion- +ization regions on the y-axis and low ionization regions on +the x-axis. In this plot, the relation is much tighter than +in other plots with the He ii-emitting regions having the +lowest value of [O i] λ6300/[O iii] λ5007 ratio for any fixed +[O iii] λ5007/[O ii] λ7325 value. +In order to investigate the possible presence of shock +ionization/excitation +in +H ii +regions +having +He ii λ4686 +lines, in Figure 13 we plot the He ii λ4686/Hβ against +[O i] λ6300/[O iii] λ5007 ratio, which is sensitive to the pres- +ence of shocks (see e.g. Figure 15 in Plat et al. 2019). +The four regions with the highest values of He ii λ4686/Hβ +(#148, 17, 144 and 108) are indeed among the H ii regions +with the highest values of the shock-sensitive ratios. +MNRAS 000, 000–000 (0000) + +14 +Y. D. Mayya et al. +Figure 12. Cartwheel H ii regions in BPT diagrams following the same symbol convention as in Figure 10; (upper left) [O iii] λ5007/Hβ +vs. [N ii] λ6583/Hα ratio; (upper right) [O iii] λ5007/Hβ vs. [S ii] λ6717+6731/Hα ratio; (bottom left) [O iii] λ5007/Hβ vs. [O i] λ6300/Hα +ratio; (bottom right) [O iii] λ5007/Hβ vs. [O i] λ6300/[O iii] λ5007 ratio. See text for details. +The regions ionized by the ULX sources are expected +to have [O iii] λ5007/Hβ> 5 and [O i] λ6300/Hα>0.1, oc- +cupying the transition region between the Active Galactic +Nuclei (AGNs) and Low-Ionization narrow-emission line re- +gions (LINERs) in the BPT diagrams (G´urpide et al. 2022). +None of the ULX sources in the Cartwheel occupy these +zones, suggesting that the ULX sources have very limited +role, if any, in the ionization of the nebula with which the +ULX source is positionally coincident. +4 +DISCUSSION +We now investigate the source of He+ ionization in the +Cartwheel using the trajectory of theoretical models of ion- +ization in various diagnostic diagrams using the theoretical +line ratios calculated by Plat et al. (2019). In particular, +we use the diagnostic diagrams involving [O iii] λ5007/Hβ +ratio plotted against [N ii] λ6583/Hα and the EW(Hβ), +[O iii] λ5007/[O ii] λ7325 +vs. +[O i] λ6300/[O iii] λ5007, +He ii λ4686/Hβ +vs. +EW(Hβ), +He ii λ4686/Hβ +vs +[O i] λ6300/[O iii] λ5007 +and +[O iii] λ5007/[O ii] λ7325 +MNRAS 000, 000–000 (0000) + +HeII in the Cartwheel galaxy +15 +Figure 13. The He ii λ4686/Hβ ratio plotted against the shock- +sensitive [O i] λ6300/[O iii] λ5007 line ratio. The meaning of the +symbols are the same as in Figure 10. +vs +[Ar iv] λ4711 + 4740/[Ar iii] λ7135. +These +diagrams +are chosen because they are sensitive to the different +mechanisms of ionization explored in this work. +4.1 +Calculation of theoretical nebular line ratios +We consider two sources of ionization: photoionization, +which is the most dominant source of ionization in H ii re- +gions, and ionization by radiative shocks. Photoionization by +stellar clusters with and without binaries is considered. Since +the objects under study are H ii regions, the ionizing source +is better modelled as a single age cluster (SSP), rather than +a population of stars formed over a long period of time. We +hence consider only SSP models. However, the dilution of +EW(Hβ) caused by the presence of any non-ionizing source +(e.g. an underlying old stellar population) inside the aper- +ture used for extraction is taken into account. +The emission line fluxes from an H ii region are com- +puted with CLOUDY v17.02 (Ferland et al. 2017) following +the approach of Gutkin et al. (2016). We use C&B for single +star models and BPASS v2.2.1 for binary population mod- +els. The stellar and interstellar medium (ISM) metallicity +is set to Z = 0.003, which corresponds to the gas phase +oxygen abundance of 12 + log (O/H)gas ≈ 8 for the dust-to- +metal mass ratio, ξd=0.3 (the solar value is 0.36) following +Gutkin et al. (2016). The ISM is considered to be of uniform +hydrogen density of nH = 102 cm−3, which along with the +rate of ionizing photons and filling factor sets the ioniza- +tion parameter U. Models are parametrized in terms of the +zero-age volume-averaged ionization parameter ⟨U⟩, which is +varied by varying the filling factor, see Gutkin et al. (2016) +for details. We note that the values of 12 + log (O/H)gas, +log (N/O)gas and ne used here are based on the mean val- +ues derived using the same MUSE dataset for the Cartwheel +ring H ii regions in a companion paper (Zaragoza-Cardiel et +al. 2022). +The H ii regions are assumed to be ionization-bounded, +but contain dust grains that compete with gas in the ab- +sorption of ionizing photons. We also discuss the effect on +the line ratios if the H ii regions are density bounded, i.e. +when the size of the H ii region is limited by the gas density, +rather than the ionizing photon rate. We account for the +possible presence of holes or cavities in the H ii regions by +means of an escape fraction of ionization photons. In addi- +tion to these two escape geometries, we follow the approach +of Ramambason et al. (2020) and compute two-zone models, +combining a low and high ionization parameter component. +These two zones are either both ionization bounded, or one +of them is density bounded. +The emission from fast radiative shocks is added using +the models of Alarie & Morisset (2019)3 for the full set of +shock velocities (100 to 1000 km s−1) available in these mod- +els. A pre-shock density of 102 cm−3 and transverse magnetic +field strength B = 1µG, which are typical values for the dif- +fuse ISM in galaxies, are used. The calculated line ratios +depend weakly on these fixed values, as compared to the +variation in shock velocities. The effects of shocks are added +at representative ages, which allows us to illustrate the effect +of shocks on line ratios as the cluster evolves. +4.2 +The line ratio sequence from photoionization +models +In Figures 14 and 15, we overplot tracks for photoionization +models for different initial log⟨U⟩ values using C&B SSP +models without binary stars, and BPASS models with bi- +nary stars. The ratios for leaky H ii regions, as well as for +H ii regions experiencing radiative shocks are also explored. +Additionally, the effect of an underlying old population is +illustrated. We also explored the effect of binaries using +BPASS models. We refer the reader to Plat et al. (2019) +for a detailed illustration of the sensitivities of the explored +parameters on the commonly observed optical and ultravi- +olet line ratios. In different plots, we plot sequences of age, +or log⟨U⟩ or shock velocities, depending on the sensitivity +of the plotted quantities on the models. These are explained +in annotations and legends of the corresponding figures. +The cluster evolution is followed up to 50 Myr with +three epochs (0, 3, and 4 Myr) marked with differently +shaped symbols (in some of the plots, the plotted range cov- +ers only the trajectory around 3–4 Myr). The initial log⟨U⟩ +between −1 and −4 are explored. In shock models, shocks +are assumed to provide 25 per cent of the observed Hβ flux. +The sequences are formed for shock velocities between 100 +to 1000 km s−1. High velocity shocks are expected in H ii +regions following the explosion of SN, which start occurring +at an SSP age of ∼4 Myr. Hence, the shock component is +added to the 4 Myr H ii region emission with log⟨U⟩ = −2.5. +These are shown in Figures 14, 16 and 17 by purple lines. In +models involving escape of ionizing photons, the sequence is +3 The ISM metallicity in the shock models corresponds to the +SMC metallicity, which is slightly lower than that used for our H ii +regions. The nitrogen to oxygen abundance ratio is also lower in +these models as compared to that used in our H ii region models. +MNRAS 000, 000–000 (0000) + +16 +Y. D. Mayya et al. +Figure 14. Theoretical models superposed on the observed points of the Cartwheel in the line ratio diagnostic diagrams illustrated in +Figure 12. Top panels: C&B model sequences, formed by varying the initial log⟨U⟩ from −1 to −4, at 0, 3 and 4 Myr (solid yellow lines). +The evolutionary tracks at log⟨U⟩=−2, −2.5 and −3 are shown by black dashed lines. Effect of regions being density-bounded (solid +green lines at log⟨U⟩=−2.5), two-zone escape (red lines joining log⟨U⟩=−1 and −3.5 at 3 Myr), and shock models (purple tracks at 4 Myr +and log⟨U⟩=−2.5) are illustrated at selected locations in the diagrams. The numbers along the two-zone model sequences correspond to +the fractional contribution to the Hβ flux from the high log⟨U⟩ zone, and the numbers in the shock models correspond to shock velocities +that range from 100 to 1000 km s−1. Bottom panels: BPASS binary SSP age sequences for two values of initial log⟨U⟩. See the inset for +identification of various lines and symbols, and Sec. 4.2 for more details of the models used in these plots. +formed by varying the fraction of Lyman continuum (LyC) +photons escaping the H ii regions. In two-zone models, the +sequence is formed by varying the fractional contribution to +the Hβ flux from the high log⟨U⟩ zone. These are shown in +Figures 14, 16 (right) and 17 with red lines marked with +dots for increase of the escape fraction between 0.1 and 0.9. +We now discuss the results on the ionization mecha- +nisms suggested by the models based on the comparison of +the locus of model parameters with observations, in selected +line-ratio diagrams. We first discuss the results based on +C&B models without binary stars, and at the end, comment +on the effect of having binary stars in the SSPs. +MNRAS 000, 000–000 (0000) + +1.0 +2.5 +Oyr +0.9 +Oyr +3Myr! +99 +3Myr +66 +0.9 +112 +4Myr +0 +96 +0 +0.8 +112 +0 +0.8% +0 +4Myr +96 +2.0 +0 +40 +0.7 +[OI]5007/[OI]7325 +0.7 +1og= +O +log= +Q +00.6 +144 +A +0.5 +4 +log=-2.5 +1og=-2.5 +1.5 +QSLA +148 +1000 +100 1000 +0.3 +Oyr +2 zones +100 +0.2 +3Myr +2 zones fesc log(U) +4Myr +2 zones fesc log(U) +0.1 +SSP, C&B, log(U> sequence +shocks + SF +log: +fesc density bounded +log +0.01 +0.Q +1.0 +1.75 +-1.50 +1.25 +-1.00 +2.5 +2.0 +1.5 +1.0 +log [NII]6583/Hα +1og [O]6300/[O1I]5007 +1.0 +2.5 +log=-1.5 +112 +log=-1.5 +99 +96 +112 +0 +9o +0 +96 +& +. +2.0 +0 +Ov +80 +40 +; [I]5007/[OI]7325 +II)5007/Hβ +0 +40 +0 +088 +0.5 +0 +IHIOI +1440. +444 +log +O +1.5 +148 +Oyr +3Myr +4Myr +15Myr +log=-2.5 +log=-2.5 +SSP, BPASS, age sequence +0.Q +1.0 +1.75 +-1.50 +-1.25 +1.00 +-2.5 +-2.0 +-1.5 +-1.0 +log [NI]6583/Hα +10g [O1]6300/[O11I]5007HeII in the Cartwheel galaxy +17 +Effect of log⟨U⟩: The density of the ISM surround- +ing the cluster, the rate of ionizing photons, and the vol- +ume filling factor of the ionized gas fix the initial log⟨U⟩ +of the models. In the top-left panel of Figure 14, it can +be seen that the observed sequence of [O iii] λ5007/Hβ val- +ues can be understood as due to a dispersion in the ini- +tial log⟨U⟩ values, with the ionizing clusters in almost all +of the H ii regions having ages between 3 and 4 Myr in +C&B models. The observed range of line ratios is cov- +ered by the models with log⟨U⟩=−1.5 to −3.0, with the +small variations in the age being responsible for the ob- +served spread in the direction perpendicular to the sequence. +The observed sequence in the [O iii] λ5007/[O ii] λ7325 vs +[O i] λ6300/[O iii] λ5007 plane (top-right) is also consistent +as a sequence of initial log⟨U⟩. However, the inferred age +from this diagram is ∼0 Myr, rather than 3–4 Myr inferred +from the [O iii] λ5007/Hβ vs [N ii] λ6583/Hα diagram. The +latter diagram is sensitive to the selective escape of photons +from either the low or the high ionization zones as will be +discussed in the two-zone models below. +Effect of dust in H ii regions: All our models include dust +inside the H ii regions. The quantity of dust is parametrized +by the dust-to-metal mass ratio ξd. The dust competes with +gas in the absorption of ionization photons following the +scheme proposed by Bottorff et al. (1998). Hence the number +of ionizing photons absorbed by hydrogen, and all properties +that depend on the ionizing photon flux, predicted by our +models is lower than those predicted for dust-free ionization- +bounded models. The optical depth of ionizing photons aris- +ing from dust is proportional to the total hydrogen column +density, which is proportional to the ionization parameter. +So as the ionization parameter increases, so does the ab- +sorption of ionizing photons by dust rather than hydrogen. +This effect can be noticed in Figures 15 and 16, where the +EW(Hβ) decreases with log⟨U⟩ at a fixed age (see also Erb +et al. 2010; Plat et al. 2019). +Effect of upper cut-off mass of the IMF: The SSPs we +used for the calculation of nebular quantities correspond to +Mu=300 M⊙. The non-detection of BB (see Sec. 2.6.1) in +our spectra suggests the absence of stars more massive than +100 M⊙ in Cartwheel H ii regions. However, the results for +Mu=100 M⊙ are identical to that for Mu=300 M⊙ after +3.2 Myr, as can be inferred from Figures 7 and 8. The ex- +pected values of He ii λ4686/Hβ before and during the WR +phase with Mu=300 M⊙ is only marginally higher than that +for Mu=100 M⊙. Thus, the results presented here are not +sensitive to the choice of upper cut-off mass as long as the +SSP contains hot massive-stars that go through the WR +phase (Mu >25 M⊙). +Effect of cluster evolution: The rate of ionizing photons +emanating from a cluster starts decreasing when the most +massive stars, also the hottest and the most luminous, end +their main sequence life time. For clusters with the high- +est mass ∼100 M⊙, this starts happening at ∼3 Myr at +Z=0.003, the metallicity corresponding to the Cartwheel. +This decreases log⟨U⟩ by ∼0.6 dex over the first 10 Myr, +which leads to a gradual decrease of the high ionization line +intensity ratios such as [O iii] λ5007/Hβ. The EW(Hβ) de- +creases slowly in the first 3 Myr reaching values ∼200 ˚A at +3 Myr, beyond which it drastically drops by more than an +order of magnitude in ∼10 Myr. The decrease of log⟨U⟩ and +the EW(Hβ) with age leads to a decrease of [O iii] λ5007/Hβ +ratio as the EW(Hβ) decreases. +The evolutionary track in C&B models for a given ini- +tial log⟨U⟩ in Figure 15 follows the observed relation for +clusters younger than ∼4 Myr beyond which the model- +predicted [O iii] λ5007/Hβ ratio is much smaller than that +expected from the observed relation. Consequently, H ii re- +gions with EW(Hβ)<50 ˚A are not expected to have de- +tectable levels of the [O iii] λ5007 line emission. The conti- +nuity of the observed sequence in this diagram for the whole +range of EW(Hβ) suggests that the Cartwheel H ii regions +are indeed ionized by clusters younger than 4 Myr, and some +physical effect is responsible for lowering the observed values +of EW(Hβ) compared to that predicted in the SSP models +we used. The ionization sequence in Figure 14 also implies +that all our H ii regions are younger than 4 Myr. Escape of +ionizing photons, presence of density-bounded H ii regions, +presence of an older underlying population, are some of the +physical processes that we have explored in this work to ex- +plain the decrease of EW(Hβ) without the corresponding +decrease in [O iii] λ5007/Hβ ratio. +Effect of an older population: H ii regions often contain +a population other than that is ionizing the surrounding gas +inside the aperture used for spectral extraction(e.g. Char- +lot & Fall 1993; Mayya & Prabhu 1996). The non-ionizing +population could be a cluster older than the ionizing clus- +ter, or it could be the underlying disk population. Given the +recent star formation history of the Cartwheel, it is likely +that the apertures used for spectral extraction (740 pc di- +ameter) contain non-ionizing clusters of ∼10 Myr or slightly +older. Results from the recent numerical simulation of the +wave propagation in the Cartwheel by Renaud et al. (2018) +support the existence of a spread of this order in the ages +of clusters in the outer ring. Multiple clusters within the +extracted apertures can be directly seen in Figure 4 at the +spatial resolution of the HST images. The presence of such +a cluster would decrease the EW(Hβ) without affecting the +line ratios, and hence would move the points horizontally +in Figure 15. The EW(Hβ) would be affected by a larger +amount for larger mass of the non-ionizing cluster. In the left +panel, we show the effect of a non-ionizing cluster of 10 Myr +age for two values of relative masses: (1) equal masses, and +(2) the non-ionizing cluster 10 times more massive than the +ionizing cluster. This effect is shown by crosses placed at +3 and 4 Myr of age for the ionizing cluster. A part of the +observed horizontal spread could be due to the presence of +different amounts of mass in old stellar clusters inside the +apertures used for extraction. +Effect of escape of ionizing photons: We have assumed +that all the ionizing photons that are emitted by the clus- +ters are either used in the ionization, or absorbed by in- +ternal dust. However, there are 2 geometries by which ioniz- +ing photons can escape the nebula without getting absorbed +by gas and dust: (1) escape through holes, and (2) escape +through density-bounded zones. The former case results in +the decrease of the intensity of all lines, hence a decrease +in the EW(Hβ), without changing the line ratios. Thus, the +escape of ionizing photons through holes would move the +points horizontally in Figure 15, producing an effect indis- +tinguishable from the presence of a non-ionizing population +discussed above. The second case is discussed below. +Density bounded models: H ii regions have an ionization +MNRAS 000, 000–000 (0000) + +18 +Y. D. Mayya et al. +Figure 15. +Comparison of the observed values for the Cartwheel H ii regions (circles and small inverted triangles; see caption of +Figure 10 for their meaning) with theoretical models. Trajectory of evolutionary tracks for evolving stellar populations at two values of +initial log⟨U⟩=−1.5 and −2.5 are shown (dashed black lines) with the points on the two tracks at selected ages joined by yellow lines. +The left panel shows the C&B SSP tracks without binaries, where we also show the effect of escape of ionizing photons for two escape +geometries (holes and density bounded) at ages 3 and 4 Myr, and the effect of adding a non-ionizing cluster of 10 Myr that is 1 and 10 +times more massive than the ionizing cluster. In the right panel, we show the BPASS SSP models including binary stars. +structure with the lines of high ionization (e.g. [O iii] λ5007) +originating in zones closer to the ionizing cluster as com- +pared to the lines of low ionization (e.g. [N ii]λ6583). In +density-bounded H ii regions, the ionizing photons are lost +due to insufficient amount of gas to trap all the ionizing +photons. Such regions would have a reduced intensity of low- +ionization lines and EW(Hβ) as compared to the values for +ionization-bounded regions. This would lead to an increase +of [O iii] λ5007/Hβ as EW(Hβ) decreases, which is just the +opposite of what is observed in Figure 15. Hence, density- +bounded models cannot explain the continuation of the se- +quence to low EW(Hβ). Furthermore, the observed sequence +in [O iii] λ5007/[O ii] λ7325 vs [O i] λ6300/[O iii] λ5007 ratio +is not consistent with density-bounded models. +Two-zone escape models: In order to reproduce the +observed large range of [O i] λ6300/[O iii] λ5007 ratios in +LyC galaxies, Ramambason et al. (2020) proposed a model +wherein the nebulae loose ionizing photons selectively from +the high-U or low-U zones. Such a configuration is naturally +expected if the ISM around the clusters is non-uniform, and +contains dense clumps and filaments. Results for two-zone +models are calculated by combining the spectrum of a high +initial log⟨U⟩ nebula with a second spectrum of a low ini- +tial log⟨U⟩ nebula, with the relative luminosity of the Hβ +lines in the two spectra used as a free parameter. This free +parameter, ω, varies between 0 and 1, which is defined as +the fractional contribution of the high log⟨U⟩ spectrum to +the total Hβ luminosity. Thus, ω=1 corresponds to a neb- +ula containing only high ionization zone and ω=0 corre- +sponds to a nebula containing only low ionization zone. The +ionization-bounded or density-bounded status of the two +zones are independent of each other. In Figure 14 (right), +we show the locus of these models by red lines for three +cases, all corresponding to 3 Myr age clusters with initial +log⟨U⟩=−1 and log⟨U⟩=−3.5 for the high and low ioniza- +tion zones, respectively. The 3 cases are: (1) both the zones +are ionization-bounded (solid line), (2) combination of an +ionization-bounded low log⟨U⟩ zone, with a density bounded +high log⟨U⟩ zone with an escape fraction, fesc ∼ 30 percent +(dashed line), and (3) combination of an ionization-bounded +high log⟨U⟩ zone, with a density bounded low log⟨U⟩ zone +with fesc ∼ 10 percent (dotted line). Values for the free pa- +rameters are chosen that best illustrate the observed trends +in Figure 14 (right). C&B ionization-bounded models (yel- +low lines) even for the youngest age lie slightly to the left of +the observed points in this figure. The tracks for ages of 3 +and 4 Myr are further away from the observed points. On the +other hand, the two zone escape models are able to explain +the dataset for cluster ages of 3 Myr — most of the observed +points lie between the dashed red lines (3 Myr track with se- +lective escape of ionization photons from high log⟨U⟩ zone) +and the yellow line for the 3 Myr track (ionization bounded +regions without any escape). Thus, with two-zone models, +it is possible to obtain consistent ages of ∼3–4 Myr for the +majority of the H ii regions in the Cartwheel, from both the +diagnostic diagrams in Figure 14. +Shocks: Some of the H ii regions, though principally pho- +toionized by the stars, can also experience shocks, especially +during the post-main sequence evolution and the death of +massive stars. In order to illustrate the effect of shocks in +MNRAS 000, 000–000 (0000) + +1.0 +1.0 +log=-1.5 +O yr +3 Myr +4 Myr +99 +15 Myr +99 +log=-1.5 +0 +112 +112 +0 +0 +0 +0 +O yr +. +0 +0 +0 +0 +.96 +96 +[15007/Hβ +3 Myr +0 +V +0 +log=-2.5 +0.5 +[1HO] +0.5 +0 +C +0 +00 +w v144 +4 Myr +0 +1400 +log=-2.5 +Oyr +3Myr +Oyr +fesc density bounded +4Myr +3Myr +fesc holes + M-old = M-young +15Myr +4Myr +SSP, BPASS, age sequence +SSP. C&B, age sequence +10 M-young +M-old +0.0 +0.0 +10 +50 +50 +100 +500 +100 +500 +10 +EW(Hβ) [A] +EW(Hβ)[A]HeII in the Cartwheel galaxy +19 +the line ratio diagrams, shock models of Alarie & Morisset +(2019) are added to the evolutionary track of SSP mod- +els at one particular epoch (4 Myr) with log⟨U⟩ = −2.5. +The shocks are parametrized by 3 parameters: the percent- +age of energy in shocks compared to photoionization en- +ergy to ionize hydrogen, the velocity of the shock and the +shock age. We used their grid of truncated shock models. +We here plot models where 25 percent of the total Hβ flux +is provided through shocks for various values of shock veloc- +ities. At each velocity, we plot the ratios as the shock prop- +agates into the ISM. Increasing the shock velocity increases +the line ratios involving high ionization, whereas the low +ionization lines become stronger as the shock propagates. +The effect of shocks on the line ratios can be best seen in +the [O iii] λ5007/[O ii] λ7325 vs. [O i] λ6300/[O iii] λ5007 di- +agram of Figure 14 (top right panel). High velocity shocks +tend to move the points to the right, mostly occupied by +H ii regions not showing He ii λ4686 line. Among the re- +gions with He ii λ4686 emission, shocks could be important +in low-ionization H ii regions such as #148. On the other +hand, shocks have very little effect in the [O iii] λ5007/Hβ +vs [N ii] λ6583/Hα plot (top left panel), and no effect on the +EW(Hβ). +Cluster evolution with binaries: We now discuss the ef- +fect of binary population on the results obtained from C&B +models. For this purpose, we use the BPASS binary mod- +els plotted in the bottom panels of Figures 14, and right +panel of Figure 15. The result that the observed sequence of +points is a sequence in log⟨U⟩ holds even after including bi- +nary stars in the SSPs. Star clusters containing binary stars +produce the ionizing photons over an extended period of +time (>15 Myr) as compared to the evolution without bina- +ries. This causes both the [O iii] λ5007/Hβ and the EW(Hβ) +to decrease more slowly with evolution as compared to the +evolution of clusters without binary stars. The evolution- +ary locus and log⟨U⟩ sequences in binary models are al- +most parallel in Figure 15, both following the ionization +sequences seen in these diagrams. Thus, under the binary +models, Cartwheel H ii regions could have an age range be- +tween 0 and 15 Myr and log⟨U⟩ range between −1.5 to −3.5. +However, at ages as late as 15 Myr, the EW(He ii λ4686) de- +creases by more than an order of magnitude (see Figure 5). +Hence, it is highly unlikely that the He ii λ4686 detections +correspond to the faint late phase, rather than the luminous +early phase. +Thus, the inclusion of binary models does not qualita- +tively change the conclusions arrived from using C&B mod- +els, which do not take into account the possible presence of +binary stars. +4.3 +On the ionization state of Cartwheel H ii +regions +The observed range of line ratios in the H ii regions of the +Cartwheel corresponds to photoionization by young clusters +(age∼3–4 Myr in C&B models or 3–15 Myr in BPASS bi- +nary models) with initial H ii-averaged ionization parameter +log⟨U⟩ lying between −1.5 to −3.5. The [O iii] λ5007/Hβ +ratio, which is a direct measure of the degree of ioniza- +tion of an H ii region, is correlated with EW(Hβ) over the +entire observed range of both the quantities. This correla- +tion suggests a systematic decrease of the [O iii] λ5007/Hβ +ratio, or equivalently log⟨U⟩, with age. For the ionization- +bounded models that we have used, log⟨U⟩ decreases only +by 0.6 dex or alternatively [O iii] λ5007/Hβ by 0.2 dex in +4 Myr in C&B SSP models and 0.1 dex in 15 Myr in BPASS +binary SSP models. The amount of change of this ratio in +SSP models depends on the metallicity, with the values in +our models being consistent with the values obtained by +Stasinska & Leitherer (1996) for the Cartwheel metallicity. +Thus, an age-dependent process of softening of log⟨U⟩ is re- +quired in order to interpret the observed correlation between +[O iii] λ5007/Hβ and EW(Hβ) primarily as an age sequence. +Kim, Kim & Ostriker (2019) found that the fraction of +the escape of the ionizing photons systematically increases +with age of the cluster due to the increased amount of feed- +back with age. The escape of ionizing photons has been long +suspected to be one of the reasons for the low EW(Hβ) of +H ii regions (e.g. Mayya & Prabhu 1996). Castellanos, Di´az +& Tenorio-Tagle (2002) determined escape fraction between +0.1–0.7 for three H ii regions they analysed. The mechanical +energy feedback to the ambient ISM can also decrease log⟨U⟩ +due to the decrease in the density following the feedback- +driven expansion of the H ii regions. Mart´ın-Manj´on et al. +(2010) found that this effect can decrease log⟨U⟩ by as much +as 3 dex at the Cartwheel metallicity. All these suggest that +the evolution-dependent feedback is driving the observed +correlation through the escape of ionizing photons and the +decrease in the ISM density. Thus, statistically, the high +excitation regions are systematically younger than the low +excitation regions. +We also investigated the effect of radiative shocks in H ii +regions primarily photoionized by clusters as described in +Sec. 4.2 above. High velocity shocks tend to increase the line +ratios involving low ionization ions such as the [O i] λ6300 +line (e.g. Stasinska et al. 2015), which would move the points +to the right of the ionization sequence in the right panels of +Figure 14. A tendency of broadening of the sequence in this +diagram for low-ionization H ii regions is seen, suggesting a +possible presence of shocks in some low log⟨U⟩, which are +relatively older, H ii regions. +4.4 +The ionizing source of He+ in the Cartwheel +H ii regions +Having addressed the ionization mechanism and physical +processes prevalent in the Cartwheel H ii regions, we now +investigate whether the same physical mechanisms account +for the observed He ii λ4686/Hβ ratio. We have chosen +two plots to verify this: He ii λ4686/Hβ vs EW(Hβ) and +He ii λ4686/Hβ vs [O i] λ6300/[O iii] λ5007. These are shown +in Figure 16 for the chosen theoretical tracks from C&B +models, along with special scenarios discussed in the para- +graphs above. We do not show the plots with BPASS bi- +nary models, as the He ii λ4686/Hβ ratios produced by these +models are systematically lower than the observed values as +illustrated in Figure 6. +4.4.1 +He++ nebulae photoionized by WR stars +The majority of He ii λ4686-emitting regions fall between +the tracks corresponding to ionization-bounded H ii regions +photoionized by clusters of age between 3 and 4 Myr. This +MNRAS 000, 000–000 (0000) + +20 +Y. D. Mayya et al. +Figure 16. (left) He ii λ4686/Hβ vs. EW(Hβ); (right) He ii λ4686/Hβ vs. [O i] λ6300/[O iii] λ5007 ratio. See the guide to the lines and +symbols in the box below the graphs and the text in the beginning of Sec. 4.2 for the explanation of parameters for the shock and 2 zone +ionization models, and see caption of Figure 10 for the meaning of symbols for the observed points (circles and small inverted triangles). +age range corresponds to the WR-phase in C&B models, as +can be inferred from Figure 6. +Escape of ionizing photons through holes, and/or the +presence of non-ionizing population: The log⟨U⟩ values +inferred +for +each +region +using +the +He ii λ4686/Hβ +vs +EW(Hβ) are systematically higher by around 1 dex as +compared to that inferred from the He ii λ4686/Hβ vs +[O i] λ6300/[O iii] λ5007 plot. This can be explained as due +to the escape of ionizing photons through holes, and/or the +presence of a spatially close older cluster in majority of the +Cartwheel H ii regions, both of which displace the tracks +horizontally without changing the flux ratios of the nebular +lines. This inference is consistent with the observed correla- +tion between [O iii] λ5007/Hβ ratio and EW(Hβ). +Is the ionization by WR stars the only way to explain +all the observed line ratios. Is the source of ionization of +He+ the same as that of other ions? In order to address +these questions, we here summarize the results from other +scenarios that we have explored. +Density-bounded models: The loci of density-bounded +H ii regions in the line ratio diagrams are shown by green +solid lines in the two panels of Figure 16 for C&B mod- +els of initial log⟨U⟩=−2.5. The length of the plotted lines +correspond to 50 percent of the ionizing photons escaping +the nebula from the density bounded zones. The locations +of the observed points, with the exception of regions #144, +#17 and #148, can be reconciled with this scenario, but for +a lower initial log⟨U⟩ value, as compared to the ionization- +bounded case. However, the observed ionization sequences +(see Figure 15) are not consistent with a low value of log⟨U⟩. +Thus, we rule out the possibility that the majority of the +Cartwheel H ii regions are density-bounded with a low ini- +tial log⟨U⟩. +Role of radiative shocks and two-zone escape models: In +Figure 16, the radiative shocks and two-zone escape mod- +els are plotted for a 3 Myr old ionizing cluster in the right +panel, with the aim of covering the observed ratios of re- +gions #144, #17 and #148. It can be inferred from the plot +that these scenarios, especially with escape from high log⟨U⟩ +zone (dashed red line) would cover the observed range of +values for the majority of the regions if the ionizing cluster +is ∼4 Myr old. In fact, we have considered such a possi- +bility to explain the behaviour of points in the right panel +of Figure 16. However, these scenarios produce higher than +the observed values of [Ar iv] λ4711+4740/[Ar iii] λ7135 line +ratio, as will be discussed later in this section. Thus, we do +not find it necessary to look beyond the WR stars to ex- +plain the observed He++ nebulae. In summary, the majority +of the He++ nebulae in the Cartwheel are photoionized by +WR stars, with the H ii regions enclosing the He++ nebulae. +Different observed quantities can be consistently explained +with ∼50 percent of the hydrogen ionizing photons escap- +ing through holes, and/or the presence of older non-ionizing +populations inside the aperture used for extraction. +MNRAS 000, 000–000 (0000) + +140 +1000 +3 Myr +250 +250 +1.5 +0 +0.8 +3 Myr +0.7 +0.6 +.00.5 +0 +..0.4 +..0.3 +'oo +100 +0100 +HelI4686/Hβ +HelI4686/Hβ +O +0.1 +0.017 +-2.0 +0 +-2.0 +0 +8 +80 +4 Myr +0 +I +0 +77 +log +8 +O yr +. +04 +99 +199 +4 Myr 1 +I +1 +1 +log=-2.5 +1 +-2.5 +112 +-2.5 +112 +I +- +O yr +1 +Iv +v +- +log=-2 +96 +96 +log=-2.5 +log=-3 +log=-3.5 +log: +-3.0 +-3.0 +10 +50 +100 +500 +-2.5 +-2.0 +-1.5 +-1.0 +EW(Hβ)[A +10g [O1]6300/[O1II]5007 +Oyr +fesc holes +3Myr + 2 zones +4Myr +2 zones fesc log(U> = -1 +SSP, C&B, log(U> sequence +2 zones fesc log = +-3.5 +fesc density bounded +shocks + SFHeII in the Cartwheel galaxy +21 +4.4.2 +He++ nebulae requiring alternative sources of +ionization +Exceptions to the above scenario are five H ii regions that +standout from the main group. Two (#99 & #112) are at +the high-EW(Hβ) end, and the other three (#144, #17 & +#148) are among the lowest EW(Hβ) H ii regions. +Ionization +by +ULX +source +and +stripped +binary +stars: +The +two +high-EW(Hβ) +regions +also +have +the +highest +[O iii] λ5007/Hβ +ratio +and +the +lowest +[O i] λ6300/[O iii] λ5007 +ratio, +all +indicating +that +these +regions have the highest ionization parameter, and the +youngest of the sample regions. We infer log⟨U⟩∼ −2.0 and +an age corresponding to the pre-WR phase, suggesting that +the hot main-sequence stars are the most likely sources of +He+ ionization in these two sources. +Region #99 is the brightest, and the most massive H ii +region in the Cartwheel with an estimated number of more +than 11,000 O stars, assuming O stars are the sole sources +of ionization of hydrogen. This region is associated with an +ULX source, and hence we discuss here the possible role +of this ULX source in the ionization of He+. The observed +[O iii] λ5007/Hβ is high enough as expected for the ioniza- +tion by the hard radiation from a ULX source. However, +the observed [O i] λ6300/Hα ratio for this regions is signifi- +cantly lower than that expected for the ionization by a ULX +source (G´urpide et al. 2022), and hence ULX cannot be the +sole or principal source of ionization of this region. The line +ratio diagrams presented suggest that the photoionization +by the main-sequence stars is a viable source of ionization. +The ULX may provide additional photons for the ioniza- +tion of He+. The observed EW(Hβ), which in spite of being +the highest among the sample regions, is still more than +a factor of two lower than that expected for single burst of +age<3 Myr, suggests the presence of underlying non-ionizing +populations. It is likely that the star formation in the region +is proceeding for more than 3 Myr or that it had a star for- +mation event in the recent past. These slightly evolved stars +had enough time to form HMXBs that are generating the +X-rays emitted by the ULX source (Plat et al. 2019). +Region #112 has the lowest value of He ii λ4686/Hβ +among the regions studied here, with a value intermediate +between that of main-sequence and WR phases. Like in the +case of #99, all line ratio diagrams presented here suggest +ionization of He+ by main sequence stars. +Regions affected by radiative shocks and/or two-zone +escape models: We now discuss the ionization mechanism +of He+ in regions #148, #17 and #144, the three low- +EW(Hβ) regions with the highest ratio of He ii λ4686/Hβ. +These three regions are at the low log⟨U⟩ end of the ion- +ization sequence. The He ii λ4686/Hβ ratios predicted by +the traditional ionization-bounded case and photoionized by +stellar radiation are much lower at these low log⟨U⟩ values +as compared to the observed values for these three regions. +Two of the various processes we have explored produce the +observed high He ii λ4686/Hβ ratios at low log⟨U⟩ values. +These are (1) radiative shock contribution, and (2) two-zone +escape models with ∼50 percent escape from the high log⟨U⟩ +zone. Regions #144, #17 and #148, in that order, lie on a +sequence of increasing shock velocities with shock veloci- +ties in the 100–1000 km s−1 range, with the He ii λ4686/Hβ +value of #148 only produced by shock models. The two- +Figure +17. +[O iii] λ5007/[O ii] λ7325 +vs. +[Ar iv] λ4711 + +4740/[Ar iii] λ7135 ratio. [Ar iv] lines are detected only in the +regions indicated by the black circles, with error bars. All red +coloured symbols (circles and triangles) are 3-σ upper limits on +the [Ar iv] λ4711 + 4740/[Ar iii] λ7135 ratio. The box below the +plot contains a guide to the models plotted. See text for details. +zone escape models can also explain their location in the +plots. Their low EW(Hβ) and low log⟨U⟩ suggest that these +three regions are more evolved than the rest of the regions, +and are likely to be in the post-WR phase. We hence favour +shock ionization associated with SN explosions as the most +likely causes of the observed high values of He ii λ4686/Hβ +ratio. SN explosions can also create escape routes for ion- +izing photons from the high log⟨U⟩ zone, and hence shocks +and two-zone escape scenarios could be co-existing. Region +#144 is associated with a ULX source whose X-ray lumi- +nosity is high enough to contribute to He+ ionization. Thus, +X-ray ionization could also be prevalent in #144. +Stasinska et al. (2015) has advocated the use of the +[Ar iv] λ4711 + 4740/[Ar iii] λ7135 ratios to test the hard- +ness of the ionizing spectrum, especially for the He ii λ4686- +emitting regions. Ar++ has an ionization potential of 40.7 ev +and hence the collisionally excited [Ar iv] lines are ex- +pected to be present in He ii λ4686-emitting regions. The +[Ar iv]λλ4711,4740 doublet is relatively faint. Nevertheless, +the doublet is detected at more than 3σ levels in 14 of the +32 He ii λ4686-emitting regions. In Figure 17, we plot the +[O iii] λ5007/[O ii] λ7325 ratios against the [Ar iv] λ4711 + +4740/[Ar iii] λ7135 line ratio. For photoionized nebulae, the +locus of points in this diagram is mainly governed by log⟨U⟩, +with the observed points lying between −3.0 +115 +更 +2.0 +[OI]5007 / [OI]7325 +log +2.5 +0.6 +O +0 +0 +0 +140 +0.5 +Y +log +500 +0.4 +1.5 +1000 +0.3 +100 +1.0 +-1.0 +0.0 +0.5 +log [ArIV]4711+40/[ArIII]7135 +Oyr +2 zones +3Myr +2 zones fesc log(U> = -1 +4Myr +o--- 2 zones fesc log = -3.5 +SSP, C&B, log sequence + shocks + SF +fesc density bounded22 +Y. D. Mayya et al. +small spread in the direction perpendicular to the sequence +formed by ranges of initial log⟨U⟩. On the other hand, the +presence of radiative shocks and/or selective escape of pho- +tons from low or high log⟨U⟩ zones moves the points to the +right (i.e. higher [Ar iv] λ4711 + 4740/[Ar iii] λ7135 ratios). +Thus, this figure is useful to discriminate the purely pho- +toionized models from the photoionized+shock models, or +the models involving selective escape of photons. Unfortu- +nately the last two cases follow similar trajectories in the +diagram, and hence cannot be distinguished. +The main-group of the He ii λ4686-emitting regions in +which [Ar iv] lines have been detected lie along the pho- +toionization sequence by the SSP models, and more impor- +tantly, are not consistent with the presence of shocks and/or +two-zone escape scenarios, independent of the age of the +ionizing clusters. Thus, this diagram helps us to break the +degeneracy seen in Figure 16. Region #99 lies clearly to +the right of the photoionization sequence, reiterating that +the hard radiation from the ULX source plays a role in the +ionization of ions that have higher than 40 eV of ioniza- +tion potential. The second region in which we cannot rule +out ionization by the ULX source is #144. The [Ar iv] lines +are not detected in this region, but the upper limit on the +[Ar iv] λ4711+4740/[Ar iii] λ7135 ratio is slightly to the right +of the photoionization sequence by the SSP models. Unfor- +tunately, we have only upper limits on the detection of the +[Ar iv] lines in the remaining two interesting regions #148 +and #17. The observed upper limits in #148 and #17 are +consistent with the presence of shocks and/or two-zone es- +cape scenarios. +4.5 +Regions with non-detection of He iiλ 4686 line +We here carry out an analysis on the nature of the H ii re- +gions where the He ii λ4686 line could not be detected at the +3-σ confidence level. Given that the WR stars are the prin- +cipal sources of He+ ionizing photons in our sample regions, +and that the WR stars appear only for a short duration in +an SSP, this fraction is expected to be a function of age. +The EW(Hβ) is an excellent proxy for age in young stellar +systems (see Figures 5), and hence we analyse the detec- +tion fraction as a function of EW(Hβ). From the analysis of +Figures 15 and 16, we arrived at the conclusion that quanti- +tatively the observed values of EW(Hβ) are systematically +smaller as compared to the values expected for ionization- +bounded H ii regions using C&B models. Continuum contri- +bution from a non-ionizing population and the escape of ion- +izing photons from the nebula are two principal mechanisms +that lead to a decrease in EW(Hβ) from those expected for +ionization-bounded H ii regions ionized by a single-age pop- +ulation. The factor by which the EW(Hβ) is reduced may +vary from region to region. For the sake of using EW(Hβ) +as a proxy for age, we assume a reduction factor anywhere +between 0 and 50% for the sample regions. +In Figure 18, we show the fraction of H ii regions de- +tected as a function of the observed EW(Hβ). The theo- +retically expected range of EW(Hβ) during the WR phase +is shown by the shaded area, which takes into account the +reduction of EW(Hβ) by 0 to 50% during the WR phase. +The distribution of observed EW(Hβ) for the whole sam- +ple (dotted histogram), and the sample of regions where +we have achieved a 3-σ sensitivity to detect the He ii λ4686 +line if they had He ii λ4686/Hβ⩾0.01 (dashed histogram), +are shown. For the whole sample (black line), the detec- +tion fraction decreases with decreasing EW(Hβ). For the +subset of H ii regions (red line) that have SNRs sufficient +to detect the He ii λ4686 line for typical values during WR +phase (He ii λ4686/Hβ⩾0.01), the detection fraction peaks +at EW(Hβ)∼60 ˚A. The peak value reaches as high as 90%. +The EW(Hβ) at the peak value corresponds to that during +the WR phase. This suggests that our dataset is sensitive +enough to detect almost all (∼90%) He++ nebulae ionized +by the WR stars. Before the onset of the WR phase (at the +high EW(Hβ)-end), the detection fraction is nonzero. The +He ii λ4686 line is detected in two of the three H ii regions, in- +dicating that the He ii λ4686/Hβ ratio might be higher than +the values predicted in the current SSPs during the main +sequence phase of stars. The ULX is the likely source of ad- +ditional ionization in one of these (#99), whereas stripped +binary stars, which are not included in the C&B models, +could be the possible source of the weak ionization in the +other (#112). +The rest of the non-detections (26) corresponds to low- +EW(Hβ) regions. In the C&B models, these regions corre- +spond to the post-WR phase. However, the inclusion of the +binary channel for the formation of WR stars in the BPASS +models extends the duration of WR phase to these low +EW(Hβ) values, with the expected ratio of He ii λ4686/Hβ +lower than that during the WR phase in C&B models. Un- +fortunately, we do not reach the sensitivity to detect the +He ii λ4686 line with He ii λ4686/Hβ<0.01 for regions with +EW(Hβ)< 40 ˚A regions, and hence the non-detection of the +He ii λ4686 line could be due to the absence of He++ ions in +these regions, or that we do not reach the sensitivity level re- +quired to detect weak ionization from the WR stars formed +through the binary channel. +In general, the H ii regions with He ii λ4686-line detec- +tion have higher ionization parameter as compared to the +H ii regions without the He ii λ4686-line detection, at each +EW(Hβ) bin. +The location of H ii regions with non-detection of +He ii λ4686 line in Figure 17 suggests shock and/or two- +zone escape scenarios are more prevalent in these regions +as compared to the He ii λ4686-emitting main-group H ii re- +gions. These processes are strong enough to increase the +He ii λ4686 lines above the detectable limits in only three +cases (#144, #17 and #148). +5 +CONCLUSIONS +We have carried out a search for He ii λ4686 nebular line +in the Cartwheel H ii regions using the VLT/MUSE dat- +acube. We detect the He ii λ4686 line in 32 H ii regions, with +a mean value of He ii λ4686/Hβ=0.010±0.003. All the detec- +tions are situated in the star-forming ring of the Cartwheel, +with ten of these sources coinciding with the location of a +ULX source. We use commonly used diagnostic line ratios +to compare the ionization properties of H ii regions with and +without the detection of the He ii λ4686 line. The He ii λ4686 +line-emitting regions with and without the ULX sources, in +general, show similar ionization properties in the diagnos- +tic diagrams. Hence, the ULX sources are not the principal +suppliers of ionizing photons in all the H ii regions contain- +MNRAS 000, 000–000 (0000) + +HeII in the Cartwheel galaxy +23 +Figure 18. +Fraction of H ii regions with detection of the +He ii λ4686-line as a function of the EW(Hβ) for the whole sample +(black solid line) and a sample of H ii regions that had sensitivity +to detect the He ii λ4686 line if He ii λ4686/Hβ⩾0.01 (red solid +line). The distribution of the EW(Hβ) for the whole sample (dot- +ted histogram) and a sample of H ii regions that had sensitivity +to detect the He ii λ4686 line if He ii λ4686/Hβ⩾0.01 (dashed his- +togram) are also plotted, with the right axis showing the numbers. +For reference, we also show the range of EW(Hβ) when the num- +ber of WR stars in the C&B Mu = 100 M⊙ models is non-zero, +for an ionization-bounded H ii region (the shaded area). The error +on the derived fractions for the whole sample is shown, which is +based on the square-root of the number of H ii regions in each bin +(the sizes of the error bars on the red line are similar and hence +we omit them for the sake of clarity of the figure). +ing ULX sources. Analysis of the diagnostic diagrams using +C&B SSPs suggests that the majority (27) of the detec- +tions correspond to H ii regions in their WR phase, with +two and three detections corresponding to H ii regions in +their pre-WR and post-WR phases, respectively. However, +the characteristic BB indicating the presence of WR stars is +not detected in our sample regions. We illustrate that this +non-detection is due to the relatively low EWs of the BB in +SSPs for IMFs with Mu ⩽100 M⊙ at the metallicity of the +Cartwheel, even when the SSPs have sufficient number of +WR stars to provide the ionization of He+. We suggest that +main sequence stars are the major contributors to ionization +in the two pre-WR H ii regions, with an additional contribu- +tion from other hard sources. In region#99, this additional +contribution most likely comes from the ULX source. On the +other hand, the three H ii regions in the post-WR phase may +be either ionized by radiative shocks or their H ii regions are +leaky. We find a correlation between [O iii] λ5007/Hβ and +EW(Hβ), which requires a more rapid softening of the ion- +ization parameter log⟨U⟩ than that considered in C&B SSP +models. This rapid softening can be naturally explained if +the H ii regions expand as the cluster evolves due to the +feedback from massive stars. The detection frequency of the +He ii λ4686 line reaches values as high as 90% for H ii regions +that have EW(Hβ)=40–70 ˚A. These values of EW(Hβ) cor- +respond to late stages of the WR phase in the C&B mod- +els. Our dataset lacks sensitivity to detect the He ii λ4686 +line from H ii regions with EW(Hβ)<40 ˚A, when WR stars +formed from the binary channel are expected to dominate +the ionization of He+. +ACKNOWLEDGEMENTS +We thank an anonymous referee for many thoughtful com- +ments that improved the paper. We also thank Gerardo +Ramos-Larios who helped us in preparing the images ap- +pearing in Figures 1 and 4, and CONACyT for the research +grant CB-A1-S-25070 (YDM). This work is based on data +obtained from the ESO Science Archive Facility, program +ID: 60.A-9333. Observations made with the NASA/ESA +Hubble Space Telescope were obtained from the data archive +at the Space Telescope Science Institute. STScI is operated +by the Association of Universities for Research in Astron- +omy, Inc. under NASA contract NAS 5-26555. +DATA AVAILABILITY +The fluxes of principal emission lines used in this work are +available in the article and in its online supplementary ma- +terial. 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A., Mayya D., +Ramos-Larios G., 2022, MNRAS, 514, 1689 +MNRAS 000, 000–000 (0000) + diff --git a/5tE1T4oBgHgl3EQfTAOS/content/tmp_files/load_file.txt b/5tE1T4oBgHgl3EQfTAOS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..58ad06a0da57d51d821918b646a82628c0547bf5 --- /dev/null +++ b/5tE1T4oBgHgl3EQfTAOS/content/tmp_files/load_file.txt @@ -0,0 +1,2150 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf,len=2149 +page_content='MNRAS 000, 000–000 (0000) Preprint 10 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content='0 Detection of He++ ion in the star-forming ring of the Cartwheel using MUSE data and ionizing mechanisms Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' Mayya⋆1 ID, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' Plat2 ID, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' G´omez-Gonz´alez3 ID, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' Zaragoza-Cardiel1,4 ID, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' Charlot5 ID and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' Bruzual6 ID 1Instituto Nacional de Astrof´ısica, ´Optica y Electr´onica, Luis Enrique Erro 1, Tonantzintla 72840, Puebla, Mexico 2Steward Observatory, 933 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' Cherry Avenue, University of Arizona, Tucson, AZ 85721, USA 3Institute for Physics and Astronomy, Universit¨at Potsdam, Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' 24/25, D-14476 Potsdam, Germany 4Consejo Nacional de Ciencia y Tecnolog´ıa, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' Insurgentes Sur 1582, 03940, Mexico City, Mexico 5Sorbonne Universit´e, CNRS, UMR7095, Institut d’Astrophysique de Paris, F-75014, Paris, France 6Instituto de Radioastronom´ıa y Astrof´ısica, UNAM Campus Morelia, Apartado postal 3-72, 58090 Morelia, Michoac´an, Mexico 10 January 2023 ABSTRACT We here report the detection of the nebular He ii λ4686 line in 32 H ii regions in the metal-poor collisional ring galaxy Cartwheel using the Multi-Unit Spectroscopic Ex- plorer (MUSE) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' The measured I(He ii λ4686)/I(Hβ) ratio varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content='004 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content='07, with a mean value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content='010±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' Ten of these 32 H ii regions are coinci- dent with the location of an Ultra Luminous X-ray (ULX) source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' We used the flux ratios of important diagnostic lines and results of photoionization by Simple Stellar Populations (SSPs) to investigate the likely physical mechanisms responsible for the ionization of He+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content=' We find that the majority of the regions (27) are consistent with photoionization by star clusters in their Wolf-Rayet (WR) phase with initial ioniza- tion parameter −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfTAOS/content/2301.03073v1.pdf'} +page_content='5 1 au) are +intrinsically rare. For example, it has been suggested + +RV detection of exomoons +3 +that planets could lose their satellites as they migrate +inward (Spalding et al. 2016). +The detection of exo- +moons around imaged planets with astrometry or direct +imaging is more sensitive to longer-period moons (Laz- +zoni et al. 2022). These two techniques might not be +well suited for satellites with mass ratios and separa- +tions (< 30RJup) similar to the ones orbiting the solar +system gas giants. +1.3. A promising alternative: RV detections around +directly imaged planets +Another technique has been proposed to look for +moons around directly imaged planets using RV mea- +surements of the planet itself (Vanderburg et al. 2018; +Vanderburg & Rodriguez 2021). By measuring the wob- +ble of planets caused by orbiting satellites, planetary RV +surveys is a promising alternative for finding Galilean +moons analogs. Directly-imaged companions are likely +to have a different formation and migration history com- +pared to transiting exoplanets. +They are also gener- +ally more massive and further away resulting in much +more extended Hill spheres. Models suggest that larger +planets form even larger moons following the scaling +m ∝ M 3/2, with m and M the masses of the moon and +the planet respectively (based on equation 43 in Baty- +gin & Morbidelli (2020)). As another formation path- +way, binary systems forming as the tail end of stellar +formation through gravitational instability of the proto- +stellar cloud or the protoplanetary disk could lead to a +population of easily detectable high-mass ratio satellites +and binary companions (Lazzoni et al. 2022). In sum- +mary, directly imaged companions could be more likely +to host larger moons, which could be detected from RV +monitoring of the planets themselves. +1.4. Recent technology developments +There are two aspects to optimizing the choice of a +target for exomoon searches: the RV precision that can +be achieved and the probability of the object to host a +moon. In this work, we focus on the former and study +the detectable mass ratios for exomoons as a function of +the instrument, the telescope, and the planet or brown- +dwarf properties. +Although the RV detection of exo- +moons remains challenging due to the intrinsic faintness +of planets and the light contamination from the glare +of the host star, the expertise obtained from 30 years +of stellar RV exoplanet detections is an invaluable as- +set. +Indeed, stable high-resolution spectrographs and +data analysis techniques are already demonstrating sta- +bility and performance in excess of the level of preci- +sion needed for exomoons. +Vanderburg & Rodriguez +(2021) derived the first exomoon mass upper limits with +this technique around the HR 8799 planets based on the +planetary RV time series reported by Ruffio et al. (2021) +using OSIRIS, an R = 4, 000 spectrograph, at the W. +M. Keck observatory. This study ruled out moons with +mass greater than a Jupiter mass and period less than +one day that would be orbiting the 7 Jupiter mass planet +HR 8799 c. +Recent +technological +advances +in +infrared +high- +resolution spectroscopy for high-contrast companions +are enabling the first planetary RV searches for exo- +moons (Snellen et al. 2015; Jovanovic et al. 2017; De- +lorme et al. 2021; Otten et al. 2021). The Keck Planet +Imager and Characterizer (KPIC) recently demon- +strated R = 35, 000 K-band spectroscopy of directly- +imaged exoplanets, including the HR 8799 system, mea- +suring their RVs and obtaining spin measurements for +the first time (Wang et al. 2021b). KPIC is the first +implementation of a new class of spectrographs that +combines the power of the Keck II adaptive optics sys- +tems, the stability and starlight suppression of single +mode fibers, and the high spectral resolution of the NIR- +SPEC spectrograph for the detailed study of directly +imaged planets (Delorme et al. 2021). We observed a +bright brown dwarf companion (HR 7672 B, K = 13.04, +∼ 0.7′′, Liu et al. (2002); Boccaletti et al. (2003)) as part +of the commissioning and science verification of KPIC +(Delorme et al. 2021; Wang et al. 2022a). While it is +at the boundary of the stellar regime, HR 7672 B is an +interesting benchmark companion, because it has a dy- +namically measured mass of 72.7±0.8MJup (Crepp et al. +2012; Brandt et al. 2019) and its composition should be +similar to that of the star due to its assumed formation +history from gravitational instability. Using this target, +Wang et al. (2022a) showed that accurate atmospheric +compositions could be retrieved using KPIC’s high re- +solving power and angular resolution by demonstrating +a 1.5σ consistency between the composition of HR 7672 +B and its host star (see also Xuan et al. 2022). +HR +7672 B was also observed for a full night with KPIC as +a test case for variability studies. This time series can +be used to put the deepest limits to date on the mass +of an orbiting satellite around the sub-stellar compan- +ion, which we are demonstrating in this work. KPIC +is already undergoing several upgrades including a laser +frequency comb which will enable precise RV science (Yi +et al. 2016; Jovanovic et al. 2020). The expected dou- +bling of the instrumental throughput will significantly +improve its sensitivity (Jovanovic et al. 2020; Echev- +erri et al. 2022). The next generation of high-contrast +high-resolution spectrograph such as HISPEC at the +W. M. Keck Observatory and MODHIS on the future +Thirty Meter Telescope (TMT) will undoubtedly open + +4 +Ruffio et al. +new frontiers in this field by allowing 0.98 − 2.46 µm si- +multaneous coverage at an average spectral resolution +R > 100, 000 (Mawet et al. 2019; Mawet et al. 2022). +1.5. Outline +In section 2, we present exomoon RV detection lim- +its for the brown dwarf companion HR 7672 B using +KPIC. In section 3, we then simulate observations of the +same brown dwarf and the planet HR 8799 c with next +generation facilities and compare their sensitivity to the +moons in the solar system. In section 4, we explore the +parameter space of satellites that could be detected with +TMT/MODHIS as a function of planet properties. Fi- +nally, we conclude on the prospects for RV detections of +exomoons in section 5. +2. EXOMOON LIMITS AROUND HR 7672 B WITH +KPIC +2.1. Observations and data reduction +The brown dwarf companion HR 7672 B was observed +three times in 2020 and then for a full night on July 4, +2021 with KPIC (R ∼ 35, 000) in K band (1.9−2.4 µm) +(Mawet et al. 2017; Delorme et al. 2021). These obser- +vations are detailed in Table 2. The first three epochs +included one to two hours of on-target exposures per +night and were already published in Wang et al. (2022a) +and Delorme et al. (2021). +Unfortunately, the condi- +tions on July 4, 2021 were well below average with the +companion undetectable in some individual 5-minute ex- +posures. During this one night specifically, we used an +ABAB pattern to nod the companion between two KPIC +fibers, fiber 1 and 2, to limit or identify any fiber-specific +biases. There was no nodding during the other epochs. +The data was reduced with the KPIC data reduction +pipeline (DRP)1 following the same approach described +in Wang et al. (2021b, 2022a). The first steps include +background subtraction, bad pixel correction, and the +calibration of the fiber trace location and width on the +detector for each NIRSPEC spectroscopic order. Opti- +mal extraction is then used to extract the spectra and +the wavelength solution is derived from the telluric and +stellar lines of a M giant, namely HIP 81497, taken on +the same night. For this purpose, the telluric model is +generated with the Planetary Spectrum Generator (Vil- +lanueva et al. 2018) and star is modeled by a Phoenix +model (log(g/[1 cm.s−2]) = 1; Teff = 3600 K Husser +et al. (2013)). +2.2. Forward model and likelihood +1 https://github.com/kpicteam/kpic pipeline +We use a forward modelling approach similar to Wang +et al. (2021b) and Ruffio et al. (2021) to measure the RV +of HR 7672 B, which includes a joint modelling of the +starlight and the companion signal. Wang et al. (2021c) +showed that the continuum could be included in the for- +ward model with a fourth order polynomial, therefore +not requiring the data to be high-pass filtered nor contin- +uum normalized. In this work, we model the continuum +using a spline-based linear model, which can be ana- +lytically marginalized using the general purpose python +module breads2 (Broad Repository for Exoplanet Anal- +ysis, Discovery, and Spectroscopy) based on the formal- +ism in Ruffio et al. (2019). +The spline forward mod- +eling has the advantage of being more robust to bad +pixels than a Fourier based high-pass filter and avoids +the non-linearity of a sliding-window median filter. The +spline parameters are also easier to optimize than the +coefficients of a high order polynomial for example. +We define the forward model as, +d = MRVφ + n, +(1) +where d is the data vector of size Nd, MRV is the linear +model, φ are the linear parameters, and n is a random +vector of the noise with a diagonal covariance matrix Σ. +A scaling factor for the noise is also fitted to model any +underestimation of the noise. Off-diagonal elements in +the covariance matrix are neglected here, but subsequent +data processing steps would correct for this inaccuracy. +The different column vectors of the linear model are il- +lustrated in Figure 1. The data vector and the standard +deviation of the noise used to define Σ0 are direct out- +puts of the KPIC data reduction pipeline. The variance +of the noise is multiplied by a free parameter scaling fac- +tor s2 to account for any underestimation of the noise. +KPIC includes four single mode fibers separated by +0.8′′ on a line. We can therefore acquire simultaneous +spectra of the companion and the host star, more specif- +ically the speckle field, by rotating the field of view using +the Keck II adaptive optics system front-end K-mirror +rotator. The observations of the star are used to derive +simultaneous empirical models of the transmission and +the starlight spectra used in the forward model. The +starlight is used to model the speckle noise leaking into +the fiber at the position of the companion. The wave- +length calibration is different in each fiber so the spectra +are linearly interpolated to match the sampling of the +science fiber. The planet model is defined as the spin- +broadened best fit model from Wang et al. (2022a) using +petitRADTRANS (Molli`ere et al. 2019) multiplied by +2 https://github.com/jruffio/breads + +RV detection of exomoons +5 +Moon +Planet +Mass +Mass ratio +Semi-Major axis +Period +RV semi-amplitude +(M⊕) +(RJup) +(day) +(m/s) +Io +Jupiter +1.50 × 10−2 +4.71 × 10−5 +5.90 +1.77 +0.82 +Europa +Jupiter +8.04 × 10−3 +2.53 × 10−5 +9.39 +3.55 +0.35 +Ganymede +Jupiter +2.48 × 10−2 +7.81 × 10−5 +14.97 +7.15 +0.85 +Callisto +Jupiter +1.80 × 10−2 +5.67 × 10−5 +26.33 +16.69 +0.46 +Titan +Saturn +2.25 × 10−2 +2.37 × 10−4 +17.09 +15.95 +1.32 +Titania +Uranus +5.73 × 10−4 +3.94 × 10−5 +6.10 +8.71 +0.14 +Oberon +Uranus +4.82 × 10−4 +3.32 × 10−5 +8.16 +13.46 +0.10 +Triton +Neptune +3.58 × 10−2 +2.09 × 10−4 +4.96 +-5.88 +0.92 +Kepler-1708 b-i +Kepler-1708 b +< 37 +< 0.11(2σ) +Table 1. Properties of the largest satellites orbiting the solar system gas giants from the NASA Space Science Data Coordinated +Archive (https://nssdc.gsfc.nasa.gov/planetary/). The negative period of Triton is indicating its retrograde orbit. Kepler-1708 +b-i is a transiting exomoon candidate (Kipping et al. 2022). The period and RV semi-amplitude for these moons can also be +found in Vanderburg et al. (2018). +Object +Date +Exposure time +Seeing +Throughput +HR 7672 B +2020-06-08 +11 × 10 min +0.4′′ +1% +HR 7672 B +2020-06-09 +10 × 10 min +0.6′′ +1.5% +HR 7672 B +2020-09-28 +7 × 10 min +0.4′′ +2.7% +HR 7672 B +2021-07-04 +61 × 5 min +1′′ +2% +Table 2. K-band observations of HR 7672 A and B with KPIC. The quoted throughput is the end-to-end from the top of the +atmosphere, which is a better proxy of performance than the seeing for KPIC. +the empirical telluric and instrument transmission pro- +file. The continuum of both the planet and the speckle +are modulated by a 3rd order spline model. Ten spline +nodes are used in each spectral order (∆λ ∼ 0.05 µm) for +the planet model to manage any inaccuracies in the con- +tinuum due to imperfections in the atmosphere model +fit. This number of nodes is analogous to a 200 pixel- +wide high-pass filter. The number of nodes was chosen +as a trade-off between the number of additional param- +eters and the optimal high-pass filter scale of 100 pixels +found in Xuan et al. (2022). The speckle continuum is +modeled with three spline nodes to model any speckle +crossing the fiber location as the wavelength changes. +This results in 13 linear parameters per spectral order +representing the values of the continua at the location of +the nodes (See Figure 1). This defines the linear model +MRV with dimensions Nd × 13, which is also a function +of the RV of the planet, the only non-linear parameter +fitted for here. +KPIC data features strong spectral fringing due to +the FabryP´erot cavities formed by the transmissive op- +tics inside the NIRSPEC spectrograph (Hsu et al. 2021) +and within the KPIC fiber injection unit (Finnerty et al. +2022). This effect is made worse by the high spatial co- +herence of the wavefront in KPIC. We therefore apply a +Fourier filter to the data and the forward model by zero- +ing frequencies corresponding to the fringes. A physical +model of the fringing such as Cale et al. (2019) could be +explored in the future. +The likelihood function is defined from a multivariate +Gaussian distribution as, +L(RV, φ, s2) = +1 +� +(2π)Nd|Σ0|s2Nd +exp +� +− 1 +2s2 (d − MRVφ)⊤Σ−1 +0 (d − MRVφ) +� +. +(2) +The likelihood is maximized using a linear least square +solver on a grid of RV values from −400 to 400 km/s in +steps of 0.2 km/s. The 1σ RV uncertainties are derived +from the RV posterior calculated analytically according +to Equation 10 in Ruffio et al. (2021) on this RV sam- +pling. +This method analytically marginalized the RV +posterior for the modulation of the continuum and the +noise scaling factor. The linear spline parameters used +to fit the continuum are forced to be positive. This is +theoretically inconsistent with the framework, which as- +sumes unconstrained parameters, but it does not appear +to significantly impact the RV time series. +Only the three reddest orders, out of nine in K band, +are used in this analysis. The bluest three orders (num- +bered 39-37; 1.94 − 2.09 µm) were discarded because +they feature strong saturated CO2 telluric lines that are +generally harder to model, but also make for an unsta- +ble fit due to overlapping frequencies with the fringing + +6 +Ruffio et al. +and the simple Fourier filter. The middle three orders +(2.10 − 2.27 µm) lack sufficient stellar and telluric spec- +tral lines to calibrate the wavelength precisely enough. +Thus, only the remaining three orders are used in this +analysis: 2.29−2.34 µm (order 33), 2.36−2.41 µm (order +32), and 2.44 − 2.49 µm (order 31). Order 33 includes +the carbon monoxide bandhead and therefore results in +the strongest signal-to-noise ratio (S/N) and the most +precise radial velocity measurement. +Each NIRSPEC +spectral order is fitted separately resulting in three RV +estimates for each exposure. +2.3. RV measurements +The barycentric corrected RV measurements for the +four epochs and three orders are shown in Figure 2. +Following the method described in subsection 2.2, the +median RV uncertainties in five minute exposures are +2.5 km.s−1,4.0 km.s−1,4.9 km.s−1 for order 6, 7, and 8 +respectively. We overplot the predicted radial velocity of +the brown dwarf from orbital fits to the relative astrom- +etry from Crepp et al. (2012) and RV measurements of +the host star (Crepp et al. 2012; Rosenthal et al. 2021). +The orbit fits were done with orbitize! (Blunt et al. +2020) following its RV tutorial3 and using the emcee +(Foreman-Mackey et al. 2013) sampler to obtain a pos- +terior of allowed orbits. This orbital RV of the compan- +ion in each epoch is predicted from this orbit fit and is +subsequently subtracted from the estimated RV of the +planet when running the exomoon search. Similarly to +fitting the centroid of a Gaussian (King 1983), the RV +precision goes as the typical linewidth in the spectrum +divided by the total S/N of the detection. In the case of +HR 7672 B, the large spin with v sin i = 45.0±0.5 km.s−1 +(Wang et al. 2022a) is a limiting factor in deriving more +precise RVs. The impact on the exomoon sensitivity of +other fundamental parameters such as the brightness, +age, mass, and separation from the star are discussed in +section 4 in the context of TMT/MODHIS. +2.4. Exomoon sensitivity +The open-source Python package RVSearch4 (Rosen- +thal et al. 2021) is used to look for possible exomoons +around HR 7672 B and derive the sensitivity of our +KPIC RV time series. RVSearch is a planet search algo- +rithm that was developed by the California Legacy Sur- +vey for high-precision radial velocity surveys (Howard & +Fulton 2016; Rosenthal et al. 2021; Fulton et al. 2021). +Planets are detected from periodograms, which are ex- +3 https://orbitize.readthedocs.io/en/latest/tutorials/ +RV MCMC Tutorial.html +4 https://github.com/California-Planet-Search/rvsearch +pressed as the difference in Bayesian Information Cri- +terion (BIC) between a model including the planet and +a model without it (Rosenthal et al. 2021). The ∆BIC +can be used to select the model that best represents the +data, or, in other words, determine if a planet is neces- +sary to explain the observations. Planet candidates are +detected by iteratively adding additional planet signal +to the model (Rosenthal et al. 2021). For each iterative +search, the algorithm fits a detection threshold to the +periodogram using the power law noise model described +in Howard & Fulton (2016). To characterize the search +completeness of a dataset, RVSearch performs injection- +recovery tests, drawing many synthetic planet signals, +injecting them in the data, and checking whether their +signals surpass the last detection threshold. The sim- +ulated signals were injected as described in (Rosenthal +et al. 2021) with period and M sin i from log-uniform +distributions, and eccentricity from an empirically cali- +brated beta distribution (Kipping 2013). +RVSearch is directly applicable to the search for ex- +omoons by replacing the properties of the star by the +ones of the planet. We assume that each spectral order +in NIRSPEC has a different zero RV point due to pos- +sible inconsistencies between them. This can be done +with RVSearch, which linearly solves for offsets between +subsets of RVs, and uses a wide, Gaussian, uninforma- +tive prior on white noise for each subset. +This fea- +ture is usually used to fit data from different instru- +ments. Two analyses are performed, first only using the +long night of observations (07/04/2021) and then all the +available data. The latter provides a longer time base- +line. +The resulting periodograms and exomoon com- +pleteness are shown in Figure 3. By combining the four +epochs, the observations are sensitive to satellites with +a mass ratio of 1% at semi-major axes similar to that +of Io (6RJup) around Jupiter or 4% at the distance of +Callisto (15RJup). While these are encouraging results, +the smallest detectable satellites would be as large as +Jupiter due to the already large mass of HR 7672 B. As +shown in section 4, targeting smaller brown dwarfs and +planets does not generally allow the detection of moons +with smaller absolute masses, because the S/N drops +faster than the mass of the object due to the decreas- +ing brightness. If satellites around HR 7672 B were to +orbit within ∼ 10RJup of the brown-dwarf, they would +likely fall within the Roche radius (See Figure 4). Such +satellites would be tidally disrupted and likely result in +the formation of rings around the planet. It is possible +that this issue would prevent the formation of a reso- +nant chain of satellites if the inner edge of the decretion +disk falls within the Roche limit. This is for example +cited as a possibility to explain the difference between + +RV detection of exomoons +7 +2.30 +2.31 +2.32 +2.33 +200 +0 +200 +400 +600 +Data number +Data +Combined model +Planet model +Starlight model +Residuals +Data uncertainty +2.30 +2.31 +2.32 +2.33 +200 +0 +200 +400 +600 +Data number +Planet model +Sub-components +Single sub-component +2.30 +2.31 +2.32 +2.33 + ( m) +20 +0 +20 +40 +60 +80 +100 +Data number +Starlight model +Sub-components +Single sub-component +Figure 1. Illustration of the forward model used to derive the RV of HR 7672 B. This figure shows a single NIRSPEC order +overlapping with the CO bandhead. (Top panel) A planet and a starlight model are jointly fitted to the data to account for +the diffracted starlight contamination at the location of the companion. The data uncertainty measured by the KPIC DRP +(shaded grey) slightly underestimates the amplitude of the residuals. (Center panel) The planet model is itself made of a linear +combination of ten spline modes to model the continuum of the companion spectrum. (Bottom panel) The starlight intensity +is also fitted with a spline using three nodes to account for speckles crossing at the location of the fiber. This flexible model of +the continuum is an alternative to high-pass filtering and continuum normalization of high-resolution spectra. +the Galilean and the Saturnian satellite systems in Baty- +gin & Morbidelli (2020). At the other end of possible +satellite semi-major axes, stable orbits can generally ex- +ist up to one half of the Hill sphere for prograde orbits +(Shen & Tremaine 2008). The Hill sphere of HR 7672 +B being rH ≈ 5.6 au = 1.2 × 104RJup, time series like +these ones will not be sensitive to the vast majority of +possible orbits without observations spanning years or +decades. +3. FUTURE PROSPECTS FOR HR 7672 B AND HR +8799 C +3.1. Simulations +In this section, we simulate observations from current +and future instrumentation at the Keck observatory and +TMT to estimate the properties of putative satellites +that should be detectable using planetary RVs. We use +an instrument and observation simulator called PSIsim +5, which was first developed for the Planet Systems Im- +ager (PSI Fitzgerald et al. 2022) instrument concept for +TMT, and then expanded to include other instruments +and telescopes. PSIsim is first used to estimate the RV +precision. Then, RV times series are simulated assuming +6 full nights of observations over 25 days, and the ex- +omoon sensitivity is finally computed using RVSearch. +These simulations are meant to represent an ideal sce- +nario in terms of instrument performance and telescope +time allocation. +We simulate observations of two substellar compan- +ions, the brown-dwarf companion HR 7672 B and the +planet HR 8799 c, with four generations of instru- +ments. +An exhaustive analysis of all directly imaged +companions is beyond the scope of this work so HR +5 https://github.com/planetarysystemsimager/psisim + +8 +Ruffio et al. +0.0 +0.5 +1.0 +1.5 +2.0 +time (h) +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +RV (km/s) +06-08-2020 MJD=59399.308804+t/24 +0.0 +0.5 +1.0 +1.5 +time (h) +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +RV (km/s) +06-09-2020 MJD=59008.421342+t/24 +0.00 +0.25 +0.50 +0.75 +1.00 +time (h) +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +RV (km/s) +09-28-2020 MJD=59009.452114+t/24 +0 +1 +2 +3 +4 +5 +6 +7 +time (h) +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +RV (km/s) +07-04-2021 MJD=59120.258465+t/24 +Prediction +fiber 1 order 6 +fiber 2 order 6 +fiber 1 order 7 +fiber 2 order 7 +fiber 1 order 8 +fiber 2 order 8 +Figure 2. Measured RVs of HR 7672 B with KPIC. The grey lines are predicted RVs from one hundred posterior samples of +the orbital motion of the brown dwarf. +8799 c was chosen as a representative example of the +field with a planetary mass. +HR 8799 is also the +only other high-contrast system with published RV time +series and exomoon upper limits (Vanderburg & Ro- +driguez 2021). The four instruments considered in this +work are Keck/KPIC I, Keck/KPIC II, Keck/HISPEC, +and TMT/MODHIS. KPIC I corresponds to obser- +vations carried out pre-2022A (Delorme et al. 2021). +KPIC II refers to the series of upgrades started dur- +ing the first semester of 2022 with the primary goal +of doubling the instrument throughput Jovanovic et al. +(2020); Echeverri et al. (2022). +The High-resolution +Infrared Spectrograph for Exoplanet Characterization +(HISPEC) is expected to provide Y-K (0.98 − 2.46 µm) +spectroscopy at a spectral resolution of R > 100, 000 +(Mawet et al. 2019). +The Multi-Object Diffraction- +limited High-resolution Infrared Spectrograph (MOD- +HIS) is a similar instrument to HISPEC planned for the +future TMT. A broader range of exoplanet masses is +explored in section 4 for this latter TMT instrument. +PSIsim includes full budgets of the throughput and +thermal background for each instrument, telescope, and +the Earth atmosphere. +The Strehl ratio is calculated +based on a empirically calibrated model of the adaptive +optics’ performance under median seeing conditions for +Maunakea. For KPIC I and KPIC II, we assumed Keck +AO’s current performance with the infrared Pyramid +Wavefront Sensor described in Bond et al. (2020). For +HISPEC, we assumed extreme-AO performance as pre- +dicted for the upcoming HAKA high-density deformable +mirror upgrade (W.M. Keck Observatory, private com- +munication). The star is modeled with a PHOENIX +model (Husser et al. 2013) and the substellar companion +with a BT-Settl atmospheric model grid6 (Allard et al. +2012a). Table 3 includes the input parameters and the +predicted RV precision for these simulations. The simu- +lations include a level of systematics at 1% of the contin- +uum, which is modeled by an additional white Gaussian +noise. Otherwise, the estimated RV precision assumes a +perfect data reduction. +6 https://phoenix.ens-lyon.fr/Grids/BT-Settl/CIFIST2011c/ + +RV detection of exomoons +9 +10 +2 +10 +1 +100 +101 +102 +Period (day) +10 +8 +6 +4 +2 +BIC +All data, BICthresh = 13.1 +One night, BICthresh = 10.5 +(a) Periodogram +100 +101 +102 +Semi-major axis (RJup) +10 +3 +10 +2 +10 +1 +100 +MMoon sini/MBD +Injection recovered +Injection missed +50% completeness - One night +50% completeness - All data +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Probability of detection +(b) Completeness +Figure 3. Exomoon detection limits around HR 7672 B with the Keck Planet Imager and Characterizer (KPIC) using the +open-source python module RVsearch (Rosenthal et al. 2021). (Left) Periodogram of the RV times series shown in Figure 2 +expressed a ∆BIC comparing a model with and a model without a planet. The empirical detection threshold is indicated in the +legend. (Right) Exomoon completeness derived from injection and recovery tests. The periodogram and the completeness are +shown for two cases: the single full night of observations on 07/04/2022 and all the available data including three additional +epochs with 1-2 hours of data each. The variable conditions on 07/04/2022 led to HR 7672 B to not be detected during portions +of the night, or in the RV precision to get significantly worse. By simulating RV time series, we estimate that the lost data only +affected the final sensitivity by 20%. +The predictions from PSI-sim are about a factor two +more sensitive than existing measurements with KPIC I +(See Table 3). This difference can first be explained by +uncorrected wavefront errors reducing the throughput, +both non-common path aberrations and uncorrected at- +mospheric turbulence. +Then, our current data analy- +sis framework remains limited in its ability to model +KPIC systematics. As explained in subsection 2.2, only +the redder orders of NIRSPEC are being reduced due +to strong telluric lines in the bluer orders, and an im- +perfect Fourier filtering is used to remove the fringing. +The gap between the simulations and the measurements +should decrease as observing strategies and data reduc- +tion frameworks are improved. +The final expected exomoon sensitivity of the four in- +struments is shown for the two companions in Figure 4. +For a fixed time sampling of the RV series, the minimum +detectable mass ratio is approximately proportional to +the RV semi amplitude of the signal, which is also pro- +portional to the RV precision of the instrument, so the +improvement for each generation of instrument can be +read from the simulated RV precision shown at the bot- +tom of Table 3. These simulations are compared to other +detection techniques in Appendix A, specifically astro- +metric monitoring of the companion or spatially resolv- +ing the moon through imaging. We separately discuss +the possibility of detecting transiting exomoons using +the Rossiter-McLaughlin (RM) effect in subsection 5.2. +3.2. Comparing to solar system moons +The mass ratios of the largest gas giant satellites in +the solar system are also shown in Figure 4 for compar- +ison. The higher planet masses, M, of directly imaged +planets and brown dwarfs compared to the solar sys- +tem could yield significantly bigger moons, so we also +include scaled-up mass ratios, q, according to q ∝ +√ +M +(Batygin & Morbidelli 2020). While the CPD does scale +with the Hill Sphere, we do not expect the semi-major +axis of satellites to depend on this parameter. Indeed, +young moons are thought to migrate toward the planet +during their formation due to the interaction with the +gas. The migration is stopped at the inner radius of the +CPD which is set by the magnetic field of the planet +(Batygin & Morbidelli 2020). In this work, we therefore +keep the semi-major axis of the solar system satellites +constant. A caveat is that large moons could be suscep- +tible to tidal forces if they form or migrate too close to +the planet within the Roche limit. The Roche limit is +calculated using the mass-radius relationship from Chen + +10 +Ruffio et al. +& Kipping (2017) and their associated Python package7. +However, this relationship does not account for the fact +that young objects are likely inflated. +Parameters +Star - Phoenix model +HR 7672 +HR 8799 +Apparent K mag +4.4a +5.2a +Effective temperature (Teff) +6000 Kb +7400 Kc +Surface gravity (log(g)) +4.5b +4.5c +Spin (vsin(i); km/s) +5.6d +49e +Companion - BTsettl model +HR 7672 B +HR 8799 c +Mass +73MJupb +7MJupf +Apparent K mag +13.0b +16.1g +Effective temperature (Teff) +1800 Kb +1200 Kh +Surface gravity (log(g)) +5.5b +4.0h +Spin (vsin(i)) +45 km.s−1b +10 km.s−1i +Separation +0.72′′j +0.95′′j +Telescope and instrument +airmass +1.2 +water vapor column +1.5 mm +integration time (tint) +5 min +Predicted RV sensitivity (m.s−1) +assuming 0.6′′ − 1.0′′ seeing +HR 7672 B +HR 8799 c +Keck/KPIC I (measured) +∼ 2, 000k +∼ 7, 000i +Keck/KPIC I (simulated) +800-1400 +3,000-5,000 +Keck/KPIC II +500-800 +2,000-3,000 +Keck/HISPEC +∼ 200 +100-200 +TMT/MODHIS +30-40 +10-20 +Table 3. Radial velocity (RV) precision simulations of cur- +rent and future instrumentation for two substellar compan- +ions: HR 7672 B and HR 8799 c. (Top) Representative pa- +rameters for the telescope, instrument, star, and companions +used in the PSIsim simulations. (Bottom) Predicted RV sen- +sitivity for values of seeing ranging from 0.6′′ to 1.0′′. +References—a: Cutri et al. (2003), b: Wang et al. +(2022a), c: Wang et al. (2020), d: Luck (2017), e: Royer +et al. (2007), f: Wang et al. (2018), g: Currie et al. (2011), +h: Wang et al. (2018), i: Wang et al. (2021b), j: +http://whereistheplanet.com/ (Wang et al. 2021a), k: This +work +4. FUTURE EXOMOON SENSITIVITY OF +TMT/MODHIS +Looking to the future, we expect substantial gains +in RV precision by using the next generation of high- +resolution spectrographs on large telescopes. +These +7 https://github.com/chenjj2/forecaster +gains in RV precision will lead to enhanced sensitivity +to systems with lower mass, close in exomoons, which +would form in a similar way to the Galilean moons +around Jupiter. +Using the same framework as in section 3, we calcu- +late the RV sensitivity for a variety of simulated planets +that could exist around a host star with the properties +of HR 8799 referenced in Table 3. We modeled planets +with varying effective temperatures and apparent mag- +nitudes, fixing the separation between the planet and +star to 700 mas and the surface gravity of the planet to +log(g) = 4.5cm.s−2, and used PSIsim to calculate the +RV sensitivity. The effect of the starlight contamination +on RV sensitivity can be neglected for the type of di- +rectly imaged planets that are known today and would +be observed with TMT. The RV sensitivity vary by less +than 20 percent for planets that lie beyond 500 mas and +have a flux ratio greater than ∼ 3 × 10−6. +On aver- +age, for every 0.5 dex change in surface gravity on the +planet, the RV sensitivity changes by ±0.7 m/s. Fig- +ure 5 (a) shows the RV sensitivity MODHIS could have +for a single, two hour exposure, for planets of varying +effective temperatures and apparent magnitudes around +an HR 8799 like star. The RV sensitivity of MODHIS +driven by the brightness of the planet more than than its +temperature. However, the RV sensitivity is decreased +for planets with temperatures between 1500 and 1700 +K using the BT-settl model grid due to the L-T transi- +tion. At these temperatures, clouds form in the upper +layers of the atmosphere, shrouding detectable spectral +lines. For a given planet temperature and magnitude, +the RV precision of TMT/MODHIS Figure 5 (a) can +be compared to the RV semi amplitude in Figure 5 (b) +as a function of the planet mass, the mass ratio, and +the period of the satellite. However, such a comparison +assumes multiple epochs of observations with a given +sensitivity in order to detect a moon with a similar RV +semi amplitude. +In the following, the surface gravity, temperature, and +mass of the planet are treated more self-consistently us- +ing BT-Settl evolutionary grids (Allard et al. 2012b). +The dependence of the exomoon sensitivity to the num- +ber of observations is also made explicit by using simu- +lated RV time series. We therefore express the RV pre- +cision and exomoon sensitivity as a function of planet +mass and distance to the Sun in Figure 6. +We fixed +the age of the system to different values to represent +the parameter space occupied by different populations +of stars. The 3 Myr age group is representative of the +youngest stars, such as those found in star forming re- +gions (e.g. Ophiuchus, Taurus, etc). The 30 Myr age +group is representative of young moving groups, such + +RV detection of exomoons +11 +100 +101 +102 +103 +104 +Semi-major axis (RJup) +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +(MMoon/Mplanet) sini +Hill sphere +Roche limit +Io +Europa +Ganymede +Callisto +Titan +Triton +Titania +Oberon +KPIC I (Data) +KPIC I +KPIC II +HISPEC +MODHIS +1h +1d +1wk 1mo +1yr +10yr +100yr +Period +100 +101 +102 +103 +104 +(MMoon/MEarth) sini +(a) HR 7672 B +100 +101 +102 +103 +104 +Semi-major axis (RJup) +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +(MMoon/Mplanet) sini +Hill sphere +Roche limit +Io +Europa +Ganymede +Callisto +Titan +Triton +Titania +Oberon +KPIC I +KPIC II +HISPEC +MODHIS +1h +1d +1wk 1mo +1yr +10yr +100yr +Period +10 +1 +100 +101 +102 +103 +(MMoon/MEarth) sini +(b) HR 8799 c +Figure 4. +Future prospects for exomoon detections around the brown dwarf companion HR 7672 B (left) and planet HR 8799 +c (right). Simulated sensitivity for Keck/KPIC I, Keck/KPIC II, Keck/HISPEC, and TMT/MODHIS are shown in colored +curves assuming 6 nights of observations over 25 days. The sensitivity demonstrated in this work from ∼ 1.5 nights of KPIC +observations is labeled as KPIC I (Data). The mass ratios of the Galilean satellites are shown as black dots for comparison. +Their predicted scaled-up mass ratios, q, accounting for the larger mass, M, of the brown dwarf compared to Jupiter are shown +as grey crosses (q ∝ +√ +M; Batygin & Morbidelli (2020)). The Roche limit is computed for both a rigid and a fluid satellite +shown as the inner and outer greyed region respectively. +as Beta Pictoris Moving Group and the Tucana and +Horologium Associations. +The 300 Myr age group is +representative of the oldest directly imaged substellar +companions. The RV sensitivity decreases the further +the system is away at each distinct age. For younger +systems, there is larger decrease in sensitivity as the +mass of the planet decreases below ∼ 13 MJup. +The +large decrease in RV sensitivity once the object is below +∼ 13 MJup is due to the onset of deuterium burning for +brown dwarfs, which makes them much more luminous +than a planet of a similar mass. Another interesting fea- +ture in Figure 6 (a) is the apparent independence of the +RV precision to the brown dwarf mass above ∼ 13 MJup +at 30 Myr. This can be explained by the facts that the +RV precision is mostly driven by the brightness of the +object, and that brown dwarfs have a similar brightness +over a range of masses around this age. Indeed, larger +brown dwarfs cool faster than smaller ones resulting in +the different cooling curves to meet over a small range +of brightness around 30 Myr as illustrated in Figure 7 +in Burrows et al. (1997). +Figure 6 (b) shows the moons that could be detected +around a planet from Figure 6 (a) if they were placed +at the distance of Callisto. For each planetary mass and +distance, we create an RV time series assuming six full +eight-hour nights of observations over 25 days with er- +ror bars that represent the RV sensitivity calculated by +PSIsim. +The detection threshold was computed from +simulated data created by RVsearch as in section 3. +For more massive planets and brown dwarfs, we expect +TMT/MODHIS to reach the RV sensitivity needed to +look for close in moons with mass ratios smaller than +10−4 around brown-dwarfs, similar to the ones found in +the solar system for a median age of 30 Myr. However, to +detect moons around lower mass, directly imaged plan- +ets of the same age, we are sensitive to mass ratios of +10−3 or larger. +5. DISCUSSION +5.1. The viability of exomoon RV searches +Using KPIC, we derive the most sensitive upper limits +on the mass ratio of satellites orbiting a high-contrast +substellar companion. +We rule out satellites larger +than 1-4% the mass of the brown dwarf HR 7672 B +at separations similar to the Galilean moons. +Based +on end-to-end simulations, we predict that instruments +such as TMT/MODHIS could be two orders of mag- +nitude more sensitive. This would be sufficient to de- +tect moons forming in the CPD of a planet with mass +ratios of ∼ 10−4, albeit with a substantial investment +in observing time. +If the satellite to planet mass ra- +tio grows as q ∝ +√ +M, with M the mass of the planet, + +12 +Ruffio et al. +5 +10 +15 +20 +25 +Planet Magnitude +500 +1000 +1500 +2000 +2500 +Teff (K) +0.5 m/s +1 m/s +2 m/s +5 m/s +20 m/s +100 m/s +500 m/s +1 km/s +3 km/s +100 +101 +102 +103 +RV Sensitivity (m/s) +(a) RV sensitivity +100 +101 +102 +1 day +1 m/s +10 m/s +100 m/s +1 km/s +10 km/s +1 mo +1 m/s +10 m/s +100 m/s +1 km/s +10 +4 +10 +3 +10 +2 +10 +1 +100 +Mass ratio (q) +100 +101 +102 +Planet mass (MJup) +1 yr +0.1 m/s +1 m/s +10 m/s +100 m/s +1 km/s +10 +4 +10 +3 +10 +2 +10 +1 +100 +10 yr +0.1 m/s +1 m/s +10 m/s +100 m/s +1 km/s +(b) RV semi-amplitude +Figure 5. RV precision of MODHIS. (Left) The RV sensitivity of MODHIS for model planets around a HR 8799-like host star +using BT-Settl models (Allard et al. 2012a). The RV sensitivity was predicted using PSIsim for a single, two hour exposure. +Both the contour curves and color map indicate the RV sensitivity for a specified effective temperature and apparent magnitude +of the model planet. The RV sensitivity relies more on the brightness of the planet than its effective temperature. However, the +RV sensitivity decreases for planets with temperatures between 1500 and 1700 K due to the L-T transition. (Right) The RV +semi-amplitude for different planet masses and mass ratios. Note, increasing the exposure time will increase the RV sensitivity. +the Keck/HISPEC should be sensitive to these objects +around brown dwarfs. Any detection with HISPEC, or +lack thereof, will therefore already be capable of con- +straining CPD formation models. In order to validate +our instrument simulations, we compared them with +existing observations. +The gap in sensitivity can be +explained by imperfections in the data reductions. A +continued investment in more accurate data processing +algorithms or observing strategies is therefore required +in order to realize these predictions. Planet variability +will also be a challenge to overcome using the different +timescales and the wavelength dependence of the vari- +ability compared an exomoon signal for example (Van- +derburg et al. 2018). Measuring the variability of sub- +stellar companions would in fact be an important result +of exomoon surveys to better understand the physics of +their atmospheres (Biller 2017). +Binary formation processes favor high-mass ratios so +they would be more easily detectable than the smaller +satellites forming by accretion in the CPD. The ma- +jority of multiplicity surveys for isolated brown dwarfs +(Fontanive et al. 2018), or companion brown dwarfs +(Burgasser et al. 2005; Lazzoni et al. 2020), have +searched for visual companions, leaving the separation +regime of < 1 au underexplored. +Figure 5 (b) shows +that unresolved binary substellar companion would be +detectable with RV precision between 0.1 − 1 km.s−1, +which is already routinely achieved with KPIC. As an +example, the measured dynamical mass of the brown +dwarf companion HD 47127 B suggest that it could be +a binary (Bowler et al. 2021), but this specific compan- +ion is too faint (K ∼ 18.4) to be a practical target for +KPIC. +From Appendix A and Figure 7, we conclude that the +different detection techniques are sensitive to distinct +regions of the parameter space, and therefore comple- +mentary, not unlike exoplanet searches. +If exomoons +follow the model of solar system gas giant satellites, RV +searches could be the most promising approach due to +its sensitivity to short period moons. However, unless +the theoretical prediction that bigger planets form even +bigger moons hold true, small satellites with mass ratios +∼ 10−4 might only be detectable around brown dwarfs. +5.2. Detections of moons using the +Rossiter-McLaughlin effect +An alternative strategy to look for exomoons around +directly imaged planets using RV measurements could +be to look for transiting moons through the Rossiter- +McLaughlin (RM; +Gaudi & Winn 2007) effect on the +planet. Precise photometric calibration and stability of +high-constrast instrument is notoriously difficult (Wang +et al. 2022b), so detecting a RM event during a transit + +RV detection of exomoons +13 +2 +5 +10 +20 +30 +50 +70 +2 m/s +3 m/s +4 m/s +5 m/s +10 m/s +50 m/s +4 +10 +20 +30 +50 +70 +Planet Mass (MJup) +2 m/s +3 m/s +4 m/s +5 m/s +10 m/s +50 m/s +100 m/s +20 +40 +60 +80 +100 +120 +140 +Distance (pc) +11 +20 +30 +40 +50 +60 +70 +2 m/s +3 m/s +4 m/s +5 m/s +20 m/s +30 m/s +40 m/s +50 m/s +100 m/s +200 m/s +100 +101 +102 +103 +RV Sensitivity (m/s) +3 Myr +30 Myr +300 Myr +(a) RV sensitivity +2 +5 +10 +20 +30 +50 +70 +2.5e-5 +5e-5 +1e-4 +5e-4 +1e-3 +4 +10 +20 +30 +50 +70 +Planet Mass (MJup) +2.5e-5 +5e-5 +1e-4 +5e-4 +1e-3 +5e-3 +1e-2 +20 +40 +60 +80 +100 +120 +140 +Distance (pc) +11 +20 +30 +40 +50 +60 +70 +2.5e-5 +5e-5 +1e-4 +5e-4 +1e-3 +5e-3 +1e-2 +10 +5 +10 +4 +10 +3 +10 +2 +(MMoon/MBD) sin i +3 Myr +30 Myr +300 Myr +(b) Detectable mass ratio +Figure 6. RV precision and detectable mass ratio of MODHIS similar to Figure 5, but as a function of planet mass, distance, +and age of the system. (Left) BT-Settl evolutionary models (Allard et al. 2012b) were used to infer the mass of the planet and +distance to the system at an age of 3, 30, and 300 Myr. The minor contour lines cover an evenly spaced, 50 step log scale from +0 to 1 km/s. RV sensitivity decreases the further the system is away and the lower in mass the planet is. The large decrease in +RV sensitivity when the companion mass is below ∼ 13 MJup for young systems is due to the difference in cooling rates between +brown dwarfs and planets over time. (Right) The mass ratio detectable by MODHIS assuming a fixed semi major axis for the +moon equal to that of Callisto (≈ 26RJup). For each planetary mass and distance from panel (a), we create an RV time series +assuming six nights of observations over 25 days with error bars that represent the RV sensitivity calculated by PSIsim. +could be easier than detecting its photometric counter- +part. +An RM event consists of the subsequent masking of a +portion of the blue and red-shifted areas of the surface +of a spinning object, therefore leading to large and very +distinct deviations of the measured RV. The amplitude +of the RV signal can be hundreds of times larger than +the RV semi amplitude due to the orbital motion of the +moon. Its amplitude is proportional to the spin of the +planet, which could make it an interesting alternative +to detect the smallest moons around rapidly rotating +planets and brown dwarfs. Indeed, the RV uncertainties +scale with the spin of the object so detecting the orbital +signal of small exomoons could be more challenging. +The Galilean moons have rather small orbital periods +from days to weeks. Assuming a random inclination dis- + +14 +Ruffio et al. +tribution, the transit probability of a moon (P) is given +by the ratio of the planet radius (Rp) and the moon +semi-major axis (dm), P = Rp/dm (Borucki & Sum- +mers 1984). Therefore, the probability of a transit of +a moon at the separation of Io around Jupiter is 1:6, +and 1:27 for the farthest Galilean moon Callisto. As- +suming a full 8 hour night of observations, we estimate +the probability of observing an RM event for Galilean- +like moons around a Jupiter like planet to be around 3% +for Io, 1% for Europa, 0.3% for Ganymede, and 0.07% +for Callisto. However, the orbital periods of the moons +would be even shorter around larger substellar compan- +ions, which would increase the probabilities up to 17% +for Io, 8% for Europa, 2.6% for Ganymede, and 0.6% for +Callisto . The transits would last between ∼2-5 hours +for the Galilean moons around Jupiter, but they would +only last 15-30 minutes for similar moons around HR +7672 B. +As an example, a satellite around HR 7672 B with a +mass of 1M⊕ (q = 5×10−5) would generate a RM signal +of ∼ 300 m.s−1 compared to the ∼ 0.5 m.s−1 generated +by the orbital motion (Gaudi & Winn 2007). The am- +plitude would be ∼ 5 km.s−1 for a Neptune-size moon. +Multiple satellite systems would increase the probability +of a detection. Given the low detection probability, RM +searches could be carried out in synergy with other sci- +ence cases such as brown dwarf variability (Biller 2017). +For example, Doppler spectroscopy also favors long ob- +servations of rapidly rotating objects, which would make +for ideal datasets for exomoon RM searches. +5.3. Searching for Pandora: Habitable exomoons +Estimating the occurrence rate of Earth-sized exo- +planets in the habitable zone (HZ) of Sun-like star, +called η⊕, has been an important goal of exoplanet sur- +veys. While such planets remain challenging to detect, +the best estimates of η⊕ range between 5 − 50% to date +(Gaudi et al. 2021). However, these are not the only +Earth-sized objects that could harbor life in the HZ of +their stars. Any rocky satellites orbiting HZ gas giant +planets could also provide suitable conditions for life. +Close-in exomoons can be protected from stellar radia- +tion by the strong magnetic field of Jovian mass planets +(Heller & Zuluaga 2013). +Integrating the distribution of gas giants with an inci- +dent flux between 0.3−1.5 times the solar irradiance on +Earth for an optimistic habitable zone, or 0.3 − 1 for a +conservative habitable zone (Kasting & Harman 2013), +yields about 5 − 7 giant planets per hundred FGKM +stars. This is using the giant planet (30 − 6000M⊕ sin i) +occurrence rates derived from the California Legacy Sur- +vey as a function of stellar irradiation (figure 11; Fulton +et al. 2021). Given that each planet can have multiple +satellites, this could represent a significant number of +habitable Earth-size moons that are not accounted for +in η⊕. The occurence rate of habitable exomoons could +be constrained by measuring the population of satellites +around more distant directly imaged planets and brown- +dwarfs. +6. CONCLUSION +In this work, we aimed at evaluating the prospects +for radial velocity (RV) detections of exomoons around +self-luminous directly-imaged planets. We used real ob- +servations as well as end-to-end simulations of future +facilities at the Keck observatory and the Thirty Meter +Telescope (TMT). Using data from KPIC, we were able +to derive upper limits for satellites orbiting the brown +dwarf companion HR 7672 B at a mass ratio of 1−4% for +separations similar to the Galilean moons. Current in- +strumentation is already sensitive to unresolved binary +companions that could form through gravitational in- +stability. We demonstrate that future thirty-meter class +telescopes will likely push the sensitivity down to the +mass ratios of solar system satellites (∼ 10−4), which are +thought to form in a circumplanetary disk. We note that +second generation instruments like Keck/HISPEC on +current ten meter class telescopes might be sufficient to +detect these moons if theoretical predictions that larger +planets form even larger moons hold true. Everything +else being equal and considering the RV signal from the +orbital motion of the moon, the deepest exomoon sen- +sitivity will be reached for the brightest substellar com- +panions with the smallest spin. Small moons could also +be detected from their Rossiter-McLaughlin (RM) ef- +fect on the planetary RV signal. An RM event can be +orders of magnitude larger than the orbital signal, albeit +with percents level detection probability assuming a full +night of observation. We conclude that the detection of +exomoons from planetary RV surveys is now becoming +a reality thanks to the development of high-resolution +spectrographs dedicated to directly imaged planets. + +RV detection of exomoons +15 +J.-B. R. acknowledges support from the David and Ellen +Lee Prize Postdoctoral Fellowship. +1 +2 +Funding for KPIC has been provided by the California +Institute of Technology, the Jet Propulsion Laboratory, +the Heising-Simons Foundation through grants #2019- +1312 and #2015-129, the Simons Foundation, and the +United States National Science Foundation Grant No. +AST-1611623. +3 +4 +5 +6 +7 +8 +Ji Wang acknowledges the support by the National +Science Foundation under Grant No. 2143400. +9 +10 +The W. M. Keck Observatory is operated as a scien- +tific partnership among the California Institute of Tech- +nology, the University of California, and NASA. The +Keck Observatory was made possible by the generous +financial support of the W. M. Keck Foundation. We +also wish to recognize the very important cultural role +and reverence that the summit of Maunakea has always +had within the indigenous Hawaiian community. We are +most fortunate to have the opportunity to conduct ob- +servations from this mountain. +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +Facilities: Keck II (KPIC) +Software: +astropy8 (Astropy Collaboration et al. +2013), Matplotlib9 (Hunter 2007), PSIsim10, RVSearch11 +(Rosenthal et al. 2021), KPIC Data Reduction Pipeline12 +(Delorme et al. 2021), BREADS13 (Ruffio et al. 2021; +Agrawal 2022), +APPENDIX +8 http://www.astropy.org +9 https://matplotlib.org +10 https://github.com/planetarysystemsimager/psisim +11 https://github.com/California-Planet-Search/rvsearch +12 https://github.com/kpicteam/kpic pipeline +13 https://github.com/jruffio/breads + +16 +Ruffio et al. +100 +101 +102 +103 +104 +Semi-major axis (RJup) +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +(MMoon/Mplanet) sini +Hill sphere +Roche limit +10 as astrometry +100 as astrometry +Io +Europa +Ganymede +Callisto +Titan +Triton +Titania +Oberon +Keck +TMT +VLTI +KPIC I (Data) +KPIC I +KPIC II +HISPEC +MODHIS +1h +1d +1wk 1mo +1yr +10yr +100yr +Period +100 +101 +102 +103 +104 +(MMoon/MEarth) sini +(a) HR 7672 B +100 +101 +102 +103 +104 +Semi-major axis (RJup) +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +(MMoon/Mplanet) sini +Hill sphere +Roche limit +10 as astrometry +100 as astrometry +Io +Europa +Ganymede +Callisto +Titan +Triton +Titania +Oberon +Keck +TMT +VLTI +KPIC I +KPIC II +HISPEC +MODHIS +1h +1d +1wk 1mo +1yr +10yr +100yr +Period +10 +1 +100 +101 +102 +103 +(MMoon/MEarth) sini +(b) HR 8799 c +Figure 7. +Similar to Figure 4, but including idealized exomoon sensitivities of alternative detection techniques. The diagonal +dashed black lines represent the simplified sensitivity of VLTI/Gravity through astrometry. The vertical gray scale bars represent +the diffraction limit of different telescopes for direct imaging of satellites, namely the W. M. Keck observatory, the future Thirty +Meter Telescope (TMT), and the Very Large Telescope Interferometer (VLTI). +APPENDIX +A. COMPARING TO OTHER DETECTION METHODS +Alternative exomoon detection techniques include astrometry and direct imaging of imaged planets. Figure 7 shows +their idealized detection limits to be compared to the RV sensitivity originally presented in Figure 4. +With an +astrometric precision of 10 − 100 µas (Gravity Collaboration et al. 2021), interferometry with VLTI/GRAVITY could +be sensitive to moons further away than radial velocity, but remains limited by the orbital period of the satellite at +the furthest separations. The simplified detection limits are computed by matching the astrometric precision (σastro) +of VLTI/GRAVITY with the amplitude of the planet astrometric displacement in the sky around the center of mass. +The smallest detectable mass ratio (q) is given by +q = +� +2 ∗ +�sma +1 au +� �1 pc +d +� � 1 as +σastro +� +− 1 +�−1 +, +(A1) +with d the distance of the star to the Sun, and sma the semi major axis of the moon. We use the diffraction limit +of the telescope to illustrate the parameter space that might be accessible to direct imaging. More specifically, the +detection threshold is taken at twice the spatial resolution of the telescope (∼ 2λ/D) with D the diameter of the +telescope and λ = 2 µm. Unfortunately, estimating the brightness of low mass objects (< 1MJup) remains challenging +and will depend on the age of the system, so we arbitrarily chose a lower limit of one Jupiter mass for Keck and VLTI, +and a mass similar to the solar system ice giants for TMT. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' University of Pennsylvania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Philadelphia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' PA 19104,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' USA 10Department of Chemical & Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' University of Toronto Mississauga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 3359 Mississauga Road Mississauga ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Ontario L5L 1C6m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Canada 11Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' New Mexico State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Box 30001, MSC 4500, Las Cruces, NM 88003, USA 12UK Astronomy Technology Centre, Royal Observatory, Edinburgh EH9 3HJ, United Kingdom 13Department of Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095, USA 14W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI, USA 15Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA95064, USA 16Center for Astrophysics and Space Sciences, University of California, San Diego, La Jolla, CA 92093 17Physics and Astronomy Department, Pomona College, 333 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' College Way, Claremont, CA 91711, USA 18Earth and Planets Laboratory, Carnegie Institution for Science, Washington, DC 20015, USA Submitted to AJ ABSTRACT The detection of satellites around extrasolar planets, so called exomoons, remains a largely unex- plored territory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In this work, we study the potential of detecting these elusive objects from radial velocity monitoring of self-luminous directly imaged planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This technique is now possible thanks to the development of dedicated instruments combining the power of high-resolution spectroscopy and high-contrast imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' First, we demonstrate a sensitivity to satellites with a mass ratio of 1 − 4% at separations similar to the Galilean moons from observations of a brown-dwarf companion (HR 7672 B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Kmag=13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='7′′ separation) with the Keck Planet Imager and Characterizer (KPIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' R ∼ 35, 000 in K band) at the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Keck Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Current instrumentation is therefore already sensitive to large unresolved satellites that could be forming from gravitational instability akin to binary star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Using end-to-end simulations, we then estimate that future instruments such as MODHIS, planned for Corresponding author: Jean-Baptiste Ruffio jruffio@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='04206v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='EP] 10 Jan 2023 ID2 Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' the Thirty Meter Telescope, should be sensitive to satellites with mass ratios of ∼ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Such small moons would likely form in a circumplanetary disk similar to the Jovian satellites in the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Looking for the RossiterMcLaughlin effect could also be an interesting pathway to detecting the small- est moons on short orbital periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Future exomoon discoveries will allow precise mass measurements of the substellar companions that they orbit and provide key insight into the formation of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' They would also help constrain the population of habitable Earth-sized moons orbiting gas giants in the habitable zone of their stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Keywords: Natural satellites (Extrasolar) (483) — Direct imaging (387) — Radial velocity (1332) — Exoplanet detection methods (489) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' INTRODUCTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Exomoon formation pathways Moons similar to those around Jupiter are expected to form in circumplanetary disks (CPD) as a by-product of planet formation (Batygin & Morbidelli 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The typical CPD total dust mass relative to the planet is around 10−4 (Canup & Ward 2006) commensurate with the mass ratios of solar system satellites around the gas giants listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This is consistent with the mea- sured value of the CPD around PDS 70 c from ALMA observations (Benisty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021), which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='031MEarth and corresponds to a mass ratio of about 5 × 10−5, assuming a 2MJup planet (Benisty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' It is also possible to form larger moons from the merger of Galilean-like multiple systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This is the proposed sce- nario to explain the high-eccentricity and large mass of Saturn’s moon Titan (Asphaug & Emsenhuber 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Alternative formation pathways include the capture of satellites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=', Neptune’s moon Triton, Agnor & Hamil- ton (2006)), collisions with protoplanets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=', the Moon, Canup & Asphaug (2001)]), or even gravitational insta- bility like in the formation of brown dwarf binaries (Laz- zoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The detection of satellites that formed in a CPD remains challenging with current instrumenta- tion, but binary planets and brown dwarfs are already accessible with various techniques depending on their separation (Lazzoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Characterizing the dif- ferent populations with different mass ratios, such as binary brown-dwarf companions (ie, triple systems) and smaller CPD moons, could help inform the formation pathways of directly imaged planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Indeed, binary companions could only occur in a top-down scenario, while CPD formation could occur in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' An im- portant lesson from early exoplanet discoveries is that the solar system is not a good predictor of exoplanet de- mographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Planet formation theories also often strug- gle to account for the diversity of new discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For example, the first discoveries of hot Jupiters and the ubiquity of super-Earths and mini Neptunes were ini- tially a surprise to the community (Batalha 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' As a corollary, it would be unwise to assume that exomoon searches should be any different (Kipping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' It is therefore important that we keep pushing the dis- covery space with new observational methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Status of exomoon searches For the past decade, transiting surveys have unequiv- ocally dominated the landscape of exomoon searches (Kipping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2012) through the analysis of transit timing variations and additional transit signal from the moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' They have placed the first constraints on exo- moon occurrence rates and shown that high-mass ratio satellites are not common around short-period exoplan- ets (Kipping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Teachey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Other detection techniques have been used to look for exo- moons around directly imaged exoplanets such as as- trometry or direct imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' While these terms also re- fer to planet detection techniques, in this context, direct imaging means to spatially resolve the satellite from the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Astrometric detections refers to the measure- ment of the astrometric wobble of a planet caused by orbiting moons with precise interferometric instruments such as VLTI/GRAVITY (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' To date, only a handful of exomoon candidates have been proposed: for example two around transit- ing planets (Teachey & Kipping 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Kipping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022), one orbiting a directly-imaged brown dwarf (Laz- zoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2020), and another around an isolated plan- etary mass object (Limbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' None have been confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Most notably, the exomoon candidate Kepler 1708 b-i is a transiting 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='6 Earth radii object or- biting a Jupiter-sized planet, which, if confirmed, would be several orders of magnitude larger than the Galilean moons in terms of mass ratio (Kipping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Transiting planets generally have short orbital peri- ods and smaller Hill spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' These conditions could be less favorable to moon formation and retention, while observed transits of long period exoplanets (> 1 au) are intrinsically rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For example, it has been suggested RV detection of exomoons 3 that planets could lose their satellites as they migrate inward (Spalding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The detection of exo- moons around imaged planets with astrometry or direct imaging is more sensitive to longer-period moons (Laz- zoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' These two techniques might not be well suited for satellites with mass ratios and separa- tions (< 30RJup) similar to the ones orbiting the solar system gas giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' A promising alternative: RV detections around directly imaged planets Another technique has been proposed to look for moons around directly imaged planets using RV mea- surements of the planet itself (Vanderburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Vanderburg & Rodriguez 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' By measuring the wob- ble of planets caused by orbiting satellites, planetary RV surveys is a promising alternative for finding Galilean moons analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Directly-imaged companions are likely to have a different formation and migration history com- pared to transiting exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' They are also gener- ally more massive and further away resulting in much more extended Hill spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Models suggest that larger planets form even larger moons following the scaling m ∝ M 3/2, with m and M the masses of the moon and the planet respectively (based on equation 43 in Baty- gin & Morbidelli (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' As another formation path- way, binary systems forming as the tail end of stellar formation through gravitational instability of the proto- stellar cloud or the protoplanetary disk could lead to a population of easily detectable high-mass ratio satellites and binary companions (Lazzoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In sum- mary, directly imaged companions could be more likely to host larger moons, which could be detected from RV monitoring of the planets themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Recent technology developments There are two aspects to optimizing the choice of a target for exomoon searches: the RV precision that can be achieved and the probability of the object to host a moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In this work, we focus on the former and study the detectable mass ratios for exomoons as a function of the instrument, the telescope, and the planet or brown- dwarf properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Although the RV detection of exo- moons remains challenging due to the intrinsic faintness of planets and the light contamination from the glare of the host star, the expertise obtained from 30 years of stellar RV exoplanet detections is an invaluable as- set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Indeed, stable high-resolution spectrographs and data analysis techniques are already demonstrating sta- bility and performance in excess of the level of preci- sion needed for exomoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Vanderburg & Rodriguez (2021) derived the first exomoon mass upper limits with this technique around the HR 8799 planets based on the planetary RV time series reported by Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2021) using OSIRIS, an R = 4, 000 spectrograph, at the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Keck observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This study ruled out moons with mass greater than a Jupiter mass and period less than one day that would be orbiting the 7 Jupiter mass planet HR 8799 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Recent technological advances in infrared high- resolution spectroscopy for high-contrast companions are enabling the first planetary RV searches for exo- moons (Snellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Jovanovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' De- lorme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Otten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The Keck Planet Imager and Characterizer (KPIC) recently demon- strated R = 35, 000 K-band spectroscopy of directly- imaged exoplanets, including the HR 8799 system, mea- suring their RVs and obtaining spin measurements for the first time (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' KPIC is the first implementation of a new class of spectrographs that combines the power of the Keck II adaptive optics sys- tems, the stability and starlight suppression of single mode fibers, and the high spectral resolution of the NIR- SPEC spectrograph for the detailed study of directly imaged planets (Delorme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We observed a bright brown dwarf companion (HR 7672 B, K = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='04, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='7′′, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Boccaletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2003)) as part of the commissioning and science verification of KPIC (Delorme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' While it is at the boundary of the stellar regime, HR 7672 B is an interesting benchmark companion, because it has a dy- namically measured mass of 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='8MJup (Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2019) and its composition should be similar to that of the star due to its assumed formation history from gravitational instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Using this target, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2022a) showed that accurate atmospheric compositions could be retrieved using KPIC’s high re- solving power and angular resolution by demonstrating a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5σ consistency between the composition of HR 7672 B and its host star (see also Xuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' HR 7672 B was also observed for a full night with KPIC as a test case for variability studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This time series can be used to put the deepest limits to date on the mass of an orbiting satellite around the sub-stellar compan- ion, which we are demonstrating in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' KPIC is already undergoing several upgrades including a laser frequency comb which will enable precise RV science (Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Jovanovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The expected dou- bling of the instrumental throughput will significantly improve its sensitivity (Jovanovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Echev- erri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The next generation of high-contrast high-resolution spectrograph such as HISPEC at the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Keck Observatory and MODHIS on the future Thirty Meter Telescope (TMT) will undoubtedly open 4 Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' new frontiers in this field by allowing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='98 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='46 µm si- multaneous coverage at an average spectral resolution R > 100, 000 (Mawet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Mawet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Outline In section 2, we present exomoon RV detection lim- its for the brown dwarf companion HR 7672 B using KPIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In section 3, we then simulate observations of the same brown dwarf and the planet HR 8799 c with next generation facilities and compare their sensitivity to the moons in the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In section 4, we explore the parameter space of satellites that could be detected with TMT/MODHIS as a function of planet properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Fi- nally, we conclude on the prospects for RV detections of exomoons in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' EXOMOON LIMITS AROUND HR 7672 B WITH KPIC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Observations and data reduction The brown dwarf companion HR 7672 B was observed three times in 2020 and then for a full night on July 4, 2021 with KPIC (R ∼ 35, 000) in K band (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='9−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4 µm) (Mawet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Delorme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' These obser- vations are detailed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The first three epochs included one to two hours of on-target exposures per night and were already published in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2022a) and Delorme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Unfortunately, the condi- tions on July 4, 2021 were well below average with the companion undetectable in some individual 5-minute ex- posures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' During this one night specifically, we used an ABAB pattern to nod the companion between two KPIC fibers, fiber 1 and 2, to limit or identify any fiber-specific biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' There was no nodding during the other epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The data was reduced with the KPIC data reduction pipeline (DRP)1 following the same approach described in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2021b, 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The first steps include background subtraction, bad pixel correction, and the calibration of the fiber trace location and width on the detector for each NIRSPEC spectroscopic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Opti- mal extraction is then used to extract the spectra and the wavelength solution is derived from the telluric and stellar lines of a M giant, namely HIP 81497, taken on the same night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For this purpose, the telluric model is generated with the Planetary Spectrum Generator (Vil- lanueva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2018) and star is modeled by a Phoenix model (log(g/[1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−2]) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Teff = 3600 K Husser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Forward model and likelihood 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='com/kpicteam/kpic pipeline We use a forward modelling approach similar to Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2021b) and Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2021) to measure the RV of HR 7672 B, which includes a joint modelling of the starlight and the companion signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2021c) showed that the continuum could be included in the for- ward model with a fourth order polynomial, therefore not requiring the data to be high-pass filtered nor contin- uum normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In this work, we model the continuum using a spline-based linear model, which can be ana- lytically marginalized using the general purpose python module breads2 (Broad Repository for Exoplanet Anal- ysis, Discovery, and Spectroscopy) based on the formal- ism in Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The spline forward mod- eling has the advantage of being more robust to bad pixels than a Fourier based high-pass filter and avoids the non-linearity of a sliding-window median filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The spline parameters are also easier to optimize than the coefficients of a high order polynomial for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We define the forward model as, d = MRVφ + n, (1) where d is the data vector of size Nd, MRV is the linear model, φ are the linear parameters, and n is a random vector of the noise with a diagonal covariance matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' A scaling factor for the noise is also fitted to model any underestimation of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Off-diagonal elements in the covariance matrix are neglected here, but subsequent data processing steps would correct for this inaccuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The different column vectors of the linear model are il- lustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The data vector and the standard deviation of the noise used to define Σ0 are direct out- puts of the KPIC data reduction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The variance of the noise is multiplied by a free parameter scaling fac- tor s2 to account for any underestimation of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' KPIC includes four single mode fibers separated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='8′′ on a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We can therefore acquire simultaneous spectra of the companion and the host star, more specif- ically the speckle field, by rotating the field of view using the Keck II adaptive optics system front-end K-mirror rotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The observations of the star are used to derive simultaneous empirical models of the transmission and the starlight spectra used in the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The starlight is used to model the speckle noise leaking into the fiber at the position of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The wave- length calibration is different in each fiber so the spectra are linearly interpolated to match the sampling of the science fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The planet model is defined as the spin- broadened best fit model from Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2022a) using petitRADTRANS (Molli`ere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2019) multiplied by 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='com/jruffio/breads RV detection of exomoons 5 Moon Planet Mass Mass ratio Semi-Major axis Period RV semi-amplitude (M⊕) (RJup) (day) (m/s) Io Jupiter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='50 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='71 × 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='82 Europa Jupiter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='04 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='53 × 10−5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='39 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='35 Ganymede Jupiter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='48 × 10−2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='81 × 10−5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='97 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='85 Callisto Jupiter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='80 × 10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='67 × 10−5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='33 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='46 Titan Saturn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='25 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='37 × 10−4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='09 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='32 Titania Uranus 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='73 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='94 × 10−5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='14 Oberon Uranus 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='82 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='32 × 10−5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='16 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10 Triton Neptune 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='58 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='09 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='96 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='92 Kepler-1708 b-i Kepler-1708 b < 37 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='11(2σ) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Properties of the largest satellites orbiting the solar system gas giants from the NASA Space Science Data Coordinated Archive (https://nssdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='gov/planetary/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The negative period of Triton is indicating its retrograde orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Kepler-1708 b-i is a transiting exomoon candidate (Kipping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The period and RV semi-amplitude for these moons can also be found in Vanderburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Object Date Exposure time Seeing Throughput HR 7672 B 2020-06-08 11 × 10 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4′′ 1% HR 7672 B 2020-06-09 10 × 10 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='6′′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5% HR 7672 B 2020-09-28 7 × 10 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4′′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='7% HR 7672 B 2021-07-04 61 × 5 min 1′′ 2% Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' K-band observations of HR 7672 A and B with KPIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The quoted throughput is the end-to-end from the top of the atmosphere, which is a better proxy of performance than the seeing for KPIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' the empirical telluric and instrument transmission pro- file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The continuum of both the planet and the speckle are modulated by a 3rd order spline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Ten spline nodes are used in each spectral order (∆λ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='05 µm) for the planet model to manage any inaccuracies in the con- tinuum due to imperfections in the atmosphere model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This number of nodes is analogous to a 200 pixel- wide high-pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The number of nodes was chosen as a trade-off between the number of additional param- eters and the optimal high-pass filter scale of 100 pixels found in Xuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The speckle continuum is modeled with three spline nodes to model any speckle crossing the fiber location as the wavelength changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This results in 13 linear parameters per spectral order representing the values of the continua at the location of the nodes (See Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This defines the linear model MRV with dimensions Nd × 13, which is also a function of the RV of the planet, the only non-linear parameter fitted for here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' KPIC data features strong spectral fringing due to the FabryP´erot cavities formed by the transmissive op- tics inside the NIRSPEC spectrograph (Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021) and within the KPIC fiber injection unit (Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This effect is made worse by the high spatial co- herence of the wavefront in KPIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We therefore apply a Fourier filter to the data and the forward model by zero- ing frequencies corresponding to the fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' A physical model of the fringing such as Cale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2019) could be explored in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The likelihood function is defined from a multivariate Gaussian distribution as, L(RV, φ, s2) = 1 � (2π)Nd|Σ0|s2Nd exp � − 1 2s2 (d − MRVφ)⊤Σ−1 0 (d − MRVφ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2) The likelihood is maximized using a linear least square solver on a grid of RV values from −400 to 400 km/s in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The 1σ RV uncertainties are derived from the RV posterior calculated analytically according to Equation 10 in Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2021) on this RV sam- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This method analytically marginalized the RV posterior for the modulation of the continuum and the noise scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The linear spline parameters used to fit the continuum are forced to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This is theoretically inconsistent with the framework, which as- sumes unconstrained parameters, but it does not appear to significantly impact the RV time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Only the three reddest orders, out of nine in K band, are used in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The bluest three orders (num- bered 39-37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='94 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='09 µm) were discarded because they feature strong saturated CO2 telluric lines that are generally harder to model, but also make for an unsta- ble fit due to overlapping frequencies with the fringing 6 Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' and the simple Fourier filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The middle three orders (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='27 µm) lack sufficient stellar and telluric spec- tral lines to calibrate the wavelength precisely enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Thus, only the remaining three orders are used in this analysis: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='29−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='34 µm (order 33), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='36−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='41 µm (order 32), and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='44 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='49 µm (order 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Order 33 includes the carbon monoxide bandhead and therefore results in the strongest signal-to-noise ratio (S/N) and the most precise radial velocity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Each NIRSPEC spectral order is fitted separately resulting in three RV estimates for each exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' RV measurements The barycentric corrected RV measurements for the four epochs and three orders are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Following the method described in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2, the median RV uncertainties in five minute exposures are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='9 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1 for order 6, 7, and 8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We overplot the predicted radial velocity of the brown dwarf from orbital fits to the relative astrom- etry from Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2012) and RV measurements of the host star (Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The orbit fits were done with orbitize!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Blunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2020) following its RV tutorial3 and using the emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2013) sampler to obtain a pos- terior of allowed orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This orbital RV of the compan- ion in each epoch is predicted from this orbit fit and is subsequently subtracted from the estimated RV of the planet when running the exomoon search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Similarly to fitting the centroid of a Gaussian (King 1983), the RV precision goes as the typical linewidth in the spectrum divided by the total S/N of the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In the case of HR 7672 B, the large spin with v sin i = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1 (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022a) is a limiting factor in deriving more precise RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The impact on the exomoon sensitivity of other fundamental parameters such as the brightness, age, mass, and separation from the star are discussed in section 4 in the context of TMT/MODHIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Exomoon sensitivity The open-source Python package RVSearch4 (Rosen- thal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021) is used to look for possible exomoons around HR 7672 B and derive the sensitivity of our KPIC RV time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' RVSearch is a planet search algo- rithm that was developed by the California Legacy Sur- vey for high-precision radial velocity surveys (Howard & Fulton 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Planets are detected from periodograms, which are ex- 3 https://orbitize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='io/en/latest/tutorials/ RV MCMC Tutorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='html 4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='com/California-Planet-Search/rvsearch pressed as the difference in Bayesian Information Cri- terion (BIC) between a model including the planet and a model without it (Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The ∆BIC can be used to select the model that best represents the data, or, in other words, determine if a planet is neces- sary to explain the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Planet candidates are detected by iteratively adding additional planet signal to the model (Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For each iterative search, the algorithm fits a detection threshold to the periodogram using the power law noise model described in Howard & Fulton (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' To characterize the search completeness of a dataset, RVSearch performs injection- recovery tests, drawing many synthetic planet signals, injecting them in the data, and checking whether their signals surpass the last detection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The sim- ulated signals were injected as described in (Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021) with period and M sin i from log-uniform distributions, and eccentricity from an empirically cali- brated beta distribution (Kipping 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' RVSearch is directly applicable to the search for ex- omoons by replacing the properties of the star by the ones of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We assume that each spectral order in NIRSPEC has a different zero RV point due to pos- sible inconsistencies between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This can be done with RVSearch, which linearly solves for offsets between subsets of RVs, and uses a wide, Gaussian, uninforma- tive prior on white noise for each subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This fea- ture is usually used to fit data from different instru- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Two analyses are performed, first only using the long night of observations (07/04/2021) and then all the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The latter provides a longer time base- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The resulting periodograms and exomoon com- pleteness are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' By combining the four epochs, the observations are sensitive to satellites with a mass ratio of 1% at semi-major axes similar to that of Io (6RJup) around Jupiter or 4% at the distance of Callisto (15RJup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' While these are encouraging results, the smallest detectable satellites would be as large as Jupiter due to the already large mass of HR 7672 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' As shown in section 4, targeting smaller brown dwarfs and planets does not generally allow the detection of moons with smaller absolute masses, because the S/N drops faster than the mass of the object due to the decreas- ing brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' If satellites around HR 7672 B were to orbit within ∼ 10RJup of the brown-dwarf, they would likely fall within the Roche radius (See Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Such satellites would be tidally disrupted and likely result in the formation of rings around the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' It is possible that this issue would prevent the formation of a reso- nant chain of satellites if the inner edge of the decretion disk falls within the Roche limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This is for example cited as a possibility to explain the difference between RV detection of exomoons 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='33 200 0 200 400 600 Data number Data Combined model Planet model Starlight model Residuals Data uncertainty 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='33 200 0 200 400 600 Data number Planet model Sub-components Single sub-component 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='33 ( m) 20 0 20 40 60 80 100 Data number Starlight model Sub-components Single sub-component Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Illustration of the forward model used to derive the RV of HR 7672 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This figure shows a single NIRSPEC order overlapping with the CO bandhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Top panel) A planet and a starlight model are jointly fitted to the data to account for the diffracted starlight contamination at the location of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The data uncertainty measured by the KPIC DRP (shaded grey) slightly underestimates the amplitude of the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Center panel) The planet model is itself made of a linear combination of ten spline modes to model the continuum of the companion spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Bottom panel) The starlight intensity is also fitted with a spline using three nodes to account for speckles crossing at the location of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This flexible model of the continuum is an alternative to high-pass filtering and continuum normalization of high-resolution spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' the Galilean and the Saturnian satellite systems in Baty- gin & Morbidelli (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' At the other end of possible satellite semi-major axes, stable orbits can generally ex- ist up to one half of the Hill sphere for prograde orbits (Shen & Tremaine 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The Hill sphere of HR 7672 B being rH ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='6 au = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2 × 104RJup, time series like these ones will not be sensitive to the vast majority of possible orbits without observations spanning years or decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' FUTURE PROSPECTS FOR HR 7672 B AND HR 8799 C 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Simulations In this section, we simulate observations from current and future instrumentation at the Keck observatory and TMT to estimate the properties of putative satellites that should be detectable using planetary RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We use an instrument and observation simulator called PSIsim 5, which was first developed for the Planet Systems Im- ager (PSI Fitzgerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022) instrument concept for TMT, and then expanded to include other instruments and telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' PSIsim is first used to estimate the RV precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Then, RV times series are simulated assuming 6 full nights of observations over 25 days, and the ex- omoon sensitivity is finally computed using RVSearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' These simulations are meant to represent an ideal sce- nario in terms of instrument performance and telescope time allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We simulate observations of two substellar compan- ions, the brown-dwarf companion HR 7672 B and the planet HR 8799 c, with four generations of instru- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' An exhaustive analysis of all directly imaged companions is beyond the scope of this work so HR 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='com/planetarysystemsimager/psisim 8 Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='308804+t/24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 time (h) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='421342+t/24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='00 time (h) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 RV (km/s) 09-28-2020 MJD=59009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='452114+t/24 0 1 2 3 4 5 6 7 time (h) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 RV (km/s) 07-04-2021 MJD=59120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='258465+t/24 Prediction fiber 1 order 6 fiber 2 order 6 fiber 1 order 7 fiber 2 order 7 fiber 1 order 8 fiber 2 order 8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Measured RVs of HR 7672 B with KPIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The grey lines are predicted RVs from one hundred posterior samples of the orbital motion of the brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 8799 c was chosen as a representative example of the field with a planetary mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' HR 8799 is also the only other high-contrast system with published RV time series and exomoon upper limits (Vanderburg & Ro- driguez 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The four instruments considered in this work are Keck/KPIC I, Keck/KPIC II, Keck/HISPEC, and TMT/MODHIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' KPIC I corresponds to obser- vations carried out pre-2022A (Delorme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' KPIC II refers to the series of upgrades started dur- ing the first semester of 2022 with the primary goal of doubling the instrument throughput Jovanovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Echeverri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The High-resolution Infrared Spectrograph for Exoplanet Characterization (HISPEC) is expected to provide Y-K (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='98 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='46 µm) spectroscopy at a spectral resolution of R > 100, 000 (Mawet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The Multi-Object Diffraction- limited High-resolution Infrared Spectrograph (MOD- HIS) is a similar instrument to HISPEC planned for the future TMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' A broader range of exoplanet masses is explored in section 4 for this latter TMT instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' PSIsim includes full budgets of the throughput and thermal background for each instrument, telescope, and the Earth atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The Strehl ratio is calculated based on a empirically calibrated model of the adaptive optics’ performance under median seeing conditions for Maunakea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For KPIC I and KPIC II, we assumed Keck AO’s current performance with the infrared Pyramid Wavefront Sensor described in Bond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For HISPEC, we assumed extreme-AO performance as pre- dicted for the upcoming HAKA high-density deformable mirror upgrade (W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Keck Observatory, private com- munication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The star is modeled with a PHOENIX model (Husser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2013) and the substellar companion with a BT-Settl atmospheric model grid6 (Allard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Table 3 includes the input parameters and the predicted RV precision for these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The simu- lations include a level of systematics at 1% of the contin- uum, which is modeled by an additional white Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Otherwise, the estimated RV precision assumes a perfect data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 6 https://phoenix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='ens-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='fr/Grids/BT-Settl/CIFIST2011c/ RV detection of exomoons 9 10 2 10 1 100 101 102 Period (day) 10 8 6 4 2 BIC All data, BICthresh = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1 One night, BICthresh = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 (a) Periodogram 100 101 102 Semi-major axis (RJup) 10 3 10 2 10 1 100 MMoon sini/MBD Injection recovered Injection missed 50% completeness - One night 50% completeness - All data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0 Probability of detection (b) Completeness Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Exomoon detection limits around HR 7672 B with the Keck Planet Imager and Characterizer (KPIC) using the open-source python module RVsearch (Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Left) Periodogram of the RV times series shown in Figure 2 expressed a ∆BIC comparing a model with and a model without a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The empirical detection threshold is indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Right) Exomoon completeness derived from injection and recovery tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The periodogram and the completeness are shown for two cases: the single full night of observations on 07/04/2022 and all the available data including three additional epochs with 1-2 hours of data each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The variable conditions on 07/04/2022 led to HR 7672 B to not be detected during portions of the night, or in the RV precision to get significantly worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' By simulating RV time series, we estimate that the lost data only affected the final sensitivity by 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The predictions from PSI-sim are about a factor two more sensitive than existing measurements with KPIC I (See Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This difference can first be explained by uncorrected wavefront errors reducing the throughput, both non-common path aberrations and uncorrected at- mospheric turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Then, our current data analy- sis framework remains limited in its ability to model KPIC systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' As explained in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2, only the redder orders of NIRSPEC are being reduced due to strong telluric lines in the bluer orders, and an im- perfect Fourier filtering is used to remove the fringing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The gap between the simulations and the measurements should decrease as observing strategies and data reduc- tion frameworks are improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The final expected exomoon sensitivity of the four in- struments is shown for the two companions in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For a fixed time sampling of the RV series, the minimum detectable mass ratio is approximately proportional to the RV semi amplitude of the signal, which is also pro- portional to the RV precision of the instrument, so the improvement for each generation of instrument can be read from the simulated RV precision shown at the bot- tom of Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' These simulations are compared to other detection techniques in Appendix A, specifically astro- metric monitoring of the companion or spatially resolv- ing the moon through imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We separately discuss the possibility of detecting transiting exomoons using the Rossiter-McLaughlin (RM) effect in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Comparing to solar system moons The mass ratios of the largest gas giant satellites in the solar system are also shown in Figure 4 for compar- ison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The higher planet masses, M, of directly imaged planets and brown dwarfs compared to the solar sys- tem could yield significantly bigger moons, so we also include scaled-up mass ratios, q, according to q ∝ √ M (Batygin & Morbidelli 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' While the CPD does scale with the Hill Sphere, we do not expect the semi-major axis of satellites to depend on this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Indeed, young moons are thought to migrate toward the planet during their formation due to the interaction with the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The migration is stopped at the inner radius of the CPD which is set by the magnetic field of the planet (Batygin & Morbidelli 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In this work, we therefore keep the semi-major axis of the solar system satellites constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' A caveat is that large moons could be suscep- tible to tidal forces if they form or migrate too close to the planet within the Roche limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The Roche limit is calculated using the mass-radius relationship from Chen 10 Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' & Kipping (2017) and their associated Python package7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' However, this relationship does not account for the fact that young objects are likely inflated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Parameters Star - Phoenix model HR 7672 HR 8799 Apparent K mag 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2a Effective temperature (Teff) 6000 Kb 7400 Kc Surface gravity (log(g)) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5b 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5c Spin (vsin(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' km/s) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='6d 49e Companion - BTsettl model HR 7672 B HR 8799 c Mass 73MJupb 7MJupf Apparent K mag 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0b 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1g Effective temperature (Teff) 1800 Kb 1200 Kh Surface gravity (log(g)) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5b 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0h Spin (vsin(i)) 45 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1b 10 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1i Separation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='72′′j 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='95′′j Telescope and instrument airmass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2 water vapor column 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 mm integration time (tint) 5 min Predicted RV sensitivity (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1) assuming 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='6′′ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0′′ seeing HR 7672 B HR 8799 c Keck/KPIC I (measured) ∼ 2, 000k ∼ 7, 000i Keck/KPIC I (simulated) 800-1400 3,000-5,000 Keck/KPIC II 500-800 2,000-3,000 Keck/HISPEC ∼ 200 100-200 TMT/MODHIS 30-40 10-20 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Radial velocity (RV) precision simulations of cur- rent and future instrumentation for two substellar compan- ions: HR 7672 B and HR 8799 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Top) Representative pa- rameters for the telescope, instrument, star, and companions used in the PSIsim simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Bottom) Predicted RV sen- sitivity for values of seeing ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='6′′ to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' References—a: Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2003), b: Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2022a), c: Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2020), d: Luck (2017), e: Royer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2007), f: Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2018), g: Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2011), h: Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2018), i: Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (2021b), j: http://whereistheplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='com/ (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021a), k: This work 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' FUTURE EXOMOON SENSITIVITY OF TMT/MODHIS Looking to the future, we expect substantial gains in RV precision by using the next generation of high- resolution spectrographs on large telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' These 7 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='com/chenjj2/forecaster gains in RV precision will lead to enhanced sensitivity to systems with lower mass, close in exomoons, which would form in a similar way to the Galilean moons around Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Using the same framework as in section 3, we calcu- late the RV sensitivity for a variety of simulated planets that could exist around a host star with the properties of HR 8799 referenced in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We modeled planets with varying effective temperatures and apparent mag- nitudes, fixing the separation between the planet and star to 700 mas and the surface gravity of the planet to log(g) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−2, and used PSIsim to calculate the RV sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The effect of the starlight contamination on RV sensitivity can be neglected for the type of di- rectly imaged planets that are known today and would be observed with TMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The RV sensitivity vary by less than 20 percent for planets that lie beyond 500 mas and have a flux ratio greater than ∼ 3 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' On aver- age, for every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 dex change in surface gravity on the planet, the RV sensitivity changes by ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='7 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Fig- ure 5 (a) shows the RV sensitivity MODHIS could have for a single, two hour exposure, for planets of varying effective temperatures and apparent magnitudes around an HR 8799 like star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The RV sensitivity of MODHIS driven by the brightness of the planet more than than its temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' However, the RV sensitivity is decreased for planets with temperatures between 1500 and 1700 K using the BT-settl model grid due to the L-T transi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' At these temperatures, clouds form in the upper layers of the atmosphere, shrouding detectable spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For a given planet temperature and magnitude, the RV precision of TMT/MODHIS Figure 5 (a) can be compared to the RV semi amplitude in Figure 5 (b) as a function of the planet mass, the mass ratio, and the period of the satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' However, such a comparison assumes multiple epochs of observations with a given sensitivity in order to detect a moon with a similar RV semi amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In the following, the surface gravity, temperature, and mass of the planet are treated more self-consistently us- ing BT-Settl evolutionary grids (Allard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2012b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The dependence of the exomoon sensitivity to the num- ber of observations is also made explicit by using simu- lated RV time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We therefore express the RV pre- cision and exomoon sensitivity as a function of planet mass and distance to the Sun in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We fixed the age of the system to different values to represent the parameter space occupied by different populations of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The 3 Myr age group is representative of the youngest stars, such as those found in star forming re- gions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Ophiuchus, Taurus, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The 30 Myr age group is representative of young moving groups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' such ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='RV detection of exomoons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='100 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Titan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Triton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Titania ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Oberon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='KPIC I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='KPIC II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='HISPEC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='MODHIS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1wk 1mo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1yr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10yr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='100yr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='(MMoon/MEarth) sini ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='(b) HR 8799 c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Future prospects for exomoon detections around the brown dwarf companion HR 7672 B (left) and planet HR 8799 c (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Simulated sensitivity for Keck/KPIC I, Keck/KPIC II, Keck/HISPEC, and TMT/MODHIS are shown in colored curves assuming 6 nights of observations over 25 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The sensitivity demonstrated in this work from ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 nights of KPIC observations is labeled as KPIC I (Data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The mass ratios of the Galilean satellites are shown as black dots for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Their predicted scaled-up mass ratios, q, accounting for the larger mass, M, of the brown dwarf compared to Jupiter are shown as grey crosses (q ∝ √ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Batygin & Morbidelli (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The Roche limit is computed for both a rigid and a fluid satellite shown as the inner and outer greyed region respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' as Beta Pictoris Moving Group and the Tucana and Horologium Associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The 300 Myr age group is representative of the oldest directly imaged substellar companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The RV sensitivity decreases the further the system is away at each distinct age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For younger systems, there is larger decrease in sensitivity as the mass of the planet decreases below ∼ 13 MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The large decrease in RV sensitivity once the object is below ∼ 13 MJup is due to the onset of deuterium burning for brown dwarfs, which makes them much more luminous than a planet of a similar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Another interesting fea- ture in Figure 6 (a) is the apparent independence of the RV precision to the brown dwarf mass above ∼ 13 MJup at 30 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This can be explained by the facts that the RV precision is mostly driven by the brightness of the object, and that brown dwarfs have a similar brightness over a range of masses around this age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Indeed, larger brown dwarfs cool faster than smaller ones resulting in the different cooling curves to meet over a small range of brightness around 30 Myr as illustrated in Figure 7 in Burrows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Figure 6 (b) shows the moons that could be detected around a planet from Figure 6 (a) if they were placed at the distance of Callisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For each planetary mass and distance, we create an RV time series assuming six full eight-hour nights of observations over 25 days with er- ror bars that represent the RV sensitivity calculated by PSIsim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The detection threshold was computed from simulated data created by RVsearch as in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For more massive planets and brown dwarfs, we expect TMT/MODHIS to reach the RV sensitivity needed to look for close in moons with mass ratios smaller than 10−4 around brown-dwarfs, similar to the ones found in the solar system for a median age of 30 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' However, to detect moons around lower mass, directly imaged plan- ets of the same age, we are sensitive to mass ratios of 10−3 or larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The viability of exomoon RV searches Using KPIC, we derive the most sensitive upper limits on the mass ratio of satellites orbiting a high-contrast substellar companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We rule out satellites larger than 1-4% the mass of the brown dwarf HR 7672 B at separations similar to the Galilean moons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Based on end-to-end simulations, we predict that instruments such as TMT/MODHIS could be two orders of mag- nitude more sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This would be sufficient to de- tect moons forming in the CPD of a planet with mass ratios of ∼ 10−4, albeit with a substantial investment in observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' If the satellite to planet mass ra- tio grows as q ∝ √ M, with M the mass of the planet, 12 Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 5 10 15 20 25 Planet Magnitude 500 1000 1500 2000 2500 Teff (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 m/s 1 m/s 2 m/s 5 m/s 20 m/s 100 m/s 500 m/s 1 km/s 3 km/s 100 101 102 103 RV Sensitivity (m/s) (a) RV sensitivity 100 101 102 1 day 1 m/s 10 m/s 100 m/s 1 km/s 10 km/s 1 mo 1 m/s 10 m/s 100 m/s 1 km/s 10 4 10 3 10 2 10 1 100 Mass ratio (q) 100 101 102 Planet mass (MJup) 1 yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1 m/s 1 m/s 10 m/s 100 m/s 1 km/s 10 4 10 3 10 2 10 1 100 10 yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1 m/s 1 m/s 10 m/s 100 m/s 1 km/s (b) RV semi-amplitude Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' RV precision of MODHIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Left) The RV sensitivity of MODHIS for model planets around a HR 8799-like host star using BT-Settl models (Allard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The RV sensitivity was predicted using PSIsim for a single, two hour exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Both the contour curves and color map indicate the RV sensitivity for a specified effective temperature and apparent magnitude of the model planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The RV sensitivity relies more on the brightness of the planet than its effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' However, the RV sensitivity decreases for planets with temperatures between 1500 and 1700 K due to the L-T transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Right) The RV semi-amplitude for different planet masses and mass ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Note, increasing the exposure time will increase the RV sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' the Keck/HISPEC should be sensitive to these objects around brown dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Any detection with HISPEC, or lack thereof, will therefore already be capable of con- straining CPD formation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' In order to validate our instrument simulations, we compared them with existing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The gap in sensitivity can be explained by imperfections in the data reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' A continued investment in more accurate data processing algorithms or observing strategies is therefore required in order to realize these predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Planet variability will also be a challenge to overcome using the different timescales and the wavelength dependence of the vari- ability compared an exomoon signal for example (Van- derburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Measuring the variability of sub- stellar companions would in fact be an important result of exomoon surveys to better understand the physics of their atmospheres (Biller 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Binary formation processes favor high-mass ratios so they would be more easily detectable than the smaller satellites forming by accretion in the CPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The ma- jority of multiplicity surveys for isolated brown dwarfs (Fontanive et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2018), or companion brown dwarfs (Burgasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Lazzoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2020), have searched for visual companions, leaving the separation regime of < 1 au underexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Figure 5 (b) shows that unresolved binary substellar companion would be detectable with RV precision between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='1 − 1 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1, which is already routinely achieved with KPIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' As an example, the measured dynamical mass of the brown dwarf companion HD 47127 B suggest that it could be a binary (Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021), but this specific compan- ion is too faint (K ∼ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4) to be a practical target for KPIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' From Appendix A and Figure 7, we conclude that the different detection techniques are sensitive to distinct regions of the parameter space, and therefore comple- mentary, not unlike exoplanet searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' If exomoons follow the model of solar system gas giant satellites, RV searches could be the most promising approach due to its sensitivity to short period moons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' However, unless the theoretical prediction that bigger planets form even bigger moons hold true, small satellites with mass ratios ∼ 10−4 might only be detectable around brown dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Detections of moons using the Rossiter-McLaughlin effect An alternative strategy to look for exomoons around directly imaged planets using RV measurements could be to look for transiting moons through the Rossiter- McLaughlin (RM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Gaudi & Winn 2007) effect on the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Precise photometric calibration and stability of high-constrast instrument is notoriously difficult (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2022b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' so detecting a RM event during a transit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='RV detection of exomoons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='30 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='50 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Planet Mass (MJup) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='3 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='4 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='50 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='100 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Distance (pc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='50 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='30 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='40 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='50 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='100 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='200 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='RV Sensitivity (m/s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='3 Myr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='30 Myr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='300 Myr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='(a) RV sensitivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5e-5 5e-5 1e-4 5e-4 1e-3 4 10 20 30 50 70 Planet Mass (MJup) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5e-5 5e-5 1e-4 5e-4 1e-3 5e-3 1e-2 20 40 60 80 100 120 140 Distance (pc) 11 20 30 40 50 60 70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5e-5 5e-5 1e-4 5e-4 1e-3 5e-3 1e-2 10 5 10 4 10 3 10 2 (MMoon/MBD) sin i 3 Myr 30 Myr 300 Myr (b) Detectable mass ratio Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' RV precision and detectable mass ratio of MODHIS similar to Figure 5, but as a function of planet mass, distance, and age of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Left) BT-Settl evolutionary models (Allard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2012b) were used to infer the mass of the planet and distance to the system at an age of 3, 30, and 300 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The minor contour lines cover an evenly spaced, 50 step log scale from 0 to 1 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' RV sensitivity decreases the further the system is away and the lower in mass the planet is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The large decrease in RV sensitivity when the companion mass is below ∼ 13 MJup for young systems is due to the difference in cooling rates between brown dwarfs and planets over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' (Right) The mass ratio detectable by MODHIS assuming a fixed semi major axis for the moon equal to that of Callisto (≈ 26RJup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For each planetary mass and distance from panel (a), we create an RV time series assuming six nights of observations over 25 days with error bars that represent the RV sensitivity calculated by PSIsim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' could be easier than detecting its photometric counter- part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' An RM event consists of the subsequent masking of a portion of the blue and red-shifted areas of the surface of a spinning object, therefore leading to large and very distinct deviations of the measured RV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The amplitude of the RV signal can be hundreds of times larger than the RV semi amplitude due to the orbital motion of the moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Its amplitude is proportional to the spin of the planet, which could make it an interesting alternative to detect the smallest moons around rapidly rotating planets and brown dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Indeed, the RV uncertainties scale with the spin of the object so detecting the orbital signal of small exomoons could be more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The Galilean moons have rather small orbital periods from days to weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Assuming a random inclination dis- 14 Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' tribution, the transit probability of a moon (P) is given by the ratio of the planet radius (Rp) and the moon semi-major axis (dm), P = Rp/dm (Borucki & Sum- mers 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Therefore, the probability of a transit of a moon at the separation of Io around Jupiter is 1:6, and 1:27 for the farthest Galilean moon Callisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' As- suming a full 8 hour night of observations, we estimate the probability of observing an RM event for Galilean- like moons around a Jupiter like planet to be around 3% for Io, 1% for Europa, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='3% for Ganymede, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='07% for Callisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' However, the orbital periods of the moons would be even shorter around larger substellar compan- ions, which would increase the probabilities up to 17% for Io, 8% for Europa, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='6% for Ganymede, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='6% for Callisto .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The transits would last between ∼2-5 hours for the Galilean moons around Jupiter, but they would only last 15-30 minutes for similar moons around HR 7672 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' As an example, a satellite around HR 7672 B with a mass of 1M⊕ (q = 5×10−5) would generate a RM signal of ∼ 300 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1 compared to the ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1 generated by the orbital motion (Gaudi & Winn 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The am- plitude would be ∼ 5 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='s−1 for a Neptune-size moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Multiple satellite systems would increase the probability of a detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Given the low detection probability, RM searches could be carried out in synergy with other sci- ence cases such as brown dwarf variability (Biller 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' For example, Doppler spectroscopy also favors long ob- servations of rapidly rotating objects, which would make for ideal datasets for exomoon RM searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Searching for Pandora: Habitable exomoons Estimating the occurrence rate of Earth-sized exo- planets in the habitable zone (HZ) of Sun-like star, called η⊕, has been an important goal of exoplanet sur- veys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' While such planets remain challenging to detect, the best estimates of η⊕ range between 5 − 50% to date (Gaudi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' However, these are not the only Earth-sized objects that could harbor life in the HZ of their stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Any rocky satellites orbiting HZ gas giant planets could also provide suitable conditions for life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Close-in exomoons can be protected from stellar radia- tion by the strong magnetic field of Jovian mass planets (Heller & Zuluaga 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Integrating the distribution of gas giants with an inci- dent flux between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='3−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='5 times the solar irradiance on Earth for an optimistic habitable zone, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='3 − 1 for a conservative habitable zone (Kasting & Harman 2013), yields about 5 − 7 giant planets per hundred FGKM stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' This is using the giant planet (30 − 6000M⊕ sin i) occurrence rates derived from the California Legacy Sur- vey as a function of stellar irradiation (figure 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Given that each planet can have multiple satellites, this could represent a significant number of habitable Earth-size moons that are not accounted for in η⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The occurence rate of habitable exomoons could be constrained by measuring the population of satellites around more distant directly imaged planets and brown- dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' CONCLUSION In this work, we aimed at evaluating the prospects for radial velocity (RV) detections of exomoons around self-luminous directly-imaged planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We used real ob- servations as well as end-to-end simulations of future facilities at the Keck observatory and the Thirty Meter Telescope (TMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Using data from KPIC, we were able to derive upper limits for satellites orbiting the brown dwarf companion HR 7672 B at a mass ratio of 1−4% for separations similar to the Galilean moons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Current in- strumentation is already sensitive to unresolved binary companions that could form through gravitational in- stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We demonstrate that future thirty-meter class telescopes will likely push the sensitivity down to the mass ratios of solar system satellites (∼ 10−4), which are thought to form in a circumplanetary disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We note that second generation instruments like Keck/HISPEC on current ten meter class telescopes might be sufficient to detect these moons if theoretical predictions that larger planets form even larger moons hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Everything else being equal and considering the RV signal from the orbital motion of the moon, the deepest exomoon sen- sitivity will be reached for the brightest substellar com- panions with the smallest spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Small moons could also be detected from their Rossiter-McLaughlin (RM) ef- fect on the planetary RV signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' An RM event can be orders of magnitude larger than the orbital signal, albeit with percents level detection probability assuming a full night of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We conclude that the detection of exomoons from planetary RV surveys is now becoming a reality thanks to the development of high-resolution spectrographs dedicated to directly imaged planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' RV detection of exomoons 15 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' acknowledges support from the David and Ellen Lee Prize Postdoctoral Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 1 2 Funding for KPIC has been provided by the California Institute of Technology, the Jet Propulsion Laboratory, the Heising-Simons Foundation through grants #2019- 1312 and #2015-129, the Simons Foundation, and the United States National Science Foundation Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' AST-1611623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 3 4 5 6 7 8 Ji Wang acknowledges the support by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2143400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 9 10 The W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Keck Observatory is operated as a scien- tific partnership among the California Institute of Tech- nology, the University of California, and NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The Keck Observatory was made possible by the generous financial support of the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Keck Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We also wish to recognize the very important cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We are most fortunate to have the opportunity to conduct ob- servations from this mountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 11 12 13 14 15 16 17 18 19 20 Facilities: Keck II (KPIC) Software: astropy8 (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2013), Matplotlib9 (Hunter 2007), PSIsim10, RVSearch11 (Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021), KPIC Data Reduction Pipeline12 (Delorme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021), BREADS13 (Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Agrawal 2022), APPENDIX 8 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='astropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='org 9 https://matplotlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='org 10 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='com/planetarysystemsimager/psisim 11 https://github.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='10 as astrometry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='100 as astrometry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Io ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Europa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Ganymede ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Callisto ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Titan ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='(MMoon/MEarth) sini ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='(b) HR 8799 c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Similar to Figure 4, but including idealized exomoon sensitivities of alternative detection techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The diagonal dashed black lines represent the simplified sensitivity of VLTI/Gravity through astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The vertical gray scale bars represent the diffraction limit of different telescopes for direct imaging of satellites, namely the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Keck observatory, the future Thirty Meter Telescope (TMT), and the Very Large Telescope Interferometer (VLTI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' COMPARING TO OTHER DETECTION METHODS Alternative exomoon detection techniques include astrometry and direct imaging of imaged planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Figure 7 shows their idealized detection limits to be compared to the RV sensitivity originally presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' With an astrometric precision of 10 − 100 µas (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' 2021), interferometry with VLTI/GRAVITY could be sensitive to moons further away than radial velocity, but remains limited by the orbital period of the satellite at the furthest separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The simplified detection limits are computed by matching the astrometric precision (σastro) of VLTI/GRAVITY with the amplitude of the planet astrometric displacement in the sky around the center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' The smallest detectable mass ratio (q) is given by q = � 2 ∗ �sma 1 au � �1 pc d � � 1 as σastro � − 1 �−1 , (A1) with d the distance of the star to the Sun, and sma the semi major axis of the moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' We use the diffraction limit of the telescope to illustrate the parameter space that might be accessible to direct imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' More specifically, the detection threshold is taken at twice the spatial resolution of the telescope (∼ 2λ/D) with D the diameter of the telescope and λ = 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Unfortunately, estimating the brightness of low mass objects (< 1MJup) remains challenging and will depend on the age of the system, so we arbitrarily chose a lower limit of one Jupiter mass for Keck and VLTI, and a mass similar to the solar system ice giants for TMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQf7AjQ/content/2301.04206v1.pdf'} +page_content=' Direct imaging would be sensitive to the longest periods and largest moons.' metadata={'source': 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b/6dFKT4oBgHgl3EQfTi2q/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2147b5df9242e13481616983b44bc12ed4429ea0cd401d4ec2d1648b67c20d94 +size 165728 diff --git a/7dAyT4oBgHgl3EQfpvi2/content/tmp_files/2301.00532v1.pdf.txt b/7dAyT4oBgHgl3EQfpvi2/content/tmp_files/2301.00532v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5420017353ef3a55394f70149653f9ca3977df1d --- /dev/null +++ b/7dAyT4oBgHgl3EQfpvi2/content/tmp_files/2301.00532v1.pdf.txt @@ -0,0 +1,473 @@ + +1 +Improve the Field Strength by Adding Soft Iron +in the Hybrid Permanent magnet +Quanling Peng1,2, Jianxin Zhou1, Saike Tian1, Yingzhe Wang1 +1. Institute of High Energy Physics, Chinese Academy of sciences, Beijing, 100049, China +2. School of electronics, Nanchang institute of Technology, Jiangxi, 330013, China. + +Abstract—Permanent magnet has a small and compact structure, is especially suitable for a narrow space. With the aid of soft +iron, the magnetic field can be increased much more and the field uniformity can be well controlled. Most Permanent magnets +have a symmetry structure; the soft iron can be selected as the float magnet pole to keep its constant scalar potential, and take +roles as media to collect the direct flux from the permanent blocks, then release indirect flux in magnet aperture and to the +nearby return yoke. This paper presents the magnetic flux method to design and fabricate a hybrid permanent dipole by using +axially and radially magnetized permanent blocks. A variable gradient permanent quadrupole and a variable gradient sextupole +are designed as the extend design examples . They all consist of two nested hybrid permanent rings, where the iron poles are +used to control the field quality, collect the magnetic flux from the outer ring, block the skew quadrupole and high order +harmonics. +Index Terms—Pure permanent magnet, hybrid permanent magnet, direct flux, indirect flux, variable gradient +quadrupole. + +1. Introduction +Electromagnet or superconducting magnets, with field strength varies along with the beam energy, are widely used in +modern particle accelerators to bend or focus the particle beams. In some circumstances, where the beam energy is fixed or only +a little adjustment, permanent magnet will be a better selection, since it has advantages of small space occupation, no cooling +system, and no operation costs, one time investment can maintain a long time operation. +Permanent magnet can be made of all permanent blocks or permanent blocks mixed with soft iron, where the former called +as pure permanent magnet, later called as the hybrid permanent magnet. For pure permanent magnet design, K. Halbach had +given the design principle in 1980s [1-2]. Each permanent block can be treated as air with the current loops surrounding the +magnet or as magnetic charges on its surfaces along the easy axis [3-4]. Field strength in magnet aperture is collected the +contribution in each permanent block by its position and easy axis orientation. +An important issue is that field strength from the pure permanent magnet is weak compared with electromagnets or +superconducting magnets. According to reference [1], even with the highest residual field strength, maximum field of a dipole +that built with pure permanent blocks is less than the remnant field of Br, say about 1.4 T for the highest NdFeB magnetic +materials. On the other hand, pure permanent magnet needs permanent blocks with different easy axis orientation, that will bring +fabrication difficulties and increase the cost [5-9]. For a hybrid permanent magnet, on the other hand, with the nonlinear material +such as iron poles to collect the surrounding magnetic flux and release it into a small compact space, it can reach to a higher +magnetic field. Another way to increase the field strength is to add more permanent blocks surround the iron pole to add more +magnetic flux. Variable field permanent magnets can be built by adding auxiliary coils around the iron pole or adding an outer +concentric permanent ring surround the inner permanent ring [10]. + +2. Principle of flux method +Assume a permanent magnet has an infinite length, the 3D case can be treated in 2D case. In 2D analysis, a homogeneously +magnetized permanent block can be treated as an air space with the surface charge density =Br on its upper and lower +surfaces or a surface current distribution of density i = Hc around the other four surfaces [3-4]. Hear Br and Hc are the remnant +field and the coercively of the permanent block respectively, with Br=0Hc. For the hard permanent magnetic material, the slope +of demagnetized curve in the second quarter is near 1, or μr=1, no extra field contribution from the material magnetization. +In 2D non-current space, magnetic field can be expressed as the negative gradient of scalar potential as +V + + +B +. As shown +in Fig. 1, in a hybrid permanent magnet, the permanent block can be modeled by charge sheets at the top and bottom surfaces. +The iron pole has large relative permeability, all the surface of the magnet pole keep a constant scalar potential of V2=V0, the +return yoke and mid-plane are all kept in zero scalar potential. Since V1 and V2 are in different scalar potential, magnetic field +will be produced between the magnet pole to the mid-plane or to the return yoke, which call as useful field and stray field +respectively. +@Manuscript received Jan 2, 2023. Work supported by the accelerator research program of the Chinese Academy of Sciences Grant No. Y5294104TD. +Author email address: pengql@ihep.ac.cn. + + +2 + +Fig. 1 Different scalar potentials in a quarter of the hybrid permanent dipole, the shade region represent the iron yoke and +iron pole. +2.1 Direct flux φd +Direct flux is magnetic flux coming from the permanent magnet blocks and deposits on the iron pole, it is the source to +maintain iron pole in a higher scalar potential. The direct flux to the iron pole is + ， + + . (1) +c is the fraction of magnetic charge σ deposited on the surface of the iron pole, it equals to the scalar potential V at the +magnetic charges with respect to the potential V0 at the iron pole, D is the width of the permanent block. Assume the scalar +potential changes uniformly in the permanent block, then + + .In Fig. 1, the direct flux includes the contributions from the +positive and the negative charges, which can be expressed as + , (2) +here + + + , + + + + + . If magnet blocks directly touch on the iron pole and leave some space h1 between the +top yoke, eq. (2) can be written as: + ( + + ) + + . (3) +If the permanent block fully occupies the space between the iron pole and top yoke, then the direct flux on the iron pole is: + . (4) +In 3d case, the direct flux deposited on the iron pole is: + , (5) +S is the surface area of the permanent block. + +2.2 Indirect flux φi +Indirect flux escapes from the iron pole faces to the nearby zero scalar potential areas. As shown in Fig. 2, add permanent +magnet between the magnet pole and the side yoke, the scalar potential of the iron pole will rise up to V0, the back yoke, the top +yoke and the mid-plane still keeps zero scalar potential, parasite magnetic field will be produced between the different scalar +potential areas. Here φi1 is the expected field, φi2 is the demagnetization field for the permanent magnet, φi3 is the nearby leakage +field, total indirect flux is φi =φi1+φi2 +φi3. +Assume the expected field B0 keeps constant in the magnet gap, with + + + + , h0 is half-length of the magnet gap. +The indirect flux on the mid-plane is + + + ∬ + + . (6) +In general, indirect flux to nearby zero scalar potential is written as: + + + . (7) +S’ is the surface area of the iron pole, h is the distance between the iron pole and the zero potential. Consider the corner +effect, the scale factor f>1, + +2.3 Total magnetic flux +From ∯ , total magnetic flux around the float iron pole is zero, that is the direct flux deposits on iron pole equals +to the indirect flux leaves away from the iron pole, which can be expressed as: +0 + + +i +d + + +. (8) + + +V1=0 +h1 +Br +h2 +h +++ ++++ +V2=VO +ho +V1=0 +3 + +Fig. 2. The indirect flux calculation model. Indirect flux goes from the iron pole to the nearby zero scalar potential area. + +3. H type hybrid permanent dipole design +A hybrid permanent dipole was fabricated for magnetic material processing, aimed to produce the field higher than 2.4 T in +a 7 mm gap. Assume the half gap as h0, the expected magnetic field at the central plane is B0，then the scalar potential on the +surface of the iron pole is V0=B0h0。Fig. 3 shows the cross section of the upper half magnet, the iron yokes are displayed in +hatch. The top permanent block is magnetized along the negative y, whereas the side permanent block is radially magnetized +inward. For the lower half magnet, the side permanent block is radially magnetized outward. +The direct fluxes come from the top magnetized block and the side radially magnetized permanent ring, which can be written +as: + + . (9) +The indirect fluxes scatter from the iron pole to the nearby zero scalar potential faces, which includes parts to the mid-plane, +to the top yoke, to the side yoke and to the upper and lower corners. Here selects factor f as 1.9 to contain the corner effects. +Then total indirect flux is + + + + + + + + . (10) + NdFeB N44H material is selected for permanent blocks, the remnant field Br=1.36 T. Other related parameters are: half +magnet gap h0=3.5 mm, pole tip length h1=10 mm, distance from pole top to mid-plane h2=20 mm, distance between the top of +the side permanent blocks to the mid-plane h3=30 mm, side yoke height h4=40 mm, pole radius R2=14 mm, pole tip radius R1=5 +mm, radius of the magnet gap R3=32 mm, return yoke thickness 8 mm. Vanadium Iron is select as the magnetic pole, since it +has high saturated field as Bs=2.2 T, the return yokes are made of DT4 soft iron. By eq. 10, magnetic field produced in the +mid-plane can reach 2.42 T. OPERA-3d [13] software is used to check the field strength, the calculated peak field on the +mid-plane is 2.45 T. +In comparison, three cases were calculated when the top permanent blocks removed or replaced with soft iron. Fig 4 shows +field differences along the central mid-plane. When the top permanent block was removed, total direct fluxes were reduced, +field strength on the mid-plane will drop accordingly. What’s more, when the top permanent blocks are replaced with iron, part +of magnetic flux will directly return the top yoke, the field in the magnet gap will decrease much more. + +Fig 3. Cross section of the upper half hybrid permanent dipole. The material of the magnet pole and the side yoke are Vanadium +and DT4 iron respectively. + + +中i2 +: +i3 +Qil +DR2. +4 +5 +Y +R1 +R3 +4 + +Fig 4. Field differences on the mid-plane when the top permanent magnet replaced with air or DT4 iron. + +4. Magnet fabrication and field measurement +For technical limitation, the radial magnetized block was replaced by 6 tile-liked blocks. Fig. 5 shows the lower half magnet +assebly. In order to protect the permanent blocks, they are covered by a G10 board. Fig. 6 shows the whole magnet assembly. + + +Fig.5. Lower half of the hybrid permanent magnet assembly + +Field measurement was done by a Hall probe along the slot in the G10 board, Fig. 7 shows the field measurement result, it +has a little difference compare with the 3D field calculation. + + + +Fig. 6. Whole Hybrid permanent magnet assembly +-10 +-5 +0 +5 +10 +0 +0.5 +1 +1.5 +2 +2.5 +3 +x(mm) +Bz(T) + + +Top PM +Top air +Top iron + + +5 + + + Fig. 7. Calculated and measured field distribution along the central line in the magnet mid-plane + +5. Design variable gradient permanent quadrupole by two nested permanent rings +Variable gradient quadrupole can be built with pure or hybrid permanent magnets. Fig 8 shows a kind of pure permanent +magnet design, where the variable gradient was realized by the relative rotation between the inner and outer permanent rings, +field gradient varies from G1-G2 to G1+G2, here G1 and G2 are field gradient of the inner and outer permanent rings respectively. +The nested pure permanent rings have two disadvantages, they are the lower efficiency and the accompanied skew quadrupole +components. +First, permanent blocks in outer rings are much away from the inner, its field contribution are greatly reduced, which needs +larger size and the cost will increase accordingly. On the other hand, since permanent blocks is similar as air, a skew quadrupole +component will produce during rotation and cannot be canceled. Skew quadrupole component will give rise to work point drift, +increase the beam emittance and will eventually affect the beam life time. Same problem exists in reference [15] for two sets of +nest permanent rings that made of cylindrical permanent rods. +Another plan is using the hybrid permanent quadrupole, where several permanent blocks are replaced by iron poles, by which +to control the field quality and concentrate the magnetic flux. Fig. 9 shows a hybrid permanent quadrupole that consists of two +sets of permanent rings, the variable gradient is realized by the relative rotation between the inner and outer ring. +In a circular particle accelerator, the ramping period is in a few seconds, gradient changes for a quadrupole can go along with +that of the beam energy. According to the design idea for the conventional electromagnet, the inner surface of the iron pole in a +hybrid permanent magnet is selected as a part of hyperbola to increase the field uniformity [14]. The outer circular surface of the +iron pole is selected as wide enough to collect the magnetic flux from the outer ring. Relative rotation between the two +permanent rings does not bring extra skew quadrupole components, since the stray field is blocked by the iron pole. + + + + + + + + + + + + + + + + +Fig 8. A variable gradient quadrupole that consists of two pure permanent rings + +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +x 10 +4 +r(mm) +By(G) + + +calculaled +measured + +22 +21 +20 +23 +19 +24 +18 +8 +2 +17 +9 +33 +25 +10 +16 +11 +15 +26 +12 +32 +13 +14 +27 +31 +28 +29 +30 +6 + + +Fig. 9. Variable gradient quadrupole consists of two nest hybrid permanent rings，the dash regions are made of iron. + +Fig. 10 shows the 3D field calculation when the outer ring rotated at 60 degrees, The design parameters are: magnet aperture +40 mm, outer diameter 320 mm, magnet length is 100 mm. Taking the suitable shimmed on the iron pole surface and end plate, +all the high order harmonics can be reduced less than 5 units at different rotation angle. Using FFT function in OPERA-3d, the +quadrupole gradient and field harmonics at the reference radius of 13 mm can be found. The calculated gradient varies from 21 +T/m to 64 T/m in a 90 degrees rotation period, maximum torque is 240 N.m, which can be realized by motors with the reduction +gearbox. +Fig. 12 shows the normalized high order harmonics along the beam line, all the integral harmonics are less than 5 units. Table +1 shows the normal and skew quadrupole values at different rotation angles, where all skew quadrupoles is near to zero. + + +Fig. 10. Calculation example of a variable gradient quadrupole that consists of two nest hybrid permanent rings when the +outer ring rotation at 60 degrees. + + +Fig 11 High order harmonics along the beam line (@r=13mm ) when the outer ring rotaed at 60 degrees, all data are normalized +with the integral quadrupole strength. +-50 +0 +50 +100 +150 +200 +250 +300 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +z（mm） +unit of 10 -4 + + +A3 +B3 +A4 +B4 +A5 +B5 + + +7 + +Table 1 Normal and skew quadrupole components changes at difference rotation angles +Rotation +angles +0 +30 +60 +90 +B2(T/m) +64.21 +55.12 +35.07 +25.47 +A2(T/m) +2.0E-004 +-0.0060 +0.016 +0.027 + +6. Design variable gradient sextupole for small-angle neutron scattering detector +A variable gradient hybrid permanent sextupole was designed for the Very Small-angle Neutron Scattering instrument +(VSANS) in the China Spallation Neutron Source Science (CSNS). As shown in Fig. 12, inner permanent ring has 12 permanent +blocks and 6 Vanadium Iron poles to collect the magnetic fluxes from the inner and outer permanent rings. Magnetization angle +for each permanent block is 60 degrees relative to its central symmetrical axis, which can contribute 20% field strength compare +with the 90 degrees from the calculation. For the 12 blocks outer ring, easy axis orientation for each permanent block is parallel +or perpendicular to its central symmetrical axis. The calculated gradient varies from 7188 T/m2 to19968T/m2 in a rotation recyle +from 0 to 60 degrees. Fig. 13 shows the 3D simulation field when the outer ring rotated at 60 degrees relative to the inner ring. + + Fig. 12 , Schematic layout for the magnetic angles for inner and outer permanent ring when the outer ring at 0 degrees. The +dashed areas are the iron poles. + + +Fig 13. 3D simulated magnetic filed when the outer permanent ring rotated at 60 degrees. + For the machnical design, the inner ring is fixed on the support seat by the connected flanges at both ends, the outer ring +rotates relatve to the inner ring by a set of high speed motors with reduction gearboxes. Each iron pole is a set of 5 mm sliced +lamated Vanadium Irons with water cooling wholes to get rid of eddy current overheating during the 1.5 kHz high speed rotation. +From 3d calculation, maximum torque is 220 N.m at 45 degrees rotation for the 200 mm long nested permanent sextupole +prototype. The 200m long variable gradient sextupole has been fabricated and tested successfully. + +7 . Conclusion +In a symmetrical hybrid permanent magnet, the mid-plane can be treated as the reference zero scalar potential, whereas the +iron pole is looked as high scalar potential to collect the magnetic flux and release to the low potential area. This paper presents +how it possible to produce the expected field by using magnetic fluxes method for hybrid permanent magnet design. Through + +40 +20 +40 +20 +20 +40 +20 +-40 +8 +theoretical calculation and 3D field simulation, a permanent dipole with field strength higher than 2.4 T was fabricated and tested. +Variable gradient nested permanent quadrupole or sextupole can also be realized by using iron poles to collect the magnetic flux +and block the high order harmonics from the outer ring. For its small, compact and low operation cost, hybrid permanent magnet +can find more applications in areas such as particle accelerator, motor, medical equipment and material research. + +References +[1] K. Halbach, Design of permanent multipole magnet with oriented rare earth cobalt material, Nucl. Instr.and Meth. 169(1980), 1-8. +[2] Quanling Peng, S. M. McMurry, and J. M. D. Coey, Cylindrical Permanent-Magnet Structures Using Images in an Iron Shield, IEEE TRANSACTIONS ON +MAGNETICS, 39 (2003), 1983-1989. +[3] Quanling Peng, S. M McMurry, J.M.D. Coey, Axial Magnetic Field Produced by Axially and Radially Magnetized Permanent Rings, Journal of magnetism +and magnetic materials 268 (2004) 165-169. +[4] Quanling PENG, 2D Field Calculation of Pure Permanent Magnet by Using Current Pair Model, journal of magnetism and magnetic materials , 309 (2007) +126-131. +[5] Ross D. Schlueter, Field errors in hybrid insertion devices, LBL-36839, USA. +[6] Q. L. Peng, Z. S. Yin, et al, Construction and Tuning of BEPC mini- Permanent Quadrupoles Prototype, Nucl. Instr. and Meth. in Phys. Res. A 406 (1998), +53~57. +[7] Vikas Teotia, Sanjay Malhotra, Elina Mishra, Prashant Kumar, Design, development and characterization of tunable Permanent Magnet Quadrupole for +Drift Tube Linac, Nuclear Inst. and Methods in Physics Research, A 982 (2020) 164528. +[8] M. Kumada, Development of High Field Permanent Magnet, IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 12, NO. 1, +MARCH 2002. +[9] Wending Zhong, Ferromagnetics(II), Science press, 1998 (in Chinese). +[10] P.P. Sanchez, T.S. do Espirito Santo, E. Conforti,, G. Tosin, Concepts of tunable magnets using permanent magnetic material for synchrotron radiation +sources, Nuclear Instruments and Methods in Physics Research A 778 (2015) 67–76. +[11] Y. Iwashita, Y. Tajima, M. Ichikawa, S. Nakamura, T. Ino, S. Muto, H.M. Shimizu, Variable permanent magnet sextupole lens for focusing of pulsed cold +neutrons, Nuclear Instruments and Methods in Physics Research Section A, 586 (2008) 73-76 +[12] Junghoon Lee, Jeonghoon Yoo，Topology optimization of the permanent magnet type MRI considering the magnetic field homogeneity [J], Journal of +Magnetism and Magnetic Materials, 2010, 322:1651. +[13] Opera Manager User Guide, Version 15R1, Vector Fields Software, July 2013 +[14] Jack Tanabe, Iron Dominated Electromagnets Design, Fabrication, assembly and Measurements, SLAC-R-754, 2005. +[15] Gautam Sinha, Conceptual design of a compact high gradient quadrupole magnet of varying strength using permanent magnets, PHYSICAL REVIEW +ACCELERATORS AND BEAMS 21, 022401 (2018). + + +9 + + + diff --git a/7dAyT4oBgHgl3EQfpvi2/content/tmp_files/load_file.txt b/7dAyT4oBgHgl3EQfpvi2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1258bb7c054732db13223aa1bbc2fdcc4187dd3a --- /dev/null +++ b/7dAyT4oBgHgl3EQfpvi2/content/tmp_files/load_file.txt @@ -0,0 +1,257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf,len=256 +page_content='1 Improve the Field Strength by Adding Soft Iron in the Hybrid Permanent magnet Quanling Peng1,2, Jianxin Zhou1, Saike Tian1, Yingzhe Wang1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Institute of High Energy Physics, Chinese Academy of sciences, Beijing, 100049, China 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' School of electronics, Nanchang institute of Technology, Jiangxi, 330013, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Abstract—Permanent magnet has a small and compact structure, is especially suitable for a narrow space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' With the aid of soft iron, the magnetic field can be increased much more and the field uniformity can be well controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Most Permanent magnets have a symmetry structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' the soft iron can be selected as the float magnet pole to keep its constant scalar potential, and take roles as media to collect the direct flux from the permanent blocks, then release indirect flux in magnet aperture and to the nearby return yoke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' This paper presents the magnetic flux method to design and fabricate a hybrid permanent dipole by using axially and radially magnetized permanent blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' A variable gradient permanent quadrupole and a variable gradient sextupole are designed as the extend design examples .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' They all consist of two nested hybrid permanent rings, where the iron poles are used to control the field quality, collect the magnetic flux from the outer ring, block the skew quadrupole and high order harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Index Terms—Pure permanent magnet, hybrid permanent magnet, direct flux, indirect flux, variable gradient quadrupole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Introduction Electromagnet or superconducting magnets, with field strength varies along with the beam energy, are widely used in modern particle accelerators to bend or focus the particle beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' In some circumstances, where the beam energy is fixed or only a little adjustment, permanent magnet will be a better selection, since it has advantages of small space occupation, no cooling system, and no operation costs, one time investment can maintain a long time operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Permanent magnet can be made of all permanent blocks or permanent blocks mixed with soft iron, where the former called as pure permanent magnet, later called as the hybrid permanent magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' For pure permanent magnet design, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Halbach had given the design principle in 1980s [1-2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Each permanent block can be treated as air with the current loops surrounding the magnet or as magnetic charges on its surfaces along the easy axis [3-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Field strength in magnet aperture is collected the contribution in each permanent block by its position and easy axis orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' An important issue is that field strength from the pure permanent magnet is weak compared with electromagnets or superconducting magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' According to reference [1], even with the highest residual field strength, maximum field of a dipole that built with pure permanent blocks is less than the remnant field of Br, say about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='4 T for the highest NdFeB magnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' On the other hand, pure permanent magnet needs permanent blocks with different easy axis orientation, that will bring fabrication difficulties and increase the cost [5-9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' For a hybrid permanent magnet, on the other hand, with the nonlinear material such as iron poles to collect the surrounding magnetic flux and release it into a small compact space, it can reach to a higher magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Another way to increase the field strength is to add more permanent blocks surround the iron pole to add more magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Variable field permanent magnets can be built by adding auxiliary coils around the iron pole or adding an outer concentric permanent ring surround the inner permanent ring [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Principle of flux method Assume a permanent magnet has an infinite length, the 3D case can be treated in 2D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' In 2D analysis, a homogeneously magnetized permanent block can be treated as an air space with the surface charge density \uf073\uf0b1=\uf0b1Br on its upper and lower surfaces or a surface current distribution of density i = Hc around the other four surfaces [3-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Hear Br and Hc are the remnant field and the coercively of the permanent block respectively, with Br=\uf06d0Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' For the hard permanent magnetic material, the slope of demagnetized curve in the second quarter is near 1, or μr=1, no extra field contribution from the material magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' In 2D non-current space, magnetic field can be expressed as the negative gradient of scalar potential as V \uf02d\uf0d1 \uf03d B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 1, in a hybrid permanent magnet, the permanent block can be modeled by charge sheets at the top and bottom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The iron pole has large relative permeability, all the surface of the magnet pole keep a constant scalar potential of V2=V0, the return yoke and mid-plane are all kept in zero scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Since V1 and V2 are in different scalar potential, magnetic field will be produced between the magnet pole to the mid-plane or to the return yoke, which call as useful field and stray field respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' @Manuscript received Jan 2, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Work supported by the accelerator research program of the Chinese Academy of Sciences Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Y5294104TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Author email address: pengql@ihep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 1 Different scalar potentials in a quarter of the hybrid permanent dipole, the shade region represent the iron yoke and iron pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='1 Direct flux φd Direct flux is magnetic flux coming from the permanent magnet blocks and deposits on the iron pole, it is the source to maintain iron pole in a higher scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The direct flux to the iron pole is ， .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' (1) c is the fraction of magnetic charge σ deposited on the surface of the iron pole, it equals to the scalar potential V at the magnetic charges with respect to the potential V0 at the iron pole, D is the width of the permanent block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Assume the scalar potential changes uniformly in the permanent block, then .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 1, the direct flux includes the contributions from the positive and the negative charges, which can be expressed as , (2) here , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' If magnet blocks directly touch on the iron pole and leave some space h1 between the top yoke, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' (2) can be written as: ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' (3) If the permanent block fully occupies the space between the iron pole and top yoke, then the direct flux on the iron pole is: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' (4) In 3d case, the direct flux deposited on the iron pole is: , (5) S is the surface area of the permanent block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='2 Indirect flux φi Indirect flux escapes from the iron pole faces to the nearby zero scalar potential areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 2, add permanent magnet between the magnet pole and the side yoke, the scalar potential of the iron pole will rise up to V0, the back yoke, the top yoke and the mid-plane still keeps zero scalar potential, parasite magnetic field will be produced between the different scalar potential areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Here φi1 is the expected field, φi2 is the demagnetization field for the permanent magnet, φi3 is the nearby leakage field, total indirect flux is φi =φi1+φi2 +φi3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Assume the expected field B0 keeps constant in the magnet gap, with , h0 is half-length of the magnet gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The indirect flux on the mid-plane is ∬ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' (6) In general, indirect flux to nearby zero scalar potential is written as: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' (7) S’ is the surface area of the iron pole, h is the distance between the iron pole and the zero potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Consider the corner effect, the scale factor f>1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='3 Total magnetic flux From ∯ , total magnetic flux around the float iron pole is zero, that is the direct flux deposits on iron pole equals to the indirect flux leaves away from the iron pole, which can be expressed as: 0 \uf03d \uf02b i d \uf066 \uf066 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' (8) V1=0 h1 Br h2 h ++ +++ V2=VO ho V1=0 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The indirect flux calculation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Indirect flux goes from the iron pole to the nearby zero scalar potential area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' H type hybrid permanent dipole design A hybrid permanent dipole was fabricated for magnetic material processing, aimed to produce the field higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='4 T in a 7 mm gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Assume the half gap as h0, the expected magnetic field at the central plane is B0，then the scalar potential on the surface of the iron pole is V0=B0h0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 3 shows the cross section of the upper half magnet, the iron yokes are displayed in hatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The top permanent block is magnetized along the negative y, whereas the side permanent block is radially magnetized inward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' For the lower half magnet, the side permanent block is radially magnetized outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The direct fluxes come from the top magnetized block and the side radially magnetized permanent ring, which can be written as: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' (9) The indirect fluxes scatter from the iron pole to the nearby zero scalar potential faces, which includes parts to the mid-plane, to the top yoke, to the side yoke and to the upper and lower corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Here selects factor f as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='9 to contain the corner effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Then total indirect flux is .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' (10) NdFeB N44H material is selected for permanent blocks, the remnant field Br=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='36 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Other related parameters are: half magnet gap h0=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5 mm, pole tip length h1=10 mm, distance from pole top to mid-plane h2=20 mm, distance between the top of the side permanent blocks to the mid-plane h3=30 mm, side yoke height h4=40 mm, pole radius R2=14 mm, pole tip radius R1=5 mm, radius of the magnet gap R3=32 mm, return yoke thickness 8 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Vanadium Iron is select as the magnetic pole, since it has high saturated field as Bs=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='2 T, the return yokes are made of DT4 soft iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' By eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 10, magnetic field produced in the mid-plane can reach 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='42 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' OPERA-3d [13] software is used to check the field strength, the calculated peak field on the mid-plane is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='45 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' In comparison, three cases were calculated when the top permanent blocks removed or replaced with soft iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig 4 shows field differences along the central mid-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' When the top permanent block was removed, total direct fluxes were reduced, field strength on the mid-plane will drop accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' What’s more, when the top permanent blocks are replaced with iron, part of magnetic flux will directly return the top yoke, the field in the magnet gap will decrease much more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Cross section of the upper half hybrid permanent dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The material of the magnet pole and the side yoke are Vanadium and DT4 iron respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 中i2 : i3 Qil DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 4 5 Y R1 R3 4 Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Field differences on the mid-plane when the top permanent magnet replaced with air or DT4 iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Magnet fabrication and field measurement For technical limitation, the radial magnetized block was replaced by 6 tile-liked blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 5 shows the lower half magnet assebly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' In order to protect the permanent blocks, they are covered by a G10 board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 6 shows the whole magnet assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Lower half of the hybrid permanent magnet assembly Field measurement was done by a Hall probe along the slot in the G10 board, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 7 shows the field measurement result, it has a little difference compare with the 3D field calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Whole Hybrid permanent magnet assembly -10 -5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5 3 x(mm) Bz(T) Top PM Top air Top iron 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Calculated and measured field distribution along the central line in the magnet mid-plane 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Design variable gradient permanent quadrupole by two nested permanent rings Variable gradient quadrupole can be built with pure or hybrid permanent magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig 8 shows a kind of pure permanent magnet design, where the variable gradient was realized by the relative rotation between the inner and outer permanent rings, field gradient varies from G1-G2 to G1+G2, here G1 and G2 are field gradient of the inner and outer permanent rings respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The nested pure permanent rings have two disadvantages, they are the lower efficiency and the accompanied skew quadrupole components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' First, permanent blocks in outer rings are much away from the inner, its field contribution are greatly reduced, which needs larger size and the cost will increase accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' On the other hand, since permanent blocks is similar as air, a skew quadrupole component will produce during rotation and cannot be canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Skew quadrupole component will give rise to work point drift, increase the beam emittance and will eventually affect the beam life time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Same problem exists in reference [15] for two sets of nest permanent rings that made of cylindrical permanent rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Another plan is using the hybrid permanent quadrupole, where several permanent blocks are replaced by iron poles, by which to control the field quality and concentrate the magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 9 shows a hybrid permanent quadrupole that consists of two sets of permanent rings, the variable gradient is realized by the relative rotation between the inner and outer ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' In a circular particle accelerator, the ramping period is in a few seconds, gradient changes for a quadrupole can go along with that of the beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' According to the design idea for the conventional electromagnet, the inner surface of the iron pole in a hybrid permanent magnet is selected as a part of hyperbola to increase the field uniformity [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The outer circular surface of the iron pole is selected as wide enough to collect the magnetic flux from the outer ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Relative rotation between the two permanent rings does not bring extra skew quadrupole components, since the stray field is blocked by the iron pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' A variable gradient quadrupole that consists of two pure permanent rings 20 15 10 5 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='6 x 10 4 r(mm) By(G) calculaled measured 22 21 20 23 19 24 18 8 2 17 9 33 25 10 16 11 15 26 12 32 13 14 27 31 28 29 30 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Variable gradient quadrupole consists of two nest hybrid permanent rings，the dash regions are made of iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 10 shows the 3D field calculation when the outer ring rotated at 60 degrees, The design parameters are: magnet aperture 40 mm, outer diameter 320 mm, magnet length is 100 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Taking the suitable shimmed on the iron pole surface and end plate, all the high order harmonics can be reduced less than 5 units at different rotation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Using FFT function in OPERA-3d, the quadrupole gradient and field harmonics at the reference radius of 13 mm can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The calculated gradient varies from 21 T/m to 64 T/m in a 90 degrees rotation period, maximum torque is 240 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='m, which can be realized by motors with the reduction gearbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 12 shows the normalized high order harmonics along the beam line, all the integral harmonics are less than 5 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Table 1 shows the normal and skew quadrupole values at different rotation angles, where all skew quadrupoles is near to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Calculation example of a variable gradient quadrupole that consists of two nest hybrid permanent rings when the outer ring rotation at 60 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig 11 High order harmonics along the beam line (@r=13mm ) when the outer ring rotaed at 60 degrees, all data are normalized with the integral quadrupole strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' -50 0 50 100 150 200 250 300 -2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5 -1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5 2 z（mm） unit of 10 -4 A3 B3 A4 B4 A5 B5 7 Table 1 Normal and skew quadrupole components changes at difference rotation angles Rotation angles 0 30 60 90 B2(T/m) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='21 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='12 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='07 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='47 A2(T/m) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='0E-004 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='0060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='027 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Design variable gradient sextupole for small-angle neutron scattering detector A variable gradient hybrid permanent sextupole was designed for the Very Small-angle Neutron Scattering instrument (VSANS) in the China Spallation Neutron Source Science (CSNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 12, inner permanent ring has 12 permanent blocks and 6 Vanadium Iron poles to collect the magnetic fluxes from the inner and outer permanent rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Magnetization angle for each permanent block is 60 degrees relative to its central symmetrical axis, which can contribute 20% field strength compare with the 90 degrees from the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' For the 12 blocks outer ring, easy axis orientation for each permanent block is parallel or perpendicular to its central symmetrical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The calculated gradient varies from 7188 T/m2 to19968T/m2 in a rotation recyle from 0 to 60 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 13 shows the 3D simulation field when the outer ring rotated at 60 degrees relative to the inner ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 12 , Schematic layout for the magnetic angles for inner and outer permanent ring when the outer ring at 0 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The dashed areas are the iron poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Fig 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 3D simulated magnetic filed when the outer permanent ring rotated at 60 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' For the machnical design, the inner ring is fixed on the support seat by the connected flanges at both ends, the outer ring rotates relatve to the inner ring by a set of high speed motors with reduction gearboxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Each iron pole is a set of 5 mm sliced lamated Vanadium Irons with water cooling wholes to get rid of eddy current overheating during the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='5 kHz high speed rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' From 3d calculation, maximum torque is 220 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='m at 45 degrees rotation for the 200 mm long nested permanent sextupole prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' The 200m long variable gradient sextupole has been fabricated and tested successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Conclusion In a symmetrical hybrid permanent magnet, the mid-plane can be treated as the reference zero scalar potential, whereas the iron pole is looked as high scalar potential to collect the magnetic flux and release to the low potential area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' This paper presents how it possible to produce the expected field by using magnetic fluxes method for hybrid permanent magnet design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Through 40 20 40 20 20 40 20 -40 8 theoretical calculation and 3D field simulation, a permanent dipole with field strength higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content='4 T was fabricated and tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' Variable gradient nested permanent quadrupole or sextupole can also be realized by using iron poles to collect the magnetic flux and block the high order harmonics from the outer ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' For its small, compact and low operation cost, hybrid permanent magnet can find more applications in areas such as particle accelerator, motor, medical equipment and material research.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'} diff --git a/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf b/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d7fe072f6b4ec9b1f2e67c26ebdd5db39560745b --- /dev/null +++ b/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:78add082edf142490c20b309603f45ddae3e7a3914320225781a62644083c2a4 +size 204901 diff --git a/7dE0T4oBgHgl3EQfwQEF/vector_store/index.faiss b/7dE0T4oBgHgl3EQfwQEF/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9ac6a29c51e5c2fec7eb855298054de23b1bc440 --- /dev/null +++ b/7dE0T4oBgHgl3EQfwQEF/vector_store/index.faiss @@ -0,0 +1,3 @@ +version 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The +steady-steady 1:2 resonant mode interaction with +O(2) symmetry. +A. Corrochano1, J. Sierra-Aus´ın2,3, J. A. Martin1, +D. Fabre2, and S. Le Clainche ∗1 +1School of Aerospace Engineering, Universidad Polit´ecnica de +Madrid, Madrid 28040, Spain +2Institut de M´ecanique des Fluides de Toulouse (IMFT), Toulouse +31400, France +3DIIN, Universit´a degli Studi di Salerno, Via Giovanni Paolo II, +84084 Fisciano (SA), Italy +Abstract +In this article, a thorough characterization of the configuration com- +posed by two concentric jets at a low Reynolds number is presented. The +analysis comprises a layout with a wide range for the velocity ratio be- +tween the inner and outer jets, defined within the interval [0, 2], and also +details the influence of the distance between jets, where the wall thick- +nesses separating the two jets is [0.5, 4]. +Global linear stability analy- +sis identifies the most significant modes driving the changes in the flow +dynamics. The neutral lines revealing the critical Reynolds number con- +nected to the presence of the main (steady and unsteady) flow bifurcations, +which are presented by global azimuthal modes, show the high complexity +of the problem under study, where hysteresis and other types of complex +cycles are pointed out. Finally, the mode interaction is analysed, high- +lighting the presence of travelling waves emerging from the interaction +of steady states, and the existence of robust heteroclinic cycles that are +asymptotically stable. The high level of detail in the results presented, +makes this work as a reference for future research development in the field +of concentric jets. +1 +Introduction +Double concentric jets is a configuration enhancing the turbulent mixing of two +jets, which is used in several industrial applications where the breakup of the jet +∗Email address for correspondence: soledad.leclainche@upm.es +1 +arXiv:2301.04429v1 [physics.flu-dyn] 11 Jan 2023 + +Rin +Rout +Uin +Uout +INITIAL MERGING +ZONE +r +TRANSITIONAL +ZONE +MERGED ZONE +Outer +potential core +Inner +potential core +Outer mixing +region +Inner mixing +region +Reattachment +point +z +Figure 1: Sketch representing the three flow regimes in the near field of double +concentric jets. Figure based on the sketch presented in [12, 31]. +into droplets due to flow instabilities is presented as the key technology. Com- +bustion (i.e.: combustion chamber of rocket engines, gas turbine combustion, +internal combustion engines, etc.) and noise reduction (e.g.: in turbofan en- +gines) are the two main applications of this geometry, although the annular jets +can also be found in some other relevant applications such as ink-jet printers or +spray coating. +The qualitative picture emerging from this type of flow divides the inner +field of concentric jets in three different regions: (i) initial merging zone, (ii) +transitional zone and (iii) merged zone, as presented in fig. 1, that follows the +initial sketch presented by [12]. In the initial merging zone (i), just at the exit +of the two jets, two axisymmetric shear layers (inner and outer boundary layer) +develop and start to merge. In this region, we distinguish the inner and outer +shear layers, related with the inner and outer jet stream. Then, most of the +mixing occurs in the transitional zone (ii), that extends until the external shear +layer reaches the centreline. Finally, in the merged zone (iii), the two jets are +totally merged, modelling a single jet flow. +Several parameters define the characteristic of this flow: the inner and outer +jet velocities, the jet diameters, the shape and thickness of the wall separating +both jets, the Reynolds number, the boundary layer state and thickness at +the jet exit and the free stream turbulence. Based on these parameters, it is +possible to identify several types of flow behaviour, which can be related with +the presence of flow instabilities. +Ko & Kwan (1976) [12] postulated that the double concentric jet configura- +tion could be considered as a combination of single jets. Nevertheless, Dahm et +al. (1992) [7] revealed by means of flow visualizations, several topology patterns +as function of the outer/inner jet velocity ratio, reflecting that the dynamics of +2 + +the inner and outer jet shear layers were different from that in a single jet. More- +over, this study exhibited a complex interaction between vortices identified in +both shear layers, affecting the instability mechanism of the flow. Buresti et +al. (1992) [4] found that the outer shear layer dominated the flow dynamics for +cases in which the outer velocity was much larger than the inner velocity. These +authors also detected the presence of an alternate vortex shedding when the +wall thickness between the two jets was sufficiently large. The same mechanism +was recognised by other authors [7, 20]. Rehab et al. (1997) [25] studied in +detail the flow differences as function of the outer/inner velocity ratio, finding +two different flow regimes when the external jet diameter is much larger than +the internal jet one. When the outer/inner velocity ratio was larger than a crit- +ical value, the authors spotted a low frequency recirculation bubble at the jet +outlet. On the contrary, for outer/inner velocity ratio smaller than such critical +value, the outer (still fast) jet excites the inner jet, which ends oscillating at the +same frequency as the external jet. This is known as the lock-in phenomenon. +Moreover, the oscillation frequency detected was similar to the one defined by a +Kelvin-Helmholtz flow instability, which is generally encountered in single jets. +This lock-in phenomenon was also identified by other authors [7, 6]. +Following previous works [4, 7, 20] and paying especial attention to the sep- +arating wall thickness and the vortex shedding located behind the wall, Wallace +& Redekopp (1992) [33] showed that the wall thickness and sharpness change +the characteristic of the jet. Segalini & Talamelli (2011) [26] performed exper- +iments to inspect in detail the effects of the outer/inner velocity ratio and the +wall thickness in double concentric jets. These authors found that for small +outer/inner velocity ratios, the inner jet presents its own flow instability in the +shear layer, while a different flow instability was identified in the outer jet. On +the contrary, for large outer/inner velocity ratios, the outer shear layer drives the +flow dynamics, forcing the inner shear layer to oscillate with the same frequency, +occurring in the lock-in phenomenon previously mentioned. Finally, for similar +outer/inner velocity ratios, a Von K´arm´an vortex street was detected near the +separating wall, as also depicted by other authors [4, 7, 20]. A wake instability +affected the inner and outer shear layers, reversing the lock-in phenomenon. +Different configurations can also be found, changing the velocity ratio be- +tween jets. Williams et al. (1969) [34] worked on the influence of the exte- +rior/interior velocity ratio on noise attenuation, which was analysed experimen- +tally. It was observed that for some given configurations, more noise attenuation +was present than for the others, with a maximum between 12 and 15dB. +Talamelli & Gavarini (2006) [31] performed a local linear stability analysis, +finding that for specific wall thickness, the vortex shedding identified behind +the wall, can be related with an absolute instability that exists for some specific +outer/inner velocity ratios. The authors explained that this absolute instabil- +ity may trigger the destabilization of the flow field. This theoretical work was +verified experimentally by [21]. These authors showed once more that the wake +behind the wall separating the two jets creates a vortex shedding driving the +frequency of the external shear layer also controlling the evolution of the inner +shear layer, which can be the mechanism that triggers a global absolute insta- +3 + +bility. This passive mechanism can be considered as a potential tool for flow +control, delaying the transition to turbulence by means of controlling the near +field of the jet. Recently, Canton et al. (2017) [5] performed a global linear sta- +bility analysis to study more in detail this vortex shedding mechanism behind +the wall. They examined a concentric jet configuration with a very small wall +thickness (0.1Di, with Di the inner jet diameter), but the authors selected an +outer/inner velocity ratios where it was known that the alternate vortex shed- +ding behind the wall was driving the flow. A global unstable mode (absolute +instability) with azimuthal wavenumber m = 0 was found, confirming that the +primary instability was axisymmetric (the modes with m = 1, 2 were stable at +the flow conditions at which the study was carried out). The highest intensity +of the global mode was located in the wake of the jet, composed by an array +of counter-rotating vortex rings. The shape of the mode changes when moving +along its neutral curve, revealing through the numerical simulations a Kelvin- +Helmholtz instability over the shear-layer between the two jets and in the outer +jet at high Reynolds numbers. Nevertheless, the authors showed that the wave- +maker was located in the bubble formed upstream the separating wall, in good +agreement with the results presented by [32], who performed a similar stability +analysis in a two-dimensional configuration (wakes with co-flow). +The stability of annular jets, a limit case where the inner jets have zero +velocity, has also been investigated. In different analysis of annular jets [3, 17], +it has been illustrated that this type of axisymmetric configuration does not +behave as it appears. The m = 0 modes studied have been shown to be stable, +and the dominant mode found by both studies is helical (m = 1). In addition, +to characterise the annular jet, these investigations analyse the behaviour of +the case by adding an azimuthal component to the inflow velocity, making the +discharge of the annular jet eddy-like, comparing the evolution of the frequency +and growth rate of this m = 1 mode. +This paper expands on the work done by [5], where they use a specific ge- +ometry and vary the outer/inner velocity ratio. This paper presents a complete +characterisation of the main global modes identified in two concentric jets. The +wall thicknesses separating the two jets are defined in the interval L ∈ [0.5, 4], +and the flow is simulated for different inside/outside velocity ratios in the in- +terval Ui/Uo ∈ [0, 2], where the case with Ui/Uo = 0 represents an annular +jet. Global modes with azimuthal wavenumber m = 0 (axisymmetric modes), +m = 1 and m = 2 will be searched for. As identified in the literature [17, 3], +no axisymmetric modes (m = 0) could be identified for any of the distances, as +this is a helical case. This paper expands the conclusions found in these two +previous works, extending the results to different wall thicknesses between jets. +This part of the paper studies in detail the configuration of two concentric jets +at low Reynolds numbers. Using a linear approximation of the equations that +model the flow, the base flows will be obtained on which to apply the linear +stability analysis, by means of which it is possible to identify the most relevant +modes that influence the flow dynamics. +This work also performs a study of mode selection, as some configurations +presents interactions between different modes. +Different analysis have been +4 + +done to know the different coherent structures when there is an interaction +between modes. [30] conducted the study on the flow past a rotating sphere, +finding different coherent structures on a triple-Hopf bifurcation. Some of these +configurations are steady states, travelling waves or rotating waves. +To the authors’ knowledge, this is the first time that the characterisation +of two concentric jets is presented with such level of detail, presenting neutral +curves for a wide range of different configurations, as well as providing a deep +understanding of the flow physics through the interaction between the different +modes. +The article is organized as follows. Section 2 defines the problem and the +governing equations for the double concentric jets, as well as the linear sta- +bility equations and the methodology for mode selection. The axisymmetric +steaty-state is characterised in Section 3. In Section 4, we perform a parametric +exploration in terms of the velocity ratio between the jets and the jet distance +in order to determine the neutral curves of global stability. The results about +the mode selection are discussed in Section 5. Finally, Section 6 summarises the +main conclusions. +2 +Problem formulation +2.1 +Computational domain and general equations +The computational domain, represented in fig. 2, models a coaxial flow configu- +ration, which is composed of two inlet regions, an inner and outer pipe, both of +diameter D and length 5D, i.e. zmin = −5D. The computational domain has +an extension of zmax = 50D and rmax = 25D. The distance between the pipes +is equal to L, measured from the inner face of the outer tube to the face of the +inner jet. +The governing equations of the flow within the domain are the incompressible +Navier–Stokes equations. These are written in cylindrical coordinates (r, θ, z), +which are made dimensionless by considering D as the reference length scale +and Wo,max as the reference velocity scale, which is the maximum velocity in +the outer pipe at z = zmin. +∂U +∂t + U · ∇U = −∇P + ∇ · τ(U), +∇ · U = 0, +(1a) +with τ(U) = 1 +Re (∇U + ∇UT ), +Re = Wo,maxD +ν +. +(1b) +The dimensionless velocity vector U = (U, V, W) is composed of the radial, +azimuthal and axial components, P is the dimensionless-reduced pressure, the +dynamic viscosity ν and the viscous stress tensor τ(U). +The incompressible Navier–Stokes equations eq. (1) are complemented with +the following boundary conditions +U = (0, 0, Wi) on Γin,i and U = (0, 0, Wo) on Γin,o, +(2) +5 + +Figure 2: Computational domain of the configuration of two concentric jets, +used in StabFem. +where +Wi = δu tanh +� +bi(1 − 2r) +� +and Wo = tanh +� +bo +� +1 + +���� +r − (Router,1 + Router,2) +D +���� +�� +. +The parameter δu corresponds to the velocity ratio between the two jets, defined +as δu = Wi,max/Wo,max. The parameters bo and bi represent the boundary layer +thickness within the nozzle, which are fixed equal to 5 (as in [5]). There is a +weak influence of the boundary layer thickness on the stability properties of +the jet, and it is related to the vortex shedding regime developed upstream the +separation wall (more details may be found in [31]). Finally, no-slip boundary +condition is set on Γwall and stress-free ( +� 1 +Reτ(U) − P +� +· n = 0) boundary +condition is set on Γtop and Γout, as shown in fig. 2. +In the sequel, Navier–Stokes equations eq. (1) and the associated boundary +conditions will be written symbolically under the form +B∂Q +∂t = F(Q, ) ≡ LQ + N(Q, Q) + G(Q, η), +(3) +with the flow state vector Q = [U, P]T , η = [Re, δu]T . Such a form of the +governing equations takes into account a linear dependency on the state variable +Q through L. And a quadratic dependency on the parameters and the state +variable through operators G(·, ·) and N(·, ·). +2.2 +Asymptotic stability +2.2.1 +Linear stability analysis +In this study, the authors attempt to characterize the stable asymptotic state +from the spectral properties of the Navier–Stokes equations eq. (1). First, let us +6 + +Tmar +Router,? +D +out +Router,1 +r=0 +Z=Zmin +z=0 +z=Zmarconsider the stability of an axisymmetric steady-state solution named Q0, which +will be also referred to as trivial steady-state. For that purpose, let evaluate +a solution of eq. (1) in the neighbourhood of the trivial steady state, i.e., a +perturbed state as follows, +Q(x, t) = Q0(x, t) + εˆq(r, z)e−i(ωt−mθ). +(4) +The next step consists in the characterization of the dynamics of small-amplitude +perturbations around this base flow by expanding them over the basis of linear +eigenmodes (4). If there is a pair [iωℓ, ˆqℓ] with Im(ωℓ) > 0 (resp. the spectrum +is contained in the half of the complex plane with negative real part) there ex- +ists a basin of attraction in the phase space where the trivial steady-state Q0 is +unstable (resp. stable) [11]. The eigenpair [iωℓ, ˆqℓ] is determined as a solution +of the following eigenvalue problem, +J(ωℓ,mℓ)ˆq(zℓ) = +� +iωℓB − ∂F +∂q |q=Q0,η=0 +� +ˆq(zℓ), +(5) +where +� +∂F +∂q |q=Q0,η=0 +� +ˆq(zℓ) = Lmℓ ˆq(zℓ) + Nmℓ(Q0, ˆq(zℓ)) + Nmℓ(ˆq(zℓ), Q0). The +subscript mℓ indicates the azimuthal wavenumber used for the evaluation of the +operator. In the following, we account for eigenmodes ˆq(zℓ)(r, z) that have been +normalised in such a way ⟨ˆu(zℓ), ˆu(zℓ)⟩L2 = 1. +2.2.2 +Methodology for the study of mode selection +In the following, we briefly outline the main aspects of the methodology em- +ployed in the study of mode interaction, a comprehensive explanation is left to +appendix A. The determination of the attractor or coherent structure is explored +within the framework of equivariant bifurcation theory. The trivial steady-state +is axisymmetric, i.e. the symmetry group is the orthogonal group O(2). Near +the onset of the instability, dynamics can be reduced to those of the centre +manifold. Particularly, due to the non-uniqueness of the manifold one can al- +ways look for its simplest polynomial expression, which is known as the normal +form of the bifurcation. The reduction to the normal form is carried out via a +multiple scales expansion of the solution Q of eq. (3). The expansion considers +a two scale development of the original time t �→ t + ε2τ, here ε is the order of +magnitude of the flow disturbances, assumed to be small ε ≪ 1. In this study +we carry out a normal form reduction via a weakly non-linear expansion, where +the small parameters are +ε2 +δu = δu,c − δu ∼ ε2 and ε2 +ν = +� +νc − ν +� += +� +Re−1 +c +− Re−1� +∼ ε2. +A fast timescale t of the self-sustained instability and a slow timescale of the +evolution of the amplitudes zi(τ) are also considered in eq. (10), for i = 1, 2, 3. +The ansatz of the expansion is as follows +Q(t, τ) = Q0 + εq(ε)(t, τ) + ε2q(ε2)(t, τ) + O(ε3). +(6) +7 + +Herein, we evaluate the mode interaction between two steady symmetry break- +ing states with azimuthal wave number m1 = 1 and m2 = 2, that is, +q(ε)(t, τ) += +� +z1(τ)ˆq(z1)(r, z)e−im1θ + c.c. +� ++ +� +z2(τ)ˆq(z2)(r, z)e−im2θ + c.c. +� +. +(7) +Note that the expansion of the LHS of eq. (3) up to third order is as follows +εB∂q(ε) +∂t ++ ε2B∂q(ε2) +∂t ++ ε3� +B∂q(ε3) +∂t +� ++ O(ε4), +(8) +and the RHS respectively, +F(q, η) = F(0) + εF(ε) + ε2F(ε2) + ε3F(ε3) + O(ε4). +(9) +Then, the problem up to third order in z1 and z2 can be reduced to [1] +˙z1 += λ1z1 + e3z1z2 + z1 +� +c(1,1)|z1|2 + c(1,2)|z2|2� +, +˙z2 += λ2z2 + e4z2 +1 + z2 +� +c(2,1)|z1|2 + c(2,2)|z2|2� +. +(10) +An exhaustive analysis of the nonlinear implications of this normal form on +dynamics is left to section 5. The procedure followed for the determination of +the coefficients c(i,j) for i, j = 1, 2 and e3 and e4 is left to Appendix A. +2.2.3 +Numerical methodology for stability tools +Results presented herein follow the same numerical approach adopted by [9, +28, 27, 30], where a comparison with DNS can be found. The calculation of +the steady-state, the eigenvalue problem and the normal form expansion are +implemented in the open-source software FreeFem++. Parametric studies and +generation of figures are collected by StabFem drivers, an open-source project +available in https://gitlab.com/stabfem/StabFem. For steady-state, stabil- +ity and normal form computations, we set the stress-free boundary condition at +the outlet, which is the natural boundary condition in the variational formula- +tion. +The resolution of the steady nonlinear Navier-Stokes equations is tackled +by means of the Newton method. While, the generalised eigenvalue problem +(eq. (24)) is solved following the Arnoldi method with spectral transformations. +The normal form reduction procedure of section 2.2.2 only requires to solve a +set of linear systems, which is also carried out within StabFem. On a standard +laptop, every computation considered below can be attained within a few hours. +3 +Characterisation of the axisymmetric steady- +state +3.1 +Velocity ratio effects +We begin by characterizing the development of the axisymmetric steady-state +with varying δu at a constant Reynolds number fixed to Re = 100. Figure 3 +8 + +0 +0 +1 +r +5 +2 +4 +3 +W0 +-0.4 +-0.5 +-1 +-0 +0 +1 +r +5 +2 +4 +3 +3 +2.5 +2 +1.5 +0.5 +1 +Lr +min(W0) +δu +δu +-0.1 +-0.15 +-0.2 +-0.25 +-0.3 +-0.35 +-0.4 +0.5 1 1.5 2 +a +b +c +c +b +a +d +e +f +a +b +d +f +e +c +δ1 +u δ2 +u +0 +2 4 +0 +0.1 0.2 0.3 0.4 0.5 +0 +z +0 +0 +1 +z +r +2 4 +5 +2 +4 +3 +Figure 3: Evolution of the recirculation length (Lr) of the recirculating bubble +with respect to the velocity ratio δu between the inner and outer jet. +The +diagram of the second row on the left displays the minimum value within the +domain of the axial velocity. It is spatially localised within the recirculating +region for δu < 0.5 and near the middle wall for larger values of the velocity +ratio. Meridional projections of the axisymmetric streamfunction isolines and +the axial velocity contour in a range of (z, r) ∈ [−1, 5] × [0, 5]. +synthesises the main topological changes experiences by the steady-state. At +δu = 0, the solution (point (a) in fig. 3) represents an annular jet, which diffuses +as it travels downstream and enters the ambient fluid. This figure illustrates +that the solution curve can be divided into three segments. The first segment +comprised between 0 ≤ δu < δ1 +u is characterised by an inner jet nearly trapped +by a large recirculation region with a characteristic length Lr, which remains +almost constant with the velocity ratio. +In the second region, which ranges between δ1 +u < δu < δ2 +u and it is represented +as a shaded area in the figure, the recirculating region rapidly reduces its size. +In this region, the axial velocity of the inner jet is comparable with the axial +velocity observed in the recirculating region, which promotes mixing between +both regions. +As the velocity ratio is increased, the inner jet is sufficiently +9 + +energetic to break the recirculating region, which occurs between point (c) and +(d) in fig. 3. The final segment, that ranges between δu > δ2 +u, is characterised +by two quasi-planar jets that rapidly mix to form a larger one at around z ≈ 5. +4 +Linear stability analysis +We explore the parameter space (Re, δu, L). Herein, we examine the velocity +ratio between the jets (0 < δu < 2) and the distance between the jets (0.5 < +L < 4). Within this range of parameters, we have analysed the linear stability +properties of the flow configuration. For this purpose, we first investigate the +influence of the jet distance on the stability for the case of the annular jet +(δu = 0). +These findings are summarized in fig. 4 which displays the evolution of the +critical Reynolds number with respect to the distance (L) for the four most un- +stable modes: two steady modes with azimuthal wavenumber m = 1 and m = 2, +hereinafter referred to as modes S1 and S2, respectively. A cross-section view at +z = 1 is displayed in fig. 4 (a-b). The other two unsteady modes, named F1 and +F2 have respectively azimuthal wavenumbers m = 1 and m = 2. A cross-section +view of these two modes is displayed in fig. 4 (c-d). Please note that for the +chosen set of parameters the axisymmetric unsteady mode F0, is always found +at larger Reynolds numbers than the aforementioned modes. This is one of the +major differences with the case studied by [5], for small values of the jet distance +L, the dominant instability is an unsteady axisymmetric one, which would be +named F0 with our nomenclature. Thus, in the following, we only include the +results for the S1, S2, F1 and F2 modes. The primary instability of the annular +jet is then a steady symmetry-breaking bifurcation that leads to a jet flow with +a single symmetry plane, displayed in fig. 4 (a). On the contrary, bifurcations +that lead to the mode S2 possess two orthogonal symmetry planes, see fig. 4 (b). +As indicated in fig. 4 (g-h), these two stationary modes S1 and S2 are localised +within the recirculation bubble. For jet distances L < 2, the second mode that +bifurcates is F1 mode, depicted in fig. 4 (i). This situation corresponds to a +bifurcation scenario similar to other axisymmetric flow configurations, such as +the flow past a sphere or a disk [2, 14]. For larger distances between jets, the +scenario changes. The second bifurcation from the axisymmetric steady-state is +the F2, displayed in fig. 4 (j). Other configurations where the primary or sec- +ondary instability involves modes with azimuthal component m = 2 are swirling +jets [15] and the wake flow past a rotating sphere [29]. The unsteady modes +F1 and F2 possess a much larger spatial support than S1 and S2. They are +formed by an array of counter-rotating vortex spirals developing in the wake of +the separating duct wall. For the mode F2 the amplitude of these structures +grows downstream of the nozzle, in the axial direction, with a maximum around +z ≈ 10, after which they slowly decay. The mode F1 grows further downstream, +with a maximum around z ≈ 50. The spatial structure of these eigenmodes +resembles the axisymmetric mode of Figure 9 in [5]. Thus, the steady modes +and unsteady modes differ in their spatial support, that is, even though both +10 + +-1 +0 +1 +x +-1 +-0.5 +0 +0.5 +1 +y +-0.5 0.5 +(a) +-1 +0 +1 +x +-1 +-0.5 +0 +0.5 +1 +y +-0.5 0.5 +(b) +-4 +-2 +0 +2 +4 +x +-4 +-2 +0 +2 +4 +y +-0.5 0.5 +(c) +-4 +-2 +0 +2 +4 +x +-4 +-2 +0 +2 +4 +y +-0.5 0.5 +(d) +1 +2 +3 +4 +L +0 +100 +200 +300 +400 +500 +Re +(e) +1 +2 +3 +4 +L +0 +0.1 +0.2 +0.3 +0.4 +! +(f) +-1 +0 +1 +2 +3 +z +r +-0.2 +0 +0.2 +(g) +-1 +0 +1 +2 +3 +z +r +-0.2 +0 +0.2 +(h) +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +z +0 +5 +10 +r +-0.05 +0 +0.05 +(i) +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +z +0 +5 +10 +r +-0.05 +0 +0.05 +(j) +Figure 4: Cross-section view at z = 1 of the four unstable modes at criticality +for the annular jet case (δu = 0). The streawise component of the vorticity +vector ϖz is visualised by colours. (a) Mode S1 for L = 0.5, (b) Mode S2 for +L = 0.5, (c) Mode F1 for L = 3 and (d) Mode F2 for L = 3. (e) Linear stability +boundaries for the annular jet (δu = 0). (f) Frequency evolution of the unsteady +modes. Legend: S1 mode is displayed with a solid black line, S2 with a solid +red line and F1 and F2 modes are depicted with dashed black and red lines, +respectively. Streamwise velocity of the neutral modes for L = 3 and δu = 0 (i) +F1, (h) F2 . +11 + +0 +0.5 +1 +1.5 +2 +/u +0 +200 +400 +600 +800 +Re +S1 ! S2 +(a) +0 +0.5 +1 +1.5 +2 +/u +0 +100 +200 +300 +400 +500 +600 +Re +S1 ! S2 +(b) +0 +2 +4 +6 +z +0 +1 +2 +3 +4 +5 +6 +r +-0.2 +0 +0.2 +0.4 +(c) +0 +2 +4 +6 +z +0 +1 +2 +3 +4 +5 +6 +r +-0.5 +0 +0.5 +1 +1.5 +(d) +Figure 5: Linear stability boundaries for the concentric jets (a) L = 0.5 and (b) +L = 1. Same legend as fig. 4. Visualizations of real part of the streamwise axial +velocity of the critical modes (c) S1 and (d) S2. +steady and unsteady modes are localised in space, the support of the steady +ones is confined within the recirculation bubble. Instead, the unsteady modes +are convected far downstream until they reach a maximum. This latter char- +acteristic is classical of modes with a large transient growth, as it was noticed +by [5]. On the other hand, the nature of the steady modes is similar to the +symmetry-breaking instabilities behind the disk or the sphere. These modes are +far less sensitive to transient growth and are observable with direct numerical +simulations and experiments. +12 + +4.1 +Fixed distance between jets and variable velocity ratio +δu +In the following, we focus on the influence of the velocity ratio δu between jets +for fixed jet distances L. Figure 5 displays the neutral curve of stability for +jet distances (a) L = 0.5 and (b) L = 1. One may observe that the primary +bifurcation is not always associated to the mode S1 as it is the case for δu = 0. +For sufficiently large velocity ratios, the primary instability leads to a non- +axisymmetric steady-state with a double helix. +Another interesting feature, +which could motivate a control strategy, is the occurrence of vertical asymptotes. +This sudden change in the critical Reynolds number is due to the retraction +and disappearance of the recirculating region. For L = 0.5, this sudden change +occurs for δu ≈ 0.25, and for higher values of δu the critical Reynolds number +is around twice larger than the one of the annular jet (δu = 0). The case of jet +distance L = 1 was discussed in section 3. The sudden change in the stability +of the branch S1 occurs between δu ∈ [0.25, 0.5]. Within this narrow interval, +the primary branch of instability is the F1. At around δu = 0.4, the primary +bifurcation is again the branch F1, which becomes secondary at around δu ≈ 0.8 +in favour of the branch S2. In fig. 5 we have highlighted the codimension two +point interaction between the S1 −S2 modes, whose modes are depicted in fig. 5 +(c-d), which will be analysed in detail in section 5. Around this point, we can +observe the largest stabilisation ratio between the annular jet (δu = 0) and a +configuration of concentric jets (δu ̸= 0). +4.2 +Fixed velocity ratio δu and variable distance between +jets +Figure 6 compares the results obtained for a constant velocity ratio when vary- +ing the distance between jets. As observed before, the solution becomes more +unstable by increasing the distance between jets. The largest critical Reynolds +number is found at δu = 0. +The critical Reynolds decreases with the dis- +tance between jets L. The points of codimension two, i.e., the points where +mode switching occurs, are highlighted in fig. 6. We can appreciate that the +interaction between the branch S1 and S2 happens for every velocity ratio δu +explored, and it is the mode interaction associated to the smallest distance be- +tween jets. Additionally, for a velocity ratio δu = 0.5 there exists two points +where the branches of the linear modes S1 and F1 intersect. Another feature +of the neutral curves is the existence of turning points, which are associated +to the existence of saddle node bifurcations of the axisymmetric steady-state. +The saddle node bifurcations of the steady-state induces the existence of regions +in the neutral curves with a tongue shape. These saddle node bifurcations are +also responsible for the formation of the vertical asymptotes observed in fig. 5. +Finally, it is of interest the transition of the modes S1 and S2, which induce +the symmetry breaking of the axisymmetric steady state to slow low frequency +spiralling structures. These modes have been identified for δu = 0.5 for m = 1, +δu = 1 for m = 2, and δu = 2 for both m = 1 and m = 2. As it will be clari- +13 + +a) +b) +1 +2 +3 +4 +L +0 +200 +400 +600 +800 +1000 +Re +LS1!S2 +LS1!F1 LF1!S1 +1 +2 +3 +4 +L +0 +0.5 +1 +1.5 +! +c) +d) +1 +2 +3 +4 +L +0 +200 +400 +600 +800 +1000 +Re +LS1!S2 +2 +3 +4 +40 +60 +80 +100 +120 +1 +2 +3 +4 +L +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +! +e) +f) +1 +2 +3 +4 +L +0 +200 +400 +600 +800 +1000 +Re +LS1!S2 +1 +2 +3 +4 +L +0 +0.1 +0.2 +0.3 +0.4 +! +Figure 6: Neutral lines of the four modes found studying the configuration of +two concentric jets fixing the velocity ratio. (a-b) δu = 0.5, (c-d) δu = 1, (e-f) +δu = 2. Black lines: modes with m = 1, red lines: modes with m = 2. Straight +lines: steady modes, dashed lines: unsteady modes. +14 + +fied in section 5, these oscillations are issued from the non-linear interaction of +modes, emerging simultaneously for a specific Reynolds number, and changing +their position as the most unstable global mode of the flow. +5 +Mode interaction between two steady states. +Resonance 1 : 2 +5.1 +Normal form, basic solutions and their properties +The linear diagrams of section 4 have shown the existence of the mode in- +teraction between the modes S1 and S2. It corresponds roughly to the mode +interaction that occurs at the largest critical Reynolds number for any value of +L herein explored. In this section, we analyse the dynamics near the S1 : S2 +organizing centre. We perform a normal form reduction, which allows us to +predict non-axisymmetric steady, periodic, quasiperiodic and heteroclinic cy- +cles between non-axisymmetric states. +The mode interaction that is herein analysed corresponds to a steady-steady +bifurcation with O(2) symmetry and with strong resonance 1 : 2. Such a bifur- +cation scenario has been extensively studied in the past by [8, 10, 23, 1] and the +reflection symmetry breaking case (SO(2)) by [24]. In order to unravel the exis- +tence and the stability of the nonlinear states near the codimension two point, +let write the flow field as +q += Q0 + Re +� +r1(τ)eiφ1(τ)e−iθˆqs,1 +� ++ Re +� +r2(τ)eiφ2(τ)e−2iθˆqs,2 +� +(11) +in polar coordinates for the complex amplitudes z1 = r1eiφ1 and z2 = r2eiφ2 +where rj and φj for j = 1, 2 are the amplitude and phase of the symmetry- +breaking modes m = 1 and m = 2, respectively. The complex-amplitude normal +form eq. (10) is expressed in this reduced polar notation as follows, +˙r1 = e3r1r2 cos(χ) + r1 +� +λ(s,1) + c(1,1)r2 +1 + c(1,2)r2 +2 +� +, +(12a) +˙r2 = e4r2 +1 cos(χ) + r2 +� +λ(s,2) + c(2,1)r2 +1 + c(2,2)r2 +2 +� +, +(12b) +˙χ = − +� +2e3r2 + e4 +r2 +1 +r2 +� +sin(χ), +(12c) +where the phase χ = φ2 −2φ1 is coupled with the amplitudes r1 and r2 because +of the existence of the 1 : 2 resonance. The individual phases evolve as +˙φ1 += e3r2 sin(χ), +˙φ2 += −e4 +r2 +1 +r2 sin(χ). +(13) +Before proceeding to the analysis of the basic solutions of eq. (12), we can +simplify these equations by the rescaling +� +r1 +|e3e4|1/2 , r2 +e3 +� +→ (r1, r2), +15 + +which yields the following equivalent system +˙r1 = r1r2 cos(χ) + r1 +� +λ(s,1) + c11r2 +1 + c12r2 +2 +� +, +(14a) +˙r2 = sr2 +1 cos(χ) + r2 +� +λ(s,2) + c21r2 +1 + c22r2 +2 +� +, +(14b) +˙χ = − 1 +r2 +� +2r2 +2 + sr2 +1 +� +sin(χ), +(14c) +where the coefficients +s = sign(e3e4), +c11 = c(1,1) +|e3e4|, +c12 = c(1,2) +e2 +3 +, +c21 = c(2,1) +|e3e4|, +c22 = c(2,2) +e2 +3 +. +Finally, we consider a third normal form equivalent to the previous ones but +which removes the singularity of eqs. (12) and (14) when r2 = 0. Standing +waves (sin χ = 0) naturally encounter this type of artificial singularity, which +manifests as in eq. (14) as an instantaneous jump from one standing subspace +to the other by a π-translation. +This is the case of the heteroclinic cycles, +previously studied by [1, 23]. The third normal form, which we shall refer to as +reduced Cartesian normal form, takes advantage of the simple transformation +x = r2 cos(χ), y = r2 sin(χ) [24]: +˙r1 = r1 +� +λ(s,1) + c11r2 +1 + c12(x2 + y2) + x +� +, +(15a) +˙x = sr2 +1 + 2y2 + x +� +λ(s,2) + c21r2 +1 + c22(x2 + y2) +� +, +(15b) +˙y = −2xy + y +� +λ(s,2) + c21r2 +1 + c22(x2 + y2) +� +, +(15c) +In this final representation standing wave solutions are contained within the +invariant plane y = 0, and due to the invariance of eq. (15) under the reflection +y �→ −y, one can restrict attention, without loss of generality, to solutions with +y ≥ 0, cf [23]. +The system eq. (14) possess four types of fixed points, which are listed in +table 1. +First, the axisymmetric steady state (O) is represented by (r1, r2) = (0, 0), so +it is the trivial steady-state of the normal form. The second steady-state is what +it is denoted as pure mode (P). In the original coordinates, it corresponds to the +symmetry breaking structure associated to the mode S2. This state bifurcates +from the axisymmetric steady state (O) when λ(s,2) = 0. The third fixed point +is the mixed mode state (MM), which is listed in table 1. It corresponds to +the reflection symmetry preserving state associated to the mode S1. It may +bifurcate directly from the trivial steady state O, when λ(s,1) = 0 or from P +whenever σ+ = 0 or σ− = 0, where σ± is defined as +σ± ≡ λ(s,1) − −λ(s,2)c12 +c22 +± +� +−λ(s,2) +c22 +. +(16) +16 + +Name +Definition +Bifurcations +Comments +O +r1,O = r2,O = 0 +− +Steady axisymmetric state +P +r2 +2,P = +−λ(s,2) +c22 +, r1,P = 0 +λ(s,2) = 0 +Bifurcation from O +r1,MM = − +λ(s,1)±r2,MM+c12r2 +2,MM +c11 +λ(s,1) = 0 +Bifurcation from O +MM +PMM(r2,MM cos(χMM)) = 0 +σ± = 0 +Bifurcation from P +cos(χMM) = ±1 +cos(χT W ) = +(2c11+c12)λ(s,2)−(2c21+c22)λ(s,1) +ΣT W (2λ(s,1)+λ(s,2)) +TW +r2 +2,T W = +−(2λ(s,1)+λ(s,2)) +ΣT W +cos(χT W ) = ±1 +Bifurcation from MM +r2 +1,T W = 2r2 +2,T W +Table 1: Definition of the fixed points of the reduced polar normal form eq. (14). +σ± is defined in eq. (16), the polynomial PMM is defined in eq. (17) and ΣT W ≡ +4c11 + 2(c12 + c21) + c22. +Name +Bifurcation condition +Comments +SW +sr2 +1 − 2c11r2 +1r2,MM cos(χMM) − 2c22r3 +2,MM cos(χMM)3 = 0 +Bif. from MM +MTW +DT W − TT W IT W = 0, IT W > 0 +Bif. from TW +Table 2: Definition of the limit cycles of the reduced polar normal form eq. (14). +The representation in the reduced polar form is +r1,MM = − +λ(s,1) ± r2,MM + c12r2 +2,MM +c11 +, +cos(χMM) = ±1, +and the condition PMM(r2,MM cos(χMM)) = 0, where PMM is defined as +PMM(x) ≡ sµ1+(s+c21λ(s,1)−c(1,1)λ(s,2))x+(c21+sc12)x2+(c12c21−c11c22)x3. +(17) +Finally, the fourth fixed point of the system are travelling waves (TW). It is +surprising that the interaction between two steady-states causes a time-periodic +solution. The travelling wave emerges from MM in parity-breaking pitchfork +bifurcation that breaks the reflection symmetry when cos(χT W ) = ±1. The +TW drifts at a steady rotation rate ωT W along the group orbit, i.e., the phases +˙φ1 = r2,T W sin(χT W ) and ˙φ2 = −s +r2 +1,T W +r2,T W sin(χT W ) are non-null. +Mixed modes and travelling waves may further bifurcate into standing waves +(SW) and modulated travelling waves (MTW), respectively. These are generic +features of the 1 : 2 resonance for small values of λ(s,1) and λ(s,2), when s = +−1. In the original coordinates, SW are periodic solutions, whereas MTW are +quasiperiodic. Standing waves emerge via a Hopf bifurcation from MM when +the conditions PSW +� +r2,MM cos(χMM) +� +> 0 for +PSW(x) ≡ (2c22x3 − sr2 +1)c11 − (2c12x + 1)(c21x + s)x, +17 + +Name +Condition +Comments +Ht AGH +λ(s,1) > 0, λ(s,2) > 0, c22 < 0 +Existence +σ+ > 0, σ− < 0 +Asymptotic stability +Table 3: Definition of the conditions for the existence of the Ht AGH (robust +heteroclinic cycles connecting pure modes) of the reduced polar normal form +eq. (14). +O +P +MM +Ht AGH +SW +TW +MTW +Condition Tab. 2 +Condition Tab. 2 +Cond. +Tab. 3 +Figure 7: Schematic representation of the basic solutions of eq. (12) and their +bifurcation path. +and the one listed in table 2 are satisfied. +MTW are created when a torus +bifurcation happens on the travelling wave branch when the conditions listed in +table 2 are satisfied. +Another remarkable feature of eq. (12) is the existence of robust heteroclinic +cycles that are asymptotically stable. When s = −1, there are open sets of +parameters (see table 3) where the reduced polar normal form exhibits struc- +turally stable connections between π−translations on the circle of pure modes, +cf [1]. These structures are robust and have been observed in a large variety +of systems, [19, 18, 16, 22, 13]. In addition to these robust heteroclinic cycles +connecting pure modes, there exists more complex limit cycles connecting O, +P, MM and SW, cf [23]. These cycles are located for larger values of λ(s,1) and +λ(s,2), with possibly chaotic dynamics (Shilnikov type). In this study, we have +not identified any of these. Finally, a summary of the basic solutions and the +bifurcation path is sketched in fig. 7. +5.2 +Results of the steady-steady 1 : 2 mode interaction +Section 4.2 reported the location of mode interaction points for discrete values +of the velocity ratio δu. The location of the mode interaction between S1 and +S2 is depicted in fig. 8. It shows that the mode switching between the modes S1 +18 + +a) +b) +0 +0.5 +1 +1.5 +2 +/u +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +L +0 +0.5 +1 +1.5 +2 +/u +100 +200 +300 +400 +500 +600 +700 +800 +Re +Figure 8: Evolution of the codimension two interaction S1 − S2 in the space of +parameters (Re, L, δu). Grey points denote the points that were computed and +the red point denotes the transition from steady to unsteady with low frequency +as reported in section 4.2. +and S2 is indeed stationary only for δu < 1.5 and L < 1.3. For larger values of +the velocity ratio and the jet distance, the interaction is not purely stationary; +at least one of the linear modes oscillates with a slow frequency. It implies that +the mode selection for large velocity ratios near the codimension two points is +similar to the one reported by [15] for swirling jets. However, even when the two +primary bifurcations are non-oscillating (S1 and S2), the 1 : 2 resonance of the +azimuthal wavenumbers induces a slow frequency, what we denote as travelling +wave solutions (TW). +We consider the bifurcation sequence for δu = 1.0 and L = 1.15, which +is qualitatively similar to transitions in the range 0.5 < δu < 1.5, near the +codimension two points, which are depicted in fig. 8. At the codimension two +points for δu < 0.5, at least one of the two bifurcations is sub-critical and a +normal form reduction up to fifth order is necessary. Subcritical transition was +also noticed for a distance between jets L = 0.1 by [5], who reported high +levels of the linear gain associated to transient growth mechanisms. This last +case is out of the scope of the present manuscript. Figure 9 displays the phase +portrait of the stable attractors near the S1 : S2 interaction. +For values of +δu > 1.0, the axisymmetric steady-state loses its axisymmetry leading to a +new steady-state with symmetry m = 2, herein denoted as pure mode (P). A +reconstruction of the fluctuating flow field of such a state is performed at the +bottom right of fig. 9, which shows that the state P possesses two orthogonal +planes of symmetry. Near the codimension two point, for values of the velocity +ratio δu < 1.1, the state P is only observable, that is non-linearly stable, within +a small interval with respect to the Reynolds number. +For larger values of +the velocity ratio, the state P remains stable within the analysed interval of +Reynolds numbers. For values of the velocity ratio δu < 1.0, the bifurcation +19 + +w' = ± 0.025 +w' = ± 0.01 +w' = ± 0.01 +0.02 +0.01 +0 +-0.01 +-0.02 +w' +170 +180 +190 +200 +210 +Re +𝛿u +Rotating +Non-rotating +Non-rotating +TW +MM +P +Ht AGH +MM +TW +MTW +P +0.8 +0.9 +1.0 +1.1 +1.2 +z=1/2 +z=1 +z=1 +z=1 +z=2 +t-T/3 +t+T/3 +t +Figure 9: Phase portrait at the codimension two point S1 : S2 for parameter +values (L, δu) = (1.15, 1.0). Visualisations of blue and red surfaces in the iso- +metric views represent the respective positive and negative isocontour values of +the perturbative axial velocity indicated in the figure. +180 +185 +190 +195 +200 +Re +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +r2 +MM +TW +PM +BifTW!MTW +BifMM!TW +0 +-0.1 +0.05 +y +x +0 +0.1 +0 +0.05 +r1 +0.1 +0.1 +170 +175 +180 +185 +190 +195 +200 +Re +Figure 10: Bifurcation diagram with respect to the Reynolds number for L = +1.15 and δu = 0.8. The left diagram reports the evolution of r2 for the fixed +point solutions of the normal form. The right diagram displays the bifurcation +diagram in the Cartesian coordinates. Solid lines and dashed lines denote stable +attractors and unstable attractors, respectively. +20 + +diagram is more complex. Figure 10 displays the bifurcation diagram of the +fixed-point solutions of eq. (15) on the left diagram and the full set of solutions +of the normal form in the right diagram. The axisymmetric steady-state first +bifurcates towards a Mixed-Mode solution, which is the solution in the y = 0 +plane for the right diagram of fig. 10. A solution with a non-symmetric wake +has been reconstructed in fig. 9. The Mixed-Mode solution is only stable within +a small interval of the Reynolds number. +A secondary bifurcation, denoted +BifMM−T W , gives raise to a slowly rotating wave of the wake. The TW and the +MM solutions are identical at the bifurcation point. The phase speed is zero +at the bifurcation, thus this is not a Hopf bifurcation. It corresponds to a drift +instability that breaks the azimuthal symmetry, i.e. it starts to slowly drift. +This unusual feature, that travelling waves bifurcate from a steady solution at a +steady bifurcation, is a generic feature of the 1 : 2 resonance. A reconstruction +of the travelling wave solution is depicted on the top of fig. 9. It corresponds +to the line with non-zero y component in the right diagram of fig. 10. The TW +solution loses its stability in a tertiary bifurcation, denoted as BifT W −MT W . It +conforms to a Hopf bifurcation of the TW solution, which gives birth to a quasi- +periodic solution name Modulated Travelling Wave (MTW). A representation +of this kind of solution in the Cartesian coordinates (r1, x, y) is depicted on the +right image of fig. 10. +Eventually, the Modulated Travelling Wave experiences a global bifurcation. +That occurs when the periodic MTW solution, in the (r1, x, y) coordinates, +nearly intersects the invariant r1 = 0 and y = 0 planes. The transition sequence +is represented in the right image of fig. 10 in the Cartesian coordinates (r1, x, y). +The amplitude of the MTW limit cycle increases until the MTW arising at +the tertiary bifurcation BifT W −MT W are destroyed by meeting a heteroclinic +cycle at BifMT W −Ht. +The locus of BifMT W −Ht is reported in fig. 9 and in +good agreement with [1]. The conditions for the existence of the heteroclinic +cycles are listed in table 3. When σ− becomes negative, the cycle is attracting +and robust heteroclinic cycles are observed. It is destroyed when σ+ becomes +negative, in that case the pure modes are no longer saddles which breaks the +heteroclinic connection. Figure 11 displays the instantaneous fluctuation field +from a heteroclinic orbit connecting P and its conjugate solution P’, which is +obtained by a rotation of π/2, for parameter values Re = 200 and δu = 0.8. The +dynamics of the cycle takes place in two phases. Figure 11 depicts the motion +of the coherent structure associated to the heteroclitic cycle. Starting from the +conjugated pure mode P’, the cycle leaves the point (a), located in the vicinity +of P’, along the unstable eigenvector y, which is the stable direction of P. The +first phase consists in a rapid rotation by π/2 of the wake, it corresponds to the +sequence a-b-c-d-e displayed in fig. 11. Then it is followed by a slow approach +following the direction y and departure from the pure mode state P along the +direction r1. +The second phase consists in a rapid horizontal motion of the +wake, which is an evolution from P to P’ that takes place by the breaking of +the reflectional symmetry with respect to the vertical axis; it constitutes the +sequence e-f-g-h-i-a. Please note that equivalent motions are also possible. The +first phase of rapid counter-clockwise rotation by π/2 can be performed in the +21 + +0.1 +0.1 +0.15 +0.05 +0.05 +0 +0 +-0.1 +-0.1 +0.1 +0.05 +-0.05 +0 +0.1 +r1 +y +x +c +e +d +f +h +h +i +g +f +b +a +c +d +e +2000 +1500 +500 +1000 +r1 +x≡r2cos(χ) +y≡r2sin(χ) +0 +0 +-0.1 +0.1 +0.2 +t +sin(χ) = 0 +P' +P +b +a +g +i +w' +w' = ± 0.05 +w' = ± 0.05 +r1=0 +Figure 11: Heteroclinic cycle solution for parameter values Re = 200, δu = 0.8. +The top and bottom image sequences along the heteroclinic cycle show (from +left to right) an axial slice plane at z = 1 of the instantaneous fluctuations +of the axial velocity of the flow field as viewed from downstream, along with +a three-dimensional isometric view (d on the top and g on the bottom).The +middle diagram displays the heteroclinic cycle in the coordinates (r1, x, y). +22 + +opposite sense. It corresponds to a motion in the Cartesian coordinates along +the plane r1 along negative values of y. The sequence e-f-g-h-i-a can be replaced +by a horizontal movement in the opposite sense, which adjusts to connect the +plane y = 0 corresponding to negative values of r1, +6 +Discussion & Conclusions +This article achieves a complete description of the configuration consisting of +two coaxial jets, broadly found in industrial processes, covering a wide range +of applications such as noise reduction, mixing enhancement, or combustion +control. +The numerical procedure herein employed has been validated with +the existing literature in the case of the stability analysis (see B for a detailed +overview), and compared to DNS results, as done in [30]. The analysis comprises +a layout with a wide range of the velocity ratio (δu = Ui/Uo) between the jets, +from δu = 0 to δu = 2, as well as the distance between jets (L) enclosing values +from L = 0.5 to L = 4, substantially expanding the work of [5]. +A linear stability analysis reveals the most significant modes, consisting of +two steady modes (S1 and S2, located within the recirculation bubble) and two +unsteady ones (F1 and F2, evolving as a transient growth in the downstream +direction). +The critical Reynolds number is determined for a wide range of +velocity and distance ratios, starting with the influence of the velocity ratio. +As the relation between inner and outer velocities grows, the flow is stabilised, +increasing the critical Reynolds number. The primary instability swaps from +mode S1, characterised with one symmetry plane, to mode S2 that possesses +two symmetry planes. An abrupt divergence in the critical Reynolds number is +captured, associated with the vanishing of the recirculation region, that could +suggest a stability control strategy. +Subsequently, the effect of the distance +L between jets is analysed. The primary effect of increasing this distance is +a decrease in the critical Reynolds number for all values of δu investigated. +Additionally, the existence of saddle node bifurcations, that swap the most +unstable mode of the flow, generates turning points in the neutral curve. +The investigated bifurcation scenario starts from the codimension point, with +an axisymmetric steady state located at a velocity ratio δu = 1.0 and distance +between jets of L = 1.15. It is qualitatively equivalent to transitions found in +the range 0.5 < δu < 1.5. It reveals a break of the axisymmetry for values higher +than δu = 1.0, presenting a steady state as a pure mode P with two orthogonal +planes of symmetry. For values lower than δu = 1.0, the bifurcation diagram +exhibits a slightly complicated path. Firstly, it drives into a Mixed-Mode (MM) +solution presenting a non-symmetric wake, that is only stable for a small range +of the Reynolds number. +Subsequently, a slowly rotating wake is triggered +in the form of a Travelling Wave (TW). This unusual feature, an unsteady +state emerging from a steady state, corresponds to a drift instability commonly +found at 1 : 2 resonance. Then, the TW solution encounters a Hopf bifurcation, +developing a quasi-periodic solution in the form of a Modulated Travelling Wave +(MTW). Finally, the MTW solution undergoes a global bifurcation meeting a +23 + +heteroclinic cycle (Ht). +This heteroclinic orbit links the solution P with its +conjugate solution P’, spinning the wake from P’ to P, and moving it horizontally +from P to P’. +24 + +Acknowledgments +A.C., J.A.M. and S.L.C. acknowledge the grant PID2020-114173RB-I00 funded +by MCIN/AEI/ 10.13039/501100011033. S.L.C., J.A.M. and A.C. acknowledge +the support of Comunidad de Madrid through the call Research Grants for +Young Investigators from Universidad Polit´ecnica de Madrid. AC also acknowl- +edges the support of Universidad Polit´ecnica de Madrid, under the programme +‘Programa Propio’. +A +Normal form reduction +In this section we provide a detail explanation of the normal form reduction to +obtain the coefficients of eq. (10), we define the terms of the compact notation +of the governing equations eq. (3), which is reminded here, for the sake of +conciseness, +B∂Q +∂t = F(Q, η) ≡ LQ + N(Q, Q) + G(Q, η). +(18) +The nonlinear convective operator N(Q1, Q2) = U1 · ∇U2 accounts for the +quadratic interaction on the state variable. The linear operator on the state +variable is LQ = [∇P, ∇·U]T . The remaining term accounts for the linear vari- +ations in the state variable and the parameter vector. It is defined as G(Q, η) = +G(Q, [η1, 0]T ) + G(Q, [0, η2]T ) where G(Q, [η1, 0]T ) = η1∇ · (∇U + ∇UT ) and +G(Q, [0, η2]T ). The former operator shows the dependency on the parameter +η1, which accounts for the viscous effects. The latter operator depends on the +parameter η2, which accounts for the velocity ratio between jets and it is used +to impose the boundary condition U = (0, η2 tanh +� +bi(1 − 2r) +� +, 0) on Γin,i. In +addition, we consider the following splitting of the parameters η = ηc + ∆η. +Here ηc denotes the critical parameters ηc ≡ [Re−1 +c , δu,c]T attained when the +spectra of the Jacobian operator posses at least an eigenvalue whose real part +is zero. The distance in the parameter space to the threshold is represented by +∆η = [Re−1 +c +− Re−1, δu,c − δu]T . +A.1 +Multiple scales ansatz +The multiple scales expansion of the solution q of eq. (3) is +q(t, τ) = Q0 + εq(ε)(t, τ) + ε2q(ε2)(t, τ) + O(ε3), +(19) +where ε ≪ 1 is a small parameter. The distance in the parameter space to +the critical point ∆η = [Re−1 +c +− Re−1, δu,c − δu]T is assumed to be of second +order, i.e. ∆ηi = O(ε2) for i = 1, 2. The expansion eq. (19) considers a two +scale expansion of the original time t �→ t + ε2τ. A fast timescale t and a slow +timescale of the evolution of the amplitudes zi(τ) in eq. (19), for i = 1, 2. Note +that the expansion of the LHS eq. (3) up to third order is as follows +εB∂q(ε) +∂t ++ ε2B∂q(ε2) +∂t ++ ε3� +B∂q(ε3) +∂t ++ B∂q(ε) +∂τ +� +, +(20) +25 + +and the RHS respectively, +F(q, η) = F(0) + εF(ε) + ε2F(ε2) + ε3F(ε3). +(21) +The expansion eq. (21) will be detailed at each order. +A.1.1 +Order ε0 +The zeroth order Q0 of the multiple scales expansion eq. (19) is the steady state +of the governing equations evaluated at the threshold of instability, i.e. η = ηc, +0 = F(Q0, ηc). +(22) +A.1.2 +Order ε1 +The first order q(ε)(t, τ) of the multiple scales expansion of eq. (19) is composed +of the eigenmodes of the linearized system +q(ε)(t, τ) ≡ +� +z1(τ)e−im1θˆq1 + z2(τ)ei−m2θˆq2 + c. c. +� +. +(23) +in our case, m1 = 1 and m2 = 2. Each term ˆqℓ of the first order expansion +eq. (23) is a solution of the following linear equation +J(ωℓ,mℓ)ˆqℓ = +� +iωℓB − ∂F +∂q |q=Q0,η=ηc +� +ˆqℓ, +(24) +where ∂F +∂q |q=Q0,η=ηc ˆqℓ = Lmℓ ˆqℓ + Nmℓ(Q0, ˆqℓ) + Nmℓ(ˆqℓ, Q0). The subscript +mℓ indicates the azimuthal wavenumber used for the evaluation of the operator. +A.1.3 +Order ε2 +The second order expansion term q(ε2)(t, τ) is determined from the resolution +of a set of forced linear systems, where the forcing terms are evaluated from first +and zeroth order terms. The expansion in terms of amplitudes zi(τ) (i = 1, 2) +of q(ε2)(t, τ) is assessed from term-by-term identification of the forcing terms at +the second order. Non-linear second order terms in ε are +F(ε2) +≡ +2 +� +j,k=1 +� +zjzkN(ˆqj, ˆqk)e−i(mj+mk)θ + c.c. +� ++ +2 +� +j,k=1 +� +zjzkN(ˆqj, ˆqk)e−i(mj−mk)θ + c.c. +� ++ +2 +� +ℓ=0 +ηℓG(Q0, eℓ), +(25) +where the terms proportional to zjzk are named ˆF(zjzk) +(ϵ2) +and eℓ is an element of +the orthonormal basis of R2. +26 + +Then, we look for a second order term expanded as follows +q(ε2) ≡ +2 +� +j,k=1 +k≤j +� +zjzkˆqzjzk + zjzkˆqzjzk + c.c +� ++ +2 +� +ℓ=1 +ηℓQ(ηℓ) +0 +. +(26) +Terms ˆqz2 +j are azimuthal harmonics of the flow. The terms ˆqzjzk with j ̸= k are +coupling terms, and ˆq|zj|2 are harmonic base flow modification terms. Finally, +Q(ηℓ) +0 +are base flow corrections due to a variation of the parameter ηℓ from the +critical point. +At this order, there exists two resonant terms, the terms proportional to z1z2 +and z2 +1, which are associated with the singular Jacobian J(0,mk) for k = 1, 2. To +ensure the solvability of these terms, we must enforce compatibility conditions, +i.e. the Fredholm alternative. The resonant terms are then determined from the +resolution of the following set of bordered systems +�J(0,mk) +ˆqk +ˆq† +k +0 +� �ˆq(z(R)) +e +� += +� +ˆF(z(R)) +(ε2) +0 +� +, z(R) ∈ [z1z2, z2 +1]T , +(27) +where e = e3 for z(R) = z1z2 and e = e4 for z(R) = z2 +1. The non-resonant terms +are computed by solving the following non-degenerated forced linear systems +J(0,mj+mk)ˆqzjzk = ˆF(zjzk) +(ϵ2) +, +(28) +and +J(0,0)Q(ηℓ) +0 += G(Q0, eℓ). +(29) +A.1.4 +Order ε3 +At third order, there exists six degenerate terms. In our case, we are not inter- +ested in solving for terms of third-order, instead, we will determine the linear +and cubic coefficients of the third order normal form eq. (10) from a set of +compatibility conditions. +The linear terms λ(s,1) and λs,2 and cubic terms c(i,j) for i = 1, 2 are deter- +mined as follows +λ(s,1) = +⟨ˆq† +1, ˆF(z1) +(ε3)⟩ +⟨ˆq† +z, Bˆqz⟩ +, λ(s,2) = +⟨ˆq† +2, ˆF(z2) +(ε3)⟩ +⟨ˆq† +2, Bˆq2⟩ +, c(i,j) = +⟨ˆq† +2, ˆF(zi|zj|2) +(ε3) +⟩ +⟨ˆq† +i, Bˆqi⟩ +. +(30) +The forcing terms for the linear coefficient are +ˆF(zj) +(ε3) ≡ +2 +� +ℓ=1 +ηℓ +�� +N(ˆqj, Q(ηℓ)) +0 ++ N(Q(ηℓ) +0 +, ˆqj) +� ++ G(ˆqj, eℓ) +� +. +(31) +which allows the decomposition of λ(s,ℓ) = λ(s,ℓ),Re(Re−1 +c Re−1)+λ(s,ℓ),δu(δu,c − +δu) for ℓ = 1, 2. +27 + +Canton et al. (2017) [5] +Present work +Rec +1420 +1405 +ω +5.73 +5.72 +Table 4: Comparison of Rec and ω between previous work and the present one. +The forcing terms for the cubic coefficients are +ˆF(zj|zk|2) +(ε3) +≡ +� +N(ˆqj, ˆq|zk|2) + N(ˆq|zk|2, ˆqj) +� ++ +� +N(ˆq−k, ˆqzjzk) + N(ˆqj,k, ˆq−k) +� ++ +� +N(ˆqk, ˆqzjzk) + N(ˆqzjzk, ˆqk) +� +. +(32) +if j ̸= k and +ˆF(zj|zj|2) +(ϵ3) +≡ +� +N(ˆqj, ˆq|z|2 +j) + N(ˆq|z|2 +j, ˆqj) +� ++ +� +N(ˆq−j, ˆqz2 +j ) + N(ˆqz2 +j , ˆq−j) +� +, +(33) +for the diagonal forcing terms. +B +Validation of the code - Comparison with the +literature +The calculations made in StabFem in the sections at the main manuscript are +validated comparing the leading global mode in the geometry proposed by [5]. +Moreover, the critical Reynolds number and the frequency associated are also +analysed. In the cited work, the authors use an analogous geometry with the +following parameters: +• Radious of the inner jet Rinner = 0.5 +• Diameter of the outer jet D = 0.4 +• Distance between jets L = 0.1 +• Ratio between velocities δu = 1 +The linear stability analysis has been carried out imposing m = 0, as done +by [5], so the leading global mode will be axisymmetric. The critical Reynolds +number Rec and the frequency ω of the leading global mode are compared in +Tab. 4. As seen, few differences can be found on the critical Reynolds number +and the frequency. The relative error in the Rec calculation is 1.06% and the +one of the frequency is 0.17%. +The global mode is now calculated using StabFem and compared with the +one calculated by [5]. This mode can be found in figures 9, 10 and 11 on the +cited paper. As it can be seen, there are not substantial differences between the +direct modes, being both of them a vortex street with their biggest amplitude +28 + +0 +5 +10 +15 +20 +25 +z +0 +1 +2 +r +-0.5 0 +0.5 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +z +0 +0.5 +1 +1.5 +r +-2 +0 +2 +0 +0.2 +0.4 +0.6 +z +0.4 +0.5 +0.6 +0.7 +r +0 +0.5 +1 +Figure 12: Direct mode, adjoint mode and sensitivity of the leading global mode +studied by [5] calculated using StabFem. +situated 10 units downstream the exit of the jets. The adjoint mode is con- +centrated within the nozzle, with its biggest amplitude situated on the sharp +corners. There is not any difference between the adjoint mode calculated with +StabFem and the one in [5]. Finally, the structural sensitivity is similar to the +one computed by [5]. It is composed by two lobes in the space between the exit +of the two jets. +References +[1] Dieter Armbruster, John Guckenheimer, and Philip Holmes. 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Sci., 1:2, 1969. +31 + diff --git a/9NE3T4oBgHgl3EQfSAnZ/content/tmp_files/load_file.txt b/9NE3T4oBgHgl3EQfSAnZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9884d4e767ccec7fd74ca6238bf2ed0bb8aff1c3 --- /dev/null +++ b/9NE3T4oBgHgl3EQfSAnZ/content/tmp_files/load_file.txt @@ -0,0 +1,981 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf,len=980 +page_content='Mode selection in concentric jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The steady-steady 1:2 resonant mode interaction with O(2) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Corrochano1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Sierra-Aus´ın2,3, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Martin1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Fabre2, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Le Clainche ∗1 1School of Aerospace Engineering, Universidad Polit´ecnica de Madrid, Madrid 28040, Spain 2Institut de M´ecanique des Fluides de Toulouse (IMFT), Toulouse 31400, France 3DIIN, Universit´a degli Studi di Salerno, Via Giovanni Paolo II, 84084 Fisciano (SA), Italy Abstract In this article, a thorough characterization of the configuration com- posed by two concentric jets at a low Reynolds number is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The analysis comprises a layout with a wide range for the velocity ratio be- tween the inner and outer jets, defined within the interval [0, 2], and also details the influence of the distance between jets, where the wall thick- nesses separating the two jets is [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Global linear stability analy- sis identifies the most significant modes driving the changes in the flow dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The neutral lines revealing the critical Reynolds number con- nected to the presence of the main (steady and unsteady) flow bifurcations, which are presented by global azimuthal modes, show the high complexity of the problem under study, where hysteresis and other types of complex cycles are pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, the mode interaction is analysed, high- lighting the presence of travelling waves emerging from the interaction of steady states, and the existence of robust heteroclinic cycles that are asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The high level of detail in the results presented, makes this work as a reference for future research development in the field of concentric jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 1 Introduction Double concentric jets is a configuration enhancing the turbulent mixing of two jets, which is used in several industrial applications where the breakup of the jet ∗Email address for correspondence: soledad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='leclainche@upm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='es 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='04429v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='flu-dyn] 11 Jan 2023 Rin Rout Uin Uout INITIAL MERGING ZONE r TRANSITIONAL ZONE MERGED ZONE Outer potential core Inner potential core Outer mixing region Inner mixing region Reattachment point z Figure 1: Sketch representing the three flow regimes in the near field of double concentric jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Figure based on the sketch presented in [12, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' into droplets due to flow instabilities is presented as the key technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Com- bustion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' : combustion chamber of rocket engines, gas turbine combustion, internal combustion engines, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=') and noise reduction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' : in turbofan en- gines) are the two main applications of this geometry, although the annular jets can also be found in some other relevant applications such as ink-jet printers or spray coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The qualitative picture emerging from this type of flow divides the inner field of concentric jets in three different regions: (i) initial merging zone, (ii) transitional zone and (iii) merged zone, as presented in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 1, that follows the initial sketch presented by [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In the initial merging zone (i), just at the exit of the two jets, two axisymmetric shear layers (inner and outer boundary layer) develop and start to merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In this region, we distinguish the inner and outer shear layers, related with the inner and outer jet stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Then, most of the mixing occurs in the transitional zone (ii), that extends until the external shear layer reaches the centreline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, in the merged zone (iii), the two jets are totally merged, modelling a single jet flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Several parameters define the characteristic of this flow: the inner and outer jet velocities, the jet diameters, the shape and thickness of the wall separating both jets, the Reynolds number, the boundary layer state and thickness at the jet exit and the free stream turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Based on these parameters, it is possible to identify several types of flow behaviour, which can be related with the presence of flow instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Ko & Kwan (1976) [12] postulated that the double concentric jet configura- tion could be considered as a combination of single jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Nevertheless, Dahm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (1992) [7] revealed by means of flow visualizations, several topology patterns as function of the outer/inner jet velocity ratio, reflecting that the dynamics of 2 the inner and outer jet shear layers were different from that in a single jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' More- over, this study exhibited a complex interaction between vortices identified in both shear layers, affecting the instability mechanism of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Buresti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (1992) [4] found that the outer shear layer dominated the flow dynamics for cases in which the outer velocity was much larger than the inner velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These authors also detected the presence of an alternate vortex shedding when the wall thickness between the two jets was sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The same mechanism was recognised by other authors [7, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Rehab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (1997) [25] studied in detail the flow differences as function of the outer/inner velocity ratio, finding two different flow regimes when the external jet diameter is much larger than the internal jet one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' When the outer/inner velocity ratio was larger than a crit- ical value, the authors spotted a low frequency recirculation bubble at the jet outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' On the contrary, for outer/inner velocity ratio smaller than such critical value, the outer (still fast) jet excites the inner jet, which ends oscillating at the same frequency as the external jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This is known as the lock-in phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Moreover, the oscillation frequency detected was similar to the one defined by a Kelvin-Helmholtz flow instability, which is generally encountered in single jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This lock-in phenomenon was also identified by other authors [7, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Following previous works [4, 7, 20] and paying especial attention to the sep- arating wall thickness and the vortex shedding located behind the wall, Wallace & Redekopp (1992) [33] showed that the wall thickness and sharpness change the characteristic of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Segalini & Talamelli (2011) [26] performed exper- iments to inspect in detail the effects of the outer/inner velocity ratio and the wall thickness in double concentric jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These authors found that for small outer/inner velocity ratios, the inner jet presents its own flow instability in the shear layer, while a different flow instability was identified in the outer jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' On the contrary, for large outer/inner velocity ratios, the outer shear layer drives the flow dynamics, forcing the inner shear layer to oscillate with the same frequency, occurring in the lock-in phenomenon previously mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, for similar outer/inner velocity ratios, a Von K´arm´an vortex street was detected near the separating wall, as also depicted by other authors [4, 7, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A wake instability affected the inner and outer shear layers, reversing the lock-in phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Different configurations can also be found, changing the velocity ratio be- tween jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (1969) [34] worked on the influence of the exte- rior/interior velocity ratio on noise attenuation, which was analysed experimen- tally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It was observed that for some given configurations, more noise attenuation was present than for the others, with a maximum between 12 and 15dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Talamelli & Gavarini (2006) [31] performed a local linear stability analysis, finding that for specific wall thickness, the vortex shedding identified behind the wall, can be related with an absolute instability that exists for some specific outer/inner velocity ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The authors explained that this absolute instabil- ity may trigger the destabilization of the flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This theoretical work was verified experimentally by [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These authors showed once more that the wake behind the wall separating the two jets creates a vortex shedding driving the frequency of the external shear layer also controlling the evolution of the inner shear layer, which can be the mechanism that triggers a global absolute insta- 3 bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This passive mechanism can be considered as a potential tool for flow control, delaying the transition to turbulence by means of controlling the near field of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Recently, Canton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (2017) [5] performed a global linear sta- bility analysis to study more in detail this vortex shedding mechanism behind the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' They examined a concentric jet configuration with a very small wall thickness (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1Di, with Di the inner jet diameter), but the authors selected an outer/inner velocity ratios where it was known that the alternate vortex shed- ding behind the wall was driving the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A global unstable mode (absolute instability) with azimuthal wavenumber m = 0 was found, confirming that the primary instability was axisymmetric (the modes with m = 1, 2 were stable at the flow conditions at which the study was carried out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The highest intensity of the global mode was located in the wake of the jet, composed by an array of counter-rotating vortex rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The shape of the mode changes when moving along its neutral curve, revealing through the numerical simulations a Kelvin- Helmholtz instability over the shear-layer between the two jets and in the outer jet at high Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Nevertheless, the authors showed that the wave- maker was located in the bubble formed upstream the separating wall, in good agreement with the results presented by [32], who performed a similar stability analysis in a two-dimensional configuration (wakes with co-flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The stability of annular jets, a limit case where the inner jets have zero velocity, has also been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In different analysis of annular jets [3, 17], it has been illustrated that this type of axisymmetric configuration does not behave as it appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The m = 0 modes studied have been shown to be stable, and the dominant mode found by both studies is helical (m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In addition, to characterise the annular jet, these investigations analyse the behaviour of the case by adding an azimuthal component to the inflow velocity, making the discharge of the annular jet eddy-like, comparing the evolution of the frequency and growth rate of this m = 1 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This paper expands on the work done by [5], where they use a specific ge- ometry and vary the outer/inner velocity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This paper presents a complete characterisation of the main global modes identified in two concentric jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The wall thicknesses separating the two jets are defined in the interval L ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5, 4], and the flow is simulated for different inside/outside velocity ratios in the in- terval Ui/Uo ∈ [0, 2], where the case with Ui/Uo = 0 represents an annular jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Global modes with azimuthal wavenumber m = 0 (axisymmetric modes), m = 1 and m = 2 will be searched for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' As identified in the literature [17, 3], no axisymmetric modes (m = 0) could be identified for any of the distances, as this is a helical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This paper expands the conclusions found in these two previous works, extending the results to different wall thicknesses between jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This part of the paper studies in detail the configuration of two concentric jets at low Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Using a linear approximation of the equations that model the flow, the base flows will be obtained on which to apply the linear stability analysis, by means of which it is possible to identify the most relevant modes that influence the flow dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This work also performs a study of mode selection, as some configurations presents interactions between different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Different analysis have been 4 done to know the different coherent structures when there is an interaction between modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' [30] conducted the study on the flow past a rotating sphere, finding different coherent structures on a triple-Hopf bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Some of these configurations are steady states, travelling waves or rotating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' To the authors’ knowledge, this is the first time that the characterisation of two concentric jets is presented with such level of detail, presenting neutral curves for a wide range of different configurations, as well as providing a deep understanding of the flow physics through the interaction between the different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Section 2 defines the problem and the governing equations for the double concentric jets, as well as the linear sta- bility equations and the methodology for mode selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The axisymmetric steaty-state is characterised in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In Section 4, we perform a parametric exploration in terms of the velocity ratio between the jets and the jet distance in order to determine the neutral curves of global stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The results about the mode selection are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, Section 6 summarises the main conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 2 Problem formulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 Computational domain and general equations The computational domain, represented in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 2, models a coaxial flow configu- ration, which is composed of two inlet regions, an inner and outer pipe, both of diameter D and length 5D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' zmin = −5D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The computational domain has an extension of zmax = 50D and rmax = 25D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The distance between the pipes is equal to L, measured from the inner face of the outer tube to the face of the inner jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The governing equations of the flow within the domain are the incompressible Navier–Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These are written in cylindrical coordinates (r, θ, z), which are made dimensionless by considering D as the reference length scale and Wo,max as the reference velocity scale, which is the maximum velocity in the outer pipe at z = zmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' ∂U ∂t + U · ∇U = −∇P + ∇ · τ(U), ∇ · U = 0, (1a) with τ(U) = 1 Re (∇U + ∇UT ), Re = Wo,maxD ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (1b) The dimensionless velocity vector U = (U, V, W) is composed of the radial, azimuthal and axial components, P is the dimensionless-reduced pressure, the dynamic viscosity ν and the viscous stress tensor τ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The incompressible Navier–Stokes equations eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (1) are complemented with the following boundary conditions U = (0, 0, Wi) on Γin,i and U = (0, 0, Wo) on Γin,o, (2) 5 Figure 2: Computational domain of the configuration of two concentric jets, used in StabFem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' where Wi = δu tanh � bi(1 − 2r) � and Wo = tanh � bo � 1 + ���� r − (Router,1 + Router,2) D ���� �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The parameter δu corresponds to the velocity ratio between the two jets, defined as δu = Wi,max/Wo,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The parameters bo and bi represent the boundary layer thickness within the nozzle, which are fixed equal to 5 (as in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' There is a weak influence of the boundary layer thickness on the stability properties of the jet, and it is related to the vortex shedding regime developed upstream the separation wall (more details may be found in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, no-slip boundary condition is set on Γwall and stress-free ( � 1 Reτ(U) − P � n = 0) boundary condition is set on Γtop and Γout, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In the sequel, Navier–Stokes equations eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (1) and the associated boundary conditions will be written symbolically under the form B∂Q ∂t = F(Q, ) ≡ LQ + N(Q, Q) + G(Q, η), (3) with the flow state vector Q = [U, P]T , η = [Re, δu]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Such a form of the governing equations takes into account a linear dependency on the state variable Q through L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' And a quadratic dependency on the parameters and the state variable through operators G(·, ·) and N(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 Asymptotic stability 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 Linear stability analysis In this study, the authors attempt to characterize the stable asymptotic state from the spectral properties of the Navier–Stokes equations eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' First, let us 6 Tmar Router,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' D out Router,1 r=0 Z=Zmin z=0 z=Zmarconsider the stability of an axisymmetric steady-state solution named Q0, which will be also referred to as trivial steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For that purpose, let evaluate a solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (1) in the neighbourhood of the trivial steady state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=', a perturbed state as follows, Q(x, t) = Q0(x, t) + εˆq(r, z)e−i(ωt−mθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (4) The next step consists in the characterization of the dynamics of small-amplitude perturbations around this base flow by expanding them over the basis of linear eigenmodes (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' If there is a pair [iωℓ, ˆqℓ] with Im(ωℓ) > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' the spectrum is contained in the half of the complex plane with negative real part) there ex- ists a basin of attraction in the phase space where the trivial steady-state Q0 is unstable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' stable) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The eigenpair [iωℓ, ˆqℓ] is determined as a solution of the following eigenvalue problem, J(ωℓ,mℓ)ˆq(zℓ) = � iωℓB − ∂F ∂q |q=Q0,η=0 � ˆq(zℓ), (5) where � ∂F ∂q |q=Q0,η=0 � ˆq(zℓ) = Lmℓ ˆq(zℓ) + Nmℓ(Q0, ˆq(zℓ)) + Nmℓ(ˆq(zℓ), Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The subscript mℓ indicates the azimuthal wavenumber used for the evaluation of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In the following, we account for eigenmodes ˆq(zℓ)(r, z) that have been normalised in such a way ⟨ˆu(zℓ), ˆu(zℓ)⟩L2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 Methodology for the study of mode selection In the following, we briefly outline the main aspects of the methodology em- ployed in the study of mode interaction, a comprehensive explanation is left to appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The determination of the attractor or coherent structure is explored within the framework of equivariant bifurcation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The trivial steady-state is axisymmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' the symmetry group is the orthogonal group O(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Near the onset of the instability, dynamics can be reduced to those of the centre manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Particularly, due to the non-uniqueness of the manifold one can al- ways look for its simplest polynomial expression, which is known as the normal form of the bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The reduction to the normal form is carried out via a multiple scales expansion of the solution Q of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The expansion considers a two scale development of the original time t �→ t + ε2τ, here ε is the order of magnitude of the flow disturbances, assumed to be small ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In this study we carry out a normal form reduction via a weakly non-linear expansion, where the small parameters are ε2 δu = δu,c − δu ∼ ε2 and ε2 ν = � νc − ν � = � Re−1 c − Re−1� ∼ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A fast timescale t of the self-sustained instability and a slow timescale of the evolution of the amplitudes zi(τ) are also considered in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (10), for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The ansatz of the expansion is as follows Q(t, τ) = Q0 + εq(ε)(t, τ) + ε2q(ε2)(t, τ) + O(ε3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (6) 7 Herein, we evaluate the mode interaction between two steady symmetry break- ing states with azimuthal wave number m1 = 1 and m2 = 2, that is, q(ε)(t, τ) = � z1(τ)ˆq(z1)(r, z)e−im1θ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' � + � z2(τ)ˆq(z2)(r, z)e−im2θ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (7) Note that the expansion of the LHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (3) up to third order is as follows εB∂q(ε) ∂t + ε2B∂q(ε2) ∂t + ε3� B∂q(ε3) ∂t � + O(ε4), (8) and the RHS respectively, F(q, η) = F(0) + εF(ε) + ε2F(ε2) + ε3F(ε3) + O(ε4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (9) Then, the problem up to third order in z1 and z2 can be reduced to [1] ˙z1 = λ1z1 + e3z1z2 + z1 � c(1,1)|z1|2 + c(1,2)|z2|2� , ˙z2 = λ2z2 + e4z2 1 + z2 � c(2,1)|z1|2 + c(2,2)|z2|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (10) An exhaustive analysis of the nonlinear implications of this normal form on dynamics is left to section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The procedure followed for the determination of the coefficients c(i,j) for i, j = 1, 2 and e3 and e4 is left to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='3 Numerical methodology for stability tools Results presented herein follow the same numerical approach adopted by [9, 28, 27, 30], where a comparison with DNS can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The calculation of the steady-state, the eigenvalue problem and the normal form expansion are implemented in the open-source software FreeFem++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Parametric studies and generation of figures are collected by StabFem drivers, an open-source project available in https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='com/stabfem/StabFem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For steady-state, stabil- ity and normal form computations, we set the stress-free boundary condition at the outlet, which is the natural boundary condition in the variational formula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The resolution of the steady nonlinear Navier-Stokes equations is tackled by means of the Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' While, the generalised eigenvalue problem (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (24)) is solved following the Arnoldi method with spectral transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The normal form reduction procedure of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 only requires to solve a set of linear systems, which is also carried out within StabFem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' On a standard laptop, every computation considered below can be attained within a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 3 Characterisation of the axisymmetric steady- state 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 Velocity ratio effects We begin by characterizing the development of the axisymmetric steady-state with varying δu at a constant Reynolds number fixed to Re = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Figure 3 8 0 0 1 r 5 2 4 3 W0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 0 0 1 r 5 2 4 3 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 Lr min(W0) δu δu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 2 a b c c b a d e f a b d f e c δ1 u δ2 u 0 2 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0 z 0 0 1 z r 2 4 5 2 4 3 Figure 3: Evolution of the recirculation length (Lr) of the recirculating bubble with respect to the velocity ratio δu between the inner and outer jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The diagram of the second row on the left displays the minimum value within the domain of the axial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It is spatially localised within the recirculating region for δu < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 and near the middle wall for larger values of the velocity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Meridional projections of the axisymmetric streamfunction isolines and the axial velocity contour in a range of (z, r) ∈ [−1, 5] × [0, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' synthesises the main topological changes experiences by the steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' At δu = 0, the solution (point (a) in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 3) represents an annular jet, which diffuses as it travels downstream and enters the ambient fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This figure illustrates that the solution curve can be divided into three segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The first segment comprised between 0 ≤ δu < δ1 u is characterised by an inner jet nearly trapped by a large recirculation region with a characteristic length Lr, which remains almost constant with the velocity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In the second region, which ranges between δ1 u < δu < δ2 u and it is represented as a shaded area in the figure, the recirculating region rapidly reduces its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In this region, the axial velocity of the inner jet is comparable with the axial velocity observed in the recirculating region, which promotes mixing between both regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' As the velocity ratio is increased, the inner jet is sufficiently 9 energetic to break the recirculating region, which occurs between point (c) and (d) in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The final segment, that ranges between δu > δ2 u, is characterised by two quasi-planar jets that rapidly mix to form a larger one at around z ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4 Linear stability analysis We explore the parameter space (Re, δu, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Herein, we examine the velocity ratio between the jets (0 < δu < 2) and the distance between the jets (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 < L < 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Within this range of parameters, we have analysed the linear stability properties of the flow configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For this purpose, we first investigate the influence of the jet distance on the stability for the case of the annular jet (δu = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These findings are summarized in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4 which displays the evolution of the critical Reynolds number with respect to the distance (L) for the four most un- stable modes: two steady modes with azimuthal wavenumber m = 1 and m = 2, hereinafter referred to as modes S1 and S2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A cross-section view at z = 1 is displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4 (a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The other two unsteady modes, named F1 and F2 have respectively azimuthal wavenumbers m = 1 and m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A cross-section view of these two modes is displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4 (c-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Please note that for the chosen set of parameters the axisymmetric unsteady mode F0, is always found at larger Reynolds numbers than the aforementioned modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This is one of the major differences with the case studied by [5], for small values of the jet distance L, the dominant instability is an unsteady axisymmetric one, which would be named F0 with our nomenclature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Thus, in the following, we only include the results for the S1, S2, F1 and F2 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The primary instability of the annular jet is then a steady symmetry-breaking bifurcation that leads to a jet flow with a single symmetry plane, displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' On the contrary, bifurcations that lead to the mode S2 possess two orthogonal symmetry planes, see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' As indicated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4 (g-h), these two stationary modes S1 and S2 are localised within the recirculation bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For jet distances L < 2, the second mode that bifurcates is F1 mode, depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This situation corresponds to a bifurcation scenario similar to other axisymmetric flow configurations, such as the flow past a sphere or a disk [2, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For larger distances between jets, the scenario changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The second bifurcation from the axisymmetric steady-state is the F2, displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4 (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Other configurations where the primary or sec- ondary instability involves modes with azimuthal component m = 2 are swirling jets [15] and the wake flow past a rotating sphere [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The unsteady modes F1 and F2 possess a much larger spatial support than S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' They are formed by an array of counter-rotating vortex spirals developing in the wake of the separating duct wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For the mode F2 the amplitude of these structures grows downstream of the nozzle, in the axial direction, with a maximum around z ≈ 10, after which they slowly decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The mode F1 grows further downstream, with a maximum around z ≈ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The spatial structure of these eigenmodes resembles the axisymmetric mode of Figure 9 in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Thus, the steady modes and unsteady modes differ in their spatial support, that is, even though both 10 1 0 1 x 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 (a) 1 0 1 x 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 (b) 4 2 0 2 4 x 4 2 0 2 4 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 (c) 4 2 0 2 4 x 4 2 0 2 4 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 (d) 1 2 3 4 L 0 100 200 300 400 500 Re (e) 1 2 3 4 L 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (f) 1 0 1 2 3 z r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 (g) 1 0 1 2 3 z r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 (h) 0 5 10 15 20 25 30 35 40 45 50 z 0 5 10 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 (i) 0 5 10 15 20 25 30 35 40 45 50 z 0 5 10 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 (j) Figure 4: Cross-section view at z = 1 of the four unstable modes at criticality for the annular jet case (δu = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The streawise component of the vorticity vector ϖz is visualised by colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (a) Mode S1 for L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5, (b) Mode S2 for L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5, (c) Mode F1 for L = 3 and (d) Mode F2 for L = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (e) Linear stability boundaries for the annular jet (δu = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (f) Frequency evolution of the unsteady modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Legend: S1 mode is displayed with a solid black line, S2 with a solid red line and F1 and F2 modes are depicted with dashed black and red lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Streamwise velocity of the neutral modes for L = 3 and δu = 0 (i) F1, (h) F2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 2 /u 0 200 400 600 800 Re S1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' S2 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 2 /u 0 100 200 300 400 500 600 Re S1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' S2 (b) 0 2 4 6 z 0 1 2 3 4 5 6 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 (c) 0 2 4 6 z 0 1 2 3 4 5 6 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 (d) Figure 5: Linear stability boundaries for the concentric jets (a) L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 and (b) L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Same legend as fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Visualizations of real part of the streamwise axial velocity of the critical modes (c) S1 and (d) S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' steady and unsteady modes are localised in space, the support of the steady ones is confined within the recirculation bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Instead, the unsteady modes are convected far downstream until they reach a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This latter char- acteristic is classical of modes with a large transient growth, as it was noticed by [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' On the other hand, the nature of the steady modes is similar to the symmetry-breaking instabilities behind the disk or the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These modes are far less sensitive to transient growth and are observable with direct numerical simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 Fixed distance between jets and variable velocity ratio δu In the following, we focus on the influence of the velocity ratio δu between jets for fixed jet distances L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Figure 5 displays the neutral curve of stability for jet distances (a) L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 and (b) L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' One may observe that the primary bifurcation is not always associated to the mode S1 as it is the case for δu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For sufficiently large velocity ratios, the primary instability leads to a non- axisymmetric steady-state with a double helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Another interesting feature, which could motivate a control strategy, is the occurrence of vertical asymptotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This sudden change in the critical Reynolds number is due to the retraction and disappearance of the recirculating region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5, this sudden change occurs for δu ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='25, and for higher values of δu the critical Reynolds number is around twice larger than the one of the annular jet (δu = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The case of jet distance L = 1 was discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The sudden change in the stability of the branch S1 occurs between δu ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Within this narrow interval, the primary branch of instability is the F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' At around δu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4, the primary bifurcation is again the branch F1, which becomes secondary at around δu ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='8 in favour of the branch S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 5 we have highlighted the codimension two point interaction between the S1 −S2 modes, whose modes are depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 5 (c-d), which will be analysed in detail in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Around this point, we can observe the largest stabilisation ratio between the annular jet (δu = 0) and a configuration of concentric jets (δu ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 Fixed velocity ratio δu and variable distance between jets Figure 6 compares the results obtained for a constant velocity ratio when vary- ing the distance between jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' As observed before, the solution becomes more unstable by increasing the distance between jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The largest critical Reynolds number is found at δu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The critical Reynolds decreases with the dis- tance between jets L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The points of codimension two, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=', the points where mode switching occurs, are highlighted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' We can appreciate that the interaction between the branch S1 and S2 happens for every velocity ratio δu explored, and it is the mode interaction associated to the smallest distance be- tween jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Additionally, for a velocity ratio δu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 there exists two points where the branches of the linear modes S1 and F1 intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Another feature of the neutral curves is the existence of turning points, which are associated to the existence of saddle node bifurcations of the axisymmetric steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The saddle node bifurcations of the steady-state induces the existence of regions in the neutral curves with a tongue shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These saddle node bifurcations are also responsible for the formation of the vertical asymptotes observed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, it is of interest the transition of the modes S1 and S2, which induce the symmetry breaking of the axisymmetric steady state to slow low frequency spiralling structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These modes have been identified for δu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 for m = 1, δu = 1 for m = 2, and δu = 2 for both m = 1 and m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' As it will be clari- 13 a) b) 1 2 3 4 L 0 200 400 600 800 1000 Re LS1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='S2 LS1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='F1 LF1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='S1 1 2 3 4 L 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' c) d) 1 2 3 4 L 0 200 400 600 800 1000 Re LS1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='S2 2 3 4 40 60 80 100 120 1 2 3 4 L 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' e) f) 1 2 3 4 L 0 200 400 600 800 1000 Re LS1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='S2 1 2 3 4 L 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Figure 6: Neutral lines of the four modes found studying the configuration of two concentric jets fixing the velocity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (a-b) δu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5, (c-d) δu = 1, (e-f) δu = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Black lines: modes with m = 1, red lines: modes with m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Straight lines: steady modes, dashed lines: unsteady modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 14 fied in section 5, these oscillations are issued from the non-linear interaction of modes, emerging simultaneously for a specific Reynolds number, and changing their position as the most unstable global mode of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 5 Mode interaction between two steady states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Resonance 1 : 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 Normal form, basic solutions and their properties The linear diagrams of section 4 have shown the existence of the mode in- teraction between the modes S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It corresponds roughly to the mode interaction that occurs at the largest critical Reynolds number for any value of L herein explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In this section, we analyse the dynamics near the S1 : S2 organizing centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' We perform a normal form reduction, which allows us to predict non-axisymmetric steady, periodic, quasiperiodic and heteroclinic cy- cles between non-axisymmetric states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The mode interaction that is herein analysed corresponds to a steady-steady bifurcation with O(2) symmetry and with strong resonance 1 : 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Such a bifur- cation scenario has been extensively studied in the past by [8, 10, 23, 1] and the reflection symmetry breaking case (SO(2)) by [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In order to unravel the exis- tence and the stability of the nonlinear states near the codimension two point, let write the flow field as q = Q0 + Re � r1(τ)eiφ1(τ)e−iθˆqs,1 � + Re � r2(τ)eiφ2(τ)e−2iθˆqs,2 � (11) in polar coordinates for the complex amplitudes z1 = r1eiφ1 and z2 = r2eiφ2 where rj and φj for j = 1, 2 are the amplitude and phase of the symmetry- breaking modes m = 1 and m = 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The complex-amplitude normal form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (10) is expressed in this reduced polar notation as follows, ˙r1 = e3r1r2 cos(χ) + r1 � λ(s,1) + c(1,1)r2 1 + c(1,2)r2 2 � , (12a) ˙r2 = e4r2 1 cos(χ) + r2 � λ(s,2) + c(2,1)r2 1 + c(2,2)r2 2 � , (12b) ˙χ = − � 2e3r2 + e4 r2 1 r2 � sin(χ), (12c) where the phase χ = φ2 −2φ1 is coupled with the amplitudes r1 and r2 because of the existence of the 1 : 2 resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The individual phases evolve as ˙φ1 = e3r2 sin(χ), ˙φ2 = −e4 r2 1 r2 sin(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (13) Before proceeding to the analysis of the basic solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (12), we can simplify these equations by the rescaling � r1 |e3e4|1/2 , r2 e3 � → (r1, r2), 15 which yields the following equivalent system ˙r1 = r1r2 cos(χ) + r1 � λ(s,1) + c11r2 1 + c12r2 2 � , (14a) ˙r2 = sr2 1 cos(χ) + r2 � λ(s,2) + c21r2 1 + c22r2 2 � , (14b) ˙χ = − 1 r2 � 2r2 2 + sr2 1 � sin(χ), (14c) where the coefficients s = sign(e3e4), c11 = c(1,1) |e3e4|, c12 = c(1,2) e2 3 , c21 = c(2,1) |e3e4|, c22 = c(2,2) e2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, we consider a third normal form equivalent to the previous ones but which removes the singularity of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (12) and (14) when r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Standing waves (sin χ = 0) naturally encounter this type of artificial singularity, which manifests as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (14) as an instantaneous jump from one standing subspace to the other by a π-translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This is the case of the heteroclinic cycles, previously studied by [1, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The third normal form, which we shall refer to as reduced Cartesian normal form, takes advantage of the simple transformation x = r2 cos(χ), y = r2 sin(χ) [24]: ˙r1 = r1 � λ(s,1) + c11r2 1 + c12(x2 + y2) + x � , (15a) ˙x = sr2 1 + 2y2 + x � λ(s,2) + c21r2 1 + c22(x2 + y2) � , (15b) ˙y = −2xy + y � λ(s,2) + c21r2 1 + c22(x2 + y2) � , (15c) In this final representation standing wave solutions are contained within the invariant plane y = 0, and due to the invariance of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (15) under the reflection y �→ −y, one can restrict attention, without loss of generality, to solutions with y ≥ 0, cf [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The system eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (14) possess four types of fixed points, which are listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' First, the axisymmetric steady state (O) is represented by (r1, r2) = (0, 0), so it is the trivial steady-state of the normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The second steady-state is what it is denoted as pure mode (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In the original coordinates, it corresponds to the symmetry breaking structure associated to the mode S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This state bifurcates from the axisymmetric steady state (O) when λ(s,2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The third fixed point is the mixed mode state (MM), which is listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It corresponds to the reflection symmetry preserving state associated to the mode S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It may bifurcate directly from the trivial steady state O, when λ(s,1) = 0 or from P whenever σ+ = 0 or σ− = 0, where σ± is defined as σ± ≡ λ(s,1) − −λ(s,2)c12 c22 ± � −λ(s,2) c22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (16) 16 Name Definition Bifurcations Comments O r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='O = r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='O = 0 − Steady axisymmetric state P r2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='P = −λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2) c22 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='P = 0 λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2) = 0 Bifurcation from O r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='MM = − λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1)±r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='MM+c12r2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='MM c11 λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1) = 0 Bifurcation from O MM PMM(r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='MM cos(χMM)) = 0 σ± = 0 Bifurcation from P cos(χMM) = ±1 cos(χT W ) = (2c11+c12)λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2)−(2c21+c22)λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1) ΣT W (2λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1)+λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2)) TW r2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='T W = −(2λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1)+λ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2)) ΣT W cos(χT W ) = ±1 Bifurcation from MM r2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='T W = 2r2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='T W Table 1: Definition of the fixed points of the reduced polar normal form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' σ± is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (16), the polynomial PMM is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (17) and ΣT W ≡ 4c11 + 2(c12 + c21) + c22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Name Bifurcation condition Comments SW sr2 1 − 2c11r2 1r2,MM cos(χMM) − 2c22r3 2,MM cos(χMM)3 = 0 Bif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' from MM MTW DT W − TT W IT W = 0, IT W > 0 Bif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' from TW Table 2: Definition of the limit cycles of the reduced polar normal form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The representation in the reduced polar form is r1,MM = − λ(s,1) ± r2,MM + c12r2 2,MM c11 , cos(χMM) = ±1, and the condition PMM(r2,MM cos(χMM)) = 0, where PMM is defined as PMM(x) ≡ sµ1+(s+c21λ(s,1)−c(1,1)λ(s,2))x+(c21+sc12)x2+(c12c21−c11c22)x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (17) Finally, the fourth fixed point of the system are travelling waves (TW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It is surprising that the interaction between two steady-states causes a time-periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The travelling wave emerges from MM in parity-breaking pitchfork bifurcation that breaks the reflection symmetry when cos(χT W ) = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The TW drifts at a steady rotation rate ωT W along the group orbit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=', the phases ˙φ1 = r2,T W sin(χT W ) and ˙φ2 = −s r2 1,T W r2,T W sin(χT W ) are non-null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Mixed modes and travelling waves may further bifurcate into standing waves (SW) and modulated travelling waves (MTW), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These are generic features of the 1 : 2 resonance for small values of λ(s,1) and λ(s,2), when s = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In the original coordinates, SW are periodic solutions, whereas MTW are quasiperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Standing waves emerge via a Hopf bifurcation from MM when the conditions PSW � r2,MM cos(χMM) � > 0 for PSW(x) ≡ (2c22x3 − sr2 1)c11 − (2c12x + 1)(c21x + s)x, 17 Name Condition Comments Ht AGH λ(s,1) > 0, λ(s,2) > 0, c22 < 0 Existence σ+ > 0, σ− < 0 Asymptotic stability Table 3: Definition of the conditions for the existence of the Ht AGH (robust heteroclinic cycles connecting pure modes) of the reduced polar normal form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' O P MM Ht AGH SW TW MTW Condition Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 2 Condition Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 2 Cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 3 Figure 7: Schematic representation of the basic solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (12) and their bifurcation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' and the one listed in table 2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' MTW are created when a torus bifurcation happens on the travelling wave branch when the conditions listed in table 2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Another remarkable feature of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (12) is the existence of robust heteroclinic cycles that are asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' When s = −1, there are open sets of parameters (see table 3) where the reduced polar normal form exhibits struc- turally stable connections between π−translations on the circle of pure modes, cf [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These structures are robust and have been observed in a large variety of systems, [19, 18, 16, 22, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In addition to these robust heteroclinic cycles connecting pure modes, there exists more complex limit cycles connecting O, P, MM and SW, cf [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' These cycles are located for larger values of λ(s,1) and λ(s,2), with possibly chaotic dynamics (Shilnikov type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In this study, we have not identified any of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, a summary of the basic solutions and the bifurcation path is sketched in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 Results of the steady-steady 1 : 2 mode interaction Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 reported the location of mode interaction points for discrete values of the velocity ratio δu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The location of the mode interaction between S1 and S2 is depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It shows that the mode switching between the modes S1 18 a) b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 2 /u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='8 L 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 2 /u 100 200 300 400 500 600 700 800 Re Figure 8: Evolution of the codimension two interaction S1 − S2 in the space of parameters (Re, L, δu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Grey points denote the points that were computed and the red point denotes the transition from steady to unsteady with low frequency as reported in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' and S2 is indeed stationary only for δu < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 and L < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For larger values of the velocity ratio and the jet distance, the interaction is not purely stationary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' at least one of the linear modes oscillates with a slow frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It implies that the mode selection for large velocity ratios near the codimension two points is similar to the one reported by [15] for swirling jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' However, even when the two primary bifurcations are non-oscillating (S1 and S2), the 1 : 2 resonance of the azimuthal wavenumbers induces a slow frequency, what we denote as travelling wave solutions (TW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' We consider the bifurcation sequence for δu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='0 and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='15, which is qualitatively similar to transitions in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 < δu < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5, near the codimension two points, which are depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' At the codimension two points for δu < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5, at least one of the two bifurcations is sub-critical and a normal form reduction up to fifth order is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Subcritical transition was also noticed for a distance between jets L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 by [5], who reported high levels of the linear gain associated to transient growth mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This last case is out of the scope of the present manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Figure 9 displays the phase portrait of the stable attractors near the S1 : S2 interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For values of δu > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='0, the axisymmetric steady-state loses its axisymmetry leading to a new steady-state with symmetry m = 2, herein denoted as pure mode (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A reconstruction of the fluctuating flow field of such a state is performed at the bottom right of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 9, which shows that the state P possesses two orthogonal planes of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Near the codimension two point, for values of the velocity ratio δu < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1, the state P is only observable, that is non-linearly stable, within a small interval with respect to the Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For larger values of the velocity ratio, the state P remains stable within the analysed interval of Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For values of the velocity ratio δu < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content="0, the bifurcation 19 w' = ± 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content="025 w' = ± 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content="01 w' = ± 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content="02 w' 170 180 190 200 210 Re 𝛿u Rotating Non-rotating Non-rotating TW MM P Ht AGH MM TW MTW P 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 z=1/2 z=1 z=1 z=1 z=2 t-T/3 t+T/3 t Figure 9: Phase portrait at the codimension two point S1 : S2 for parameter values (L, δu) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='15, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Visualisations of blue and red surfaces in the iso- metric views represent the respective positive and negative isocontour values of the perturbative axial velocity indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 180 185 190 195 200 Re 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='12 r2 MM TW PM BifTW!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='MTW BifMM!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='TW 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 y x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 r1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 170 175 180 185 190 195 200 Re Figure 10: Bifurcation diagram with respect to the Reynolds number for L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='15 and δu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The left diagram reports the evolution of r2 for the fixed point solutions of the normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The right diagram displays the bifurcation diagram in the Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Solid lines and dashed lines denote stable attractors and unstable attractors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 20 diagram is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Figure 10 displays the bifurcation diagram of the fixed-point solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (15) on the left diagram and the full set of solutions of the normal form in the right diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The axisymmetric steady-state first bifurcates towards a Mixed-Mode solution, which is the solution in the y = 0 plane for the right diagram of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A solution with a non-symmetric wake has been reconstructed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The Mixed-Mode solution is only stable within a small interval of the Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A secondary bifurcation, denoted BifMM−T W , gives raise to a slowly rotating wave of the wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The TW and the MM solutions are identical at the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The phase speed is zero at the bifurcation, thus this is not a Hopf bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It corresponds to a drift instability that breaks the azimuthal symmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' it starts to slowly drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This unusual feature, that travelling waves bifurcate from a steady solution at a steady bifurcation, is a generic feature of the 1 : 2 resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A reconstruction of the travelling wave solution is depicted on the top of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It corresponds to the line with non-zero y component in the right diagram of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The TW solution loses its stability in a tertiary bifurcation, denoted as BifT W −MT W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It conforms to a Hopf bifurcation of the TW solution, which gives birth to a quasi- periodic solution name Modulated Travelling Wave (MTW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A representation of this kind of solution in the Cartesian coordinates (r1, x, y) is depicted on the right image of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Eventually, the Modulated Travelling Wave experiences a global bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' That occurs when the periodic MTW solution, in the (r1, x, y) coordinates, nearly intersects the invariant r1 = 0 and y = 0 planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The transition sequence is represented in the right image of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 10 in the Cartesian coordinates (r1, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The amplitude of the MTW limit cycle increases until the MTW arising at the tertiary bifurcation BifT W −MT W are destroyed by meeting a heteroclinic cycle at BifMT W −Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The locus of BifMT W −Ht is reported in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 9 and in good agreement with [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The conditions for the existence of the heteroclinic cycles are listed in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' When σ− becomes negative, the cycle is attracting and robust heteroclinic cycles are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It is destroyed when σ+ becomes negative, in that case the pure modes are no longer saddles which breaks the heteroclinic connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Figure 11 displays the instantaneous fluctuation field from a heteroclinic orbit connecting P and its conjugate solution P’, which is obtained by a rotation of π/2, for parameter values Re = 200 and δu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The dynamics of the cycle takes place in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Figure 11 depicts the motion of the coherent structure associated to the heteroclitic cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Starting from the conjugated pure mode P’, the cycle leaves the point (a), located in the vicinity of P’, along the unstable eigenvector y, which is the stable direction of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The first phase consists in a rapid rotation by π/2 of the wake, it corresponds to the sequence a-b-c-d-e displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Then it is followed by a slow approach following the direction y and departure from the pure mode state P along the direction r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The second phase consists in a rapid horizontal motion of the wake, which is an evolution from P to P’ that takes place by the breaking of the reflectional symmetry with respect to the vertical axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' it constitutes the sequence e-f-g-h-i-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Please note that equivalent motions are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The first phase of rapid counter-clockwise rotation by π/2 can be performed in the 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 r1 y x c e d f h h i g f b a c d e 2000 1500 500 1000 r1 x≡r2cos(χ) y≡r2sin(χ) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content="2 t sin(χ) = 0 P' P b a g i w' w' = ± 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content="05 w' = ± 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='05 r1=0 Figure 11: Heteroclinic cycle solution for parameter values Re = 200, δu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The top and bottom image sequences along the heteroclinic cycle show (from left to right) an axial slice plane at z = 1 of the instantaneous fluctuations of the axial velocity of the flow field as viewed from downstream, along with a three-dimensional isometric view (d on the top and g on the bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='The middle diagram displays the heteroclinic cycle in the coordinates (r1, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 22 opposite sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It corresponds to a motion in the Cartesian coordinates along the plane r1 along negative values of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The sequence e-f-g-h-i-a can be replaced by a horizontal movement in the opposite sense, which adjusts to connect the plane y = 0 corresponding to negative values of r1, 6 Discussion & Conclusions This article achieves a complete description of the configuration consisting of two coaxial jets, broadly found in industrial processes, covering a wide range of applications such as noise reduction, mixing enhancement, or combustion control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The numerical procedure herein employed has been validated with the existing literature in the case of the stability analysis (see B for a detailed overview), and compared to DNS results, as done in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The analysis comprises a layout with a wide range of the velocity ratio (δu = Ui/Uo) between the jets, from δu = 0 to δu = 2, as well as the distance between jets (L) enclosing values from L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 to L = 4, substantially expanding the work of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A linear stability analysis reveals the most significant modes, consisting of two steady modes (S1 and S2, located within the recirculation bubble) and two unsteady ones (F1 and F2, evolving as a transient growth in the downstream direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The critical Reynolds number is determined for a wide range of velocity and distance ratios, starting with the influence of the velocity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' As the relation between inner and outer velocities grows, the flow is stabilised, increasing the critical Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The primary instability swaps from mode S1, characterised with one symmetry plane, to mode S2 that possesses two symmetry planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' An abrupt divergence in the critical Reynolds number is captured, associated with the vanishing of the recirculation region, that could suggest a stability control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Subsequently, the effect of the distance L between jets is analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The primary effect of increasing this distance is a decrease in the critical Reynolds number for all values of δu investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Additionally, the existence of saddle node bifurcations, that swap the most unstable mode of the flow, generates turning points in the neutral curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The investigated bifurcation scenario starts from the codimension point, with an axisymmetric steady state located at a velocity ratio δu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='0 and distance between jets of L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It is qualitatively equivalent to transitions found in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 < δu < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It reveals a break of the axisymmetry for values higher than δu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='0, presenting a steady state as a pure mode P with two orthogonal planes of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' For values lower than δu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='0, the bifurcation diagram exhibits a slightly complicated path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Firstly, it drives into a Mixed-Mode (MM) solution presenting a non-symmetric wake, that is only stable for a small range of the Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Subsequently, a slowly rotating wake is triggered in the form of a Travelling Wave (TW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This unusual feature, an unsteady state emerging from a steady state, corresponds to a drift instability commonly found at 1 : 2 resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Then, the TW solution encounters a Hopf bifurcation, developing a quasi-periodic solution in the form of a Modulated Travelling Wave (MTW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, the MTW solution undergoes a global bifurcation meeting a 23 heteroclinic cycle (Ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This heteroclinic orbit links the solution P with its conjugate solution P’, spinning the wake from P’ to P, and moving it horizontally from P to P’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 24 Acknowledgments A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' acknowledge the grant PID2020-114173RB-I00 funded by MCIN/AEI/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' acknowledge the support of Comunidad de Madrid through the call Research Grants for Young Investigators from Universidad Polit´ecnica de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' AC also acknowl- edges the support of Universidad Polit´ecnica de Madrid, under the programme ‘Programa Propio’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A Normal form reduction In this section we provide a detail explanation of the normal form reduction to obtain the coefficients of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (10), we define the terms of the compact notation of the governing equations eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (3), which is reminded here, for the sake of conciseness, B∂Q ∂t = F(Q, η) ≡ LQ + N(Q, Q) + G(Q, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (18) The nonlinear convective operator N(Q1, Q2) = U1 · ∇U2 accounts for the quadratic interaction on the state variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The linear operator on the state variable is LQ = [∇P, ∇·U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The remaining term accounts for the linear vari- ations in the state variable and the parameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It is defined as G(Q, η) = G(Q, [η1, 0]T ) + G(Q, [0, η2]T ) where G(Q, [η1, 0]T ) = η1∇ · (∇U + ∇UT ) and G(Q, [0, η2]T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The former operator shows the dependency on the parameter η1, which accounts for the viscous effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The latter operator depends on the parameter η2, which accounts for the velocity ratio between jets and it is used to impose the boundary condition U = (0, η2 tanh � bi(1 − 2r) � , 0) on Γin,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In addition, we consider the following splitting of the parameters η = ηc + ∆η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Here ηc denotes the critical parameters ηc ≡ [Re−1 c , δu,c]T attained when the spectra of the Jacobian operator posses at least an eigenvalue whose real part is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The distance in the parameter space to the threshold is represented by ∆η = [Re−1 c − Re−1, δu,c − δu]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 Multiple scales ansatz The multiple scales expansion of the solution q of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (3) is q(t, τ) = Q0 + εq(ε)(t, τ) + ε2q(ε2)(t, τ) + O(ε3), (19) where ε ≪ 1 is a small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The distance in the parameter space to the critical point ∆η = [Re−1 c − Re−1, δu,c − δu]T is assumed to be of second order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' ∆ηi = O(ε2) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The expansion eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (19) considers a two scale expansion of the original time t �→ t + ε2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A fast timescale t and a slow timescale of the evolution of the amplitudes zi(τ) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (19), for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Note that the expansion of the LHS eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (3) up to third order is as follows εB∂q(ε) ∂t + ε2B∂q(ε2) ∂t + ε3� B∂q(ε3) ∂t + B∂q(ε) ∂τ � , (20) 25 and the RHS respectively, F(q, η) = F(0) + εF(ε) + ε2F(ε2) + ε3F(ε3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (21) The expansion eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (21) will be detailed at each order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 Order ε0 The zeroth order Q0 of the multiple scales expansion eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (19) is the steady state of the governing equations evaluated at the threshold of instability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' η = ηc, 0 = F(Q0, ηc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (22) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 Order ε1 The first order q(ε)(t, τ) of the multiple scales expansion of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (19) is composed of the eigenmodes of the linearized system q(ε)(t, τ) ≡ � z1(τ)e−im1θˆq1 + z2(τ)ei−m2θˆq2 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (23) in our case, m1 = 1 and m2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Each term ˆqℓ of the first order expansion eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (23) is a solution of the following linear equation J(ωℓ,mℓ)ˆqℓ = � iωℓB − ∂F ∂q |q=Q0,η=ηc � ˆqℓ, (24) where ∂F ∂q |q=Q0,η=ηc ˆqℓ = Lmℓ ˆqℓ + Nmℓ(Q0, ˆqℓ) + Nmℓ(ˆqℓ, Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The subscript mℓ indicates the azimuthal wavenumber used for the evaluation of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='3 Order ε2 The second order expansion term q(ε2)(t, τ) is determined from the resolution of a set of forced linear systems, where the forcing terms are evaluated from first and zeroth order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The expansion in terms of amplitudes zi(τ) (i = 1, 2) of q(ε2)(t, τ) is assessed from term-by-term identification of the forcing terms at the second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Non-linear second order terms in ε are F(ε2) ≡ 2 � j,k=1 � zjzkN(ˆqj, ˆqk)e−i(mj+mk)θ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' � + 2 � j,k=1 � zjzkN(ˆqj, ˆqk)e−i(mj−mk)θ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' � + 2 � ℓ=0 ηℓG(Q0, eℓ), (25) where the terms proportional to zjzk are named ˆF(zjzk) (ϵ2) and eℓ is an element of the orthonormal basis of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 26 Then, we look for a second order term expanded as follows q(ε2) ≡ 2 � j,k=1 k≤j � zjzkˆqzjzk + zjzkˆqzjzk + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='c � + 2 � ℓ=1 ηℓQ(ηℓ) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (26) Terms ˆqz2 j are azimuthal harmonics of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The terms ˆqzjzk with j ̸= k are coupling terms, and ˆq|zj|2 are harmonic base flow modification terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, Q(ηℓ) 0 are base flow corrections due to a variation of the parameter ηℓ from the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' At this order, there exists two resonant terms, the terms proportional to z1z2 and z2 1, which are associated with the singular Jacobian J(0,mk) for k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' To ensure the solvability of these terms, we must enforce compatibility conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' the Fredholm alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The resonant terms are then determined from the resolution of the following set of bordered systems �J(0,mk) ˆqk ˆq† k 0 � �ˆq(z(R)) e � = � ˆF(z(R)) (ε2) 0 � , z(R) ∈ [z1z2, z2 1]T , (27) where e = e3 for z(R) = z1z2 and e = e4 for z(R) = z2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The non-resonant terms are computed by solving the following non-degenerated forced linear systems J(0,mj+mk)ˆqzjzk = ˆF(zjzk) (ϵ2) , (28) and J(0,0)Q(ηℓ) 0 = G(Q0, eℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (29) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 Order ε3 At third order, there exists six degenerate terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In our case, we are not inter- ested in solving for terms of third-order, instead, we will determine the linear and cubic coefficients of the third order normal form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (10) from a set of compatibility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The linear terms λ(s,1) and λs,2 and cubic terms c(i,j) for i = 1, 2 are deter- mined as follows λ(s,1) = ⟨ˆq† 1, ˆF(z1) (ε3)⟩ ⟨ˆq† z, Bˆqz⟩ , λ(s,2) = ⟨ˆq† 2, ˆF(z2) (ε3)⟩ ⟨ˆq† 2, Bˆq2⟩ , c(i,j) = ⟨ˆq† 2, ˆF(zi|zj|2) (ε3) ⟩ ⟨ˆq† i, Bˆqi⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (30) The forcing terms for the linear coefficient are ˆF(zj) (ε3) ≡ 2 � ℓ=1 ηℓ �� N(ˆqj, Q(ηℓ)) 0 + N(Q(ηℓ) 0 , ˆqj) � + G(ˆqj, eℓ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (31) which allows the decomposition of λ(s,ℓ) = λ(s,ℓ),Re(Re−1 c Re−1)+λ(s,ℓ),δu(δu,c − δu) for ℓ = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 27 Canton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (2017) [5] Present work Rec 1420 1405 ω 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='73 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='72 Table 4: Comparison of Rec and ω between previous work and the present one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The forcing terms for the cubic coefficients are ˆF(zj|zk|2) (ε3) ≡ � N(ˆqj, ˆq|zk|2) + N(ˆq|zk|2, ˆqj) � + � N(ˆq−k, ˆqzjzk) + N(ˆqj,k, ˆq−k) � + � N(ˆqk, ˆqzjzk) + N(ˆqzjzk, ˆqk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' (32) if j ̸= k and ˆF(zj|zj|2) (ϵ3) ≡ � N(ˆqj, ˆq|z|2 j) + N(ˆq|z|2 j, ˆqj) � + � N(ˆq−j, ˆqz2 j ) + N(ˆqz2 j , ˆq−j) � , (33) for the diagonal forcing terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' B Validation of the code - Comparison with the literature The calculations made in StabFem in the sections at the main manuscript are validated comparing the leading global mode in the geometry proposed by [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Moreover, the critical Reynolds number and the frequency associated are also analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' In the cited work, the authors use an analogous geometry with the following parameters: Radious of the inner jet Rinner = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 Diameter of the outer jet D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 Distance between jets L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='1 Ratio between velocities δu = 1 The linear stability analysis has been carried out imposing m = 0, as done by [5], so the leading global mode will be axisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The critical Reynolds number Rec and the frequency ω of the leading global mode are compared in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' As seen, few differences can be found on the critical Reynolds number and the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The relative error in the Rec calculation is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='06% and the one of the frequency is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='17%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The global mode is now calculated using StabFem and compared with the one calculated by [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' This mode can be found in figures 9, 10 and 11 on the cited paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' As it can be seen, there are not substantial differences between the direct modes, being both of them a vortex street with their biggest amplitude 28 0 5 10 15 20 25 z 0 1 2 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 z 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 r 2 0 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='6 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='7 r 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content='5 1 Figure 12: Direct mode, adjoint mode and sensitivity of the leading global mode studied by [5] calculated using StabFem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' situated 10 units downstream the exit of the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' The adjoint mode is con- centrated within the nozzle, with its biggest amplitude situated on the sharp corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' There is not any difference between the adjoint mode calculated with StabFem and the one in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Finally, the structural sensitivity is similar to the one computed by [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' It is composed by two lobes in the space between the exit of the two jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' References [1] Dieter Armbruster, John Guckenheimer, and Philip Holmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Heteroclinic cycles and modulated travelling waves in systems with o (2) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Physica D: Nonlinear Phenomena, 29(3):257–282, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfSAnZ/content/2301.04429v1.pdf'} +page_content=' Auguste, D.' metadata={'source': 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McCarthy3 +Helen Frazer4 +Gustavo Carneiro5 +1 Australian Institute for Machine Learning, University of Adelaide +2 Harvard University 3 St Vincent’s Institute of Medical Research +4 St Vincent’s Hospital Melbourne 5 CVSSP, University of Surrey +Abstract +Prototypical part network (ProtoPNet) methods have +been designed to achieve interpretable classification by +associating predictions with a set of training prototypes, +which we refer to as trivial (i.e., easy-to-learn) prototypes +because they are trained to lie far from the classification +boundary in the feature space. +Note that it is possible +to make an analogy between ProtoPNet and support vec- +tor machine (SVM) given that the classification from both +methods relies on computing similarity with a set of train- +ing points (i.e., trivial prototypes in ProtoPNet, and support +vectors in SVM). However, while trivial prototypes are lo- +cated far from the classification boundary, support vectors +are located close to this boundary, and we argue that this +discrepancy with the well-established SVM theory can re- +sult in ProtoPNet models with suboptimal classification ac- +curacy. In this paper, we aim to improve the classification +accuracy of ProtoPNet with a new method to learn support +prototypes that lie near the classification boundary in the +feature space, as suggested by the SVM theory. In addition, +we target the improvement of classification interpretabil- +ity with a new model, named ST-ProtoPNet, which exploits +our support prototypes and the trivial prototypes to pro- +vide complementary interpretability information. Experi- +mental results on CUB-200-2011, Stanford Cars, and Stan- +ford Dogs datasets demonstrate that the proposed method +achieves state-of-the-art classification accuracy and pro- +duces more visually meaningful and diverse prototypes. +1. Introduction +Deep convolutional neural networks (CNN) [14, 26, 27] +have made remarkable achievements in various visual tasks, +e.g., image recognition [14] and object detection [34]. De- +spite the excellent feature extraction and discrimination +ability, CNNs are generally treated as black-box models +due to their complex architectures, high-dimensional fea- +Features +lct +lsp +lct +lsp +dct +dsp +Classification boundary +Classification boundary +(a) The learning of trivial prototypes +(b) +(b) The learning of support prototypes (ours) +(d) +Prototypes +Feature vectors +Clustering +Separation +Our loss +Classification boundary +Classification boundary +Figure 1. The difference between the learning of trivial and sup- +port prototypes. (a) Trivial prototypes: the separation loss pushes +the prototypes of different classes as far as possible from the clas- +sification boundary. (b) Support prototypes: our new closeness +loss enforces the prototypes of different classes to be as close as +possible to the classification boundary. +ture spaces, and an enormous number of learnable param- +eters. Such lack of interpretability hinders their successful +application in fields that require understandable and trans- +parent decisions [35], such as disease diagnosis [42], finan- +cial risk assessment [30], and autonomous driving [21]. +Recently, increasing attention has been dedicated to the +development of interpretable deep-learning models [1, 3, 4, +23]. +A particularly interesting strategy is the prototype- +based interpretable models, e.g., prototypical part network +(ProtoPNet) [4, 10]. These methods are inherently inter- +pretable since they can explain the model’s decisions by +showing image classification activation maps associated +with a set of class-specific image prototypes. These proto- +types are automatically learned from training samples, with +classification score being computed by comparing testing +image parts to the learned training prototypes. +ProtoPNet [4] is trained to learn a classifier from a set of +class-specific prototypes by minimising the cross-entropy +classification loss and two additional regularisation losses, +namely: 1) a clustering loss that pulls together training im- +1 +arXiv:2301.04011v1 [cs.CV] 8 Jan 2023 + +DIWGU2IOUX +X noiengmid +V +★ +B +B +V +口 +V +GSLUIua +1 +- +WGILC +■ +cages to prototypes of their own class; and 2) a separation +loss that pushes apart training images from all prototypes +of other classes. More specifically, the clustering loss min- +imises the distance of each image patch to at least one pro- +totype of its own class, while the separation loss maximises +the distance between all image patches and all other class +prototypes. The combination of these two losses pushes the +prototypes as far as possible from the classification bound- +ary, but still within the class distribution, so we call them +trivial (i.e., easy-to-learn) prototypes, as shown in Fig. 1(a). +We also display these trivial prototypes in Fig. 2(a), where +we present the ProtoPNet learning results for the two-moon +problem, depicting the training points (red and blue points) +and prototypes (green and black stars) in both the data +and feature spaces, learned with a feed-forward neural net- +work1. Notice that the trivial prototypes are located as far +as possible from the classification boundary in the feature +space. Similar to ProtoPNet, the support vector machine +(SVM) [6] classifier is trained by minimising a loss func- +tion that learns a set of support vectors. Different from the +ProtoPNet’s prototypes, these support vectors are located +as close as possible to the classification boundary, as shown +in Fig. 2(c). Given that the prototypes in ProtoPNet and +support vectors in SVM play similar roles in classification +problems, we argue that the ProtoPNet’s loss function may +lead to suboptimal classification results because of the triv- +ial prototypes being learned. +In this paper, we propose an alternative learning strat- +egy of ProtoPNet from the SVM perspective, to force the +learned prototypes to resemble support vectors of SVM and +be located as close as possible to the classification bound- +ary with the goal of increasing classification accuracy. The +strategy consists of a new closeness loss that minimises the +distance between prototypes of different classes. As shown +in Fig. 1(b), our new loss enforces the prototypes to move +closer to the classification boundary, as also demonstrated +by Fig. 2(b) that reveals the support prototypes produced by +the introduction of our new closeness loss are indeed more +similar to the support vectors. Furthermore, in order to im- +prove interpretability, we propose a new ProtoPNet classi- +fier that integrates the support and trivial prototypes (named +ST-ProtoPNet), where the goal is to produce two distinc- +tive and complementary sets of prototypes to obtain more +meaningful classification explanations. Due to the different +natures of the two sets of prototypes, they can also enable +further improvements in terms of classification accuracy. +The major contributions of this work are: +1. We provide the first study that makes an analogy be- +tween the prototype learning from ProtoPNet methods +and support vector learning from SVM, where we in- +1The network has an input layer of 2 nodes, a hidden layer of 256 nodes +(activated by tanh), and an output layer of 2 nodes (activated by sigmoid). +Figure 2. Two-moon classification results from ProtoPNet and +SVM classifiers. (a) Trivial prototypes (stars) and training sam- +ples (circles) in the feature (top) and data (bottom) spaces. (b) +Support prototypes (stars) and training samples (circles) in the fea- +ture (top) and data (bottom) spaces. Note that in (a) and (b), each +learned prototype is projected onto the nearest training sample in +the feature space. (c) Support vectors (stars) and training samples +(circles) from a Radial Basis Function (RBF) kernel based SVM. +vestigate if by following SVM’s support vector learn- +ing strategy and pulling the prototypes to be as close +as possible to the classification boundary, it is possible +to improve ProtoPNet’s classification accuracy. +2. We present a new ST-ProtoPNet method to exploit both +support and trivial prototypes for the interpretable im- +age classification, where the two sets of prototypes can +provide complementary information to improve both +interpretability and classification accuracy. +3. We conduct extensive experiments on three benchmark +datasets which show that our ST-ProtoPNet method +outperforms current state-of-the-art (SOTA) methods +in terms of classification accuracy. +In our experiments, we also demonstrate that the combi- +nation of the two types of prototypes contributes to richer +interpretability, where trivial prototypes tend to focus on +both local parts of the visual object of interest and the back- +ground, while support prototypes mainly focus on visually +similar object parts of different classes. +2 + +0.9 +0.8 +0.8 +0.7 +0.6 +0.6 +0.5 +0.4 +0.4 +0.3 +0.2 +0.2 +0.1 +★ +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.2 +1.2 +0.8 +0.8 +0.4 +0.4 +0.0 +0.0 +-0.4 +-0.4 +0.8 +-0.8 +-1.2 +0.0 +0.6 +1.2 +-0.6 +1.8 +2.4 +-1.2 +-0.6 +0.0 +0.6 +1.2 +1.8 +2.4 +(a) Trivial prototypes +(b) Support prototypes +1.2 +0.8 +0.4 +Class A: red, green +Class B: blue, black +0.0 +0.4 +-0.8 +-1.2 +-0.6 +0.0 +0.6 +1.2 +1.8 +2.4 +(c) Support vectors from SVM2. Related Work +In this section, we first review relevant studies on classi- +fication interpretability where we focus on prototype-based +methods, and then we briefly review support vector machine +(SVM) classification. Finally, we provide a short survey on +ensemble classification for interpretability. +2.1. Classification Interpretability +The interpretation of classification results produced by +deep neural networks can be achieved by a variety of +post-hoc explanation techniques, such as explanatory surro- +gates [31,39,50], counterfactual examples [13,17,41], and +saliency visualisation [38,40,49,53]. +In comparison with post-hoc explanations, prototype- +based interpretability is directly present in the model’s in- +ner computations. ProtoPNet [4] is the original work that +uses class-specific prototypes for interpretable image clas- +sification tasks. +Similar to ProtoPNet, TesNet [48] con- +structs class-specific transparent basis concepts on Grass- +mann manifold for the interpretable classification. +De- +rived from ProtoPNet, Deformable ProtoPNet [10] employs +spatially-flexible and deformable prototypes to adaptively +capture meaningful object features. In ProtoPShare [37], a +data-dependent merge-pruning method is presented to share +prototypes among classes, which can reduce the number +of prototypes used for classification. +Alternatively, Pro- +toPool [36] introduces a fully differentiable prototype as- +signment strategy to reduce the number of prototypes. In +Proto2Proto [18], a knowledge distillation method is de- +signed to transfer interpretability from a teacher ProtoPNet +to a shallow student ProtoPNet. ProtoTree [32] integrates +the prototype learning into a binary neural decision tree +that can explain its predictions by tracing a decision path +throughout the tree. ViT-NeT [22] further establishes the +prototype neural tree structure on visual transformers [11]. +Because of the ability to self-explain classification re- +sults, prototype-based interpretability (e.g., ProtoPNet) has +been widely utilised not only in the computer vision appli- +cations above, but also in medical imaging [2, 20, 46] and +face recognition [43]. However, an open question faced by +these methods is if the prototypes being learned are the op- +timal ones in terms of classification and interpretability. +2.2. SVM vs Prototype-based Classification +To better understand the optimality of prototypes, we +consider the support vector machine (SVM) [6] classifier +that finds support vectors to represent classes. More specif- +ically, SVM learns the maximum-margin classifier defined +by a classification boundary that maximises the distance to +the closest training samples, which are the support vectors +for the classes. The testing of SVM classifiers consists of +computing a weighted similarity between a testing sample +and the support vectors. It is interesting to note that the +testing of prototype-based classifiers is also based on mea- +suring the similarity between a testing image and a set of +class-specific prototypes learned from the training process. +Although the testing of SVM and prototype-based classi- +fiers are similar, their training procedures are quite differ- +ent. First, the training of a prototype-based classifier learns +a fixed number of prototypes [4, 10], while the SVM clas- +sifier learns to weight a variable number of support vectors +from the training set. Second, in prototype-based classi- +fiers [4,10], the learned prototypes tend to be as far as pos- +sible from the classification boundary, which is contrary to +the SVM training objective mentioned above. +The study of deep learning methods from an SVM the- +oretical perspective is a rich area of research [5,8,33] with +huge potential given the impact that both techniques have +had in the whole society for the last decades. However, +there are many practical questions that need to be addressed, +e.g., how to scale the kernel computation for large-scale +datasets, how to shorten the training process [33], and how +to integrate the learning of the SVM classifier with deep- +learning features. In this paper, we focus on adapting the +learning of ProtoPNet’s prototypes to make them similar to +SVM’s support vectors, by forcing prototypes to be as close +as possible to the classification boundary. +2.3. Ensemble Classification +Ensemble classification [9] is a traditional machine +learning approach that combines the results from multiple +classifiers, with the goals of improving learning generalisa- +tion and classification calibration. The use of interpretable +ensemble strategy to improve the classification accuracy has +been explored in [4,10,48], which is achieved by summing +the classification logits of multiple prototype-based clas- +sifiers (e.g., ProtoPNets trained with different CNN back- +bones). In this work, we obtain the interpretable ensemble +classification by combining the predictions of two ProtoP- +Nets with highly distinctive prototypes (i.e., support pro- +totypes and trivial ones), which is different from previous +studies [4,10,48] where the type of prototypes each individ- +ual classifier produces is very similar given that the same +objective function is employed for each classifier. More +specifically, the ensemble classification used in this paper +targets the utilisation of two complementary sets of pro- +totypes, particularly when the prototypes are learned from +quite different objective functions, such as the ones for +learning the support and trivial prototypes. +3. Preliminaries +We assume to have a training set D = {(xn, yn)}|D| +n=1, +where x ∈ X ⊂ RH×W ×R is an image with R colour +channels and y ∈ Y ⊂ {0, 1}C is a one-hot vector rep- +resentation of the image class label. +The interpretable +3 + +Input image +Ensemble +Trivial ProtoPNet +Add-on +layers +Add-on +layers +CNN +Backbone +fθ +Non-cancer +g +FC layers k +Support PPN +0.42 +0.56 +0.96 +0.96 + + +95.31 +122.58 Cancer +Similarity score +Logits +Support ProtoPNet +Prototypes +Feature vectors +FC layer +FC layer +Output +logits +Logits +Logits +fω (s) +fω (t) +fΦ (t) +fΦ (s) +P (s) +P (t) +Figure 3. The architecture of our proposed ST-ProtoPNet method +for the interpretable image classification. +ProtoPNet [4, 10] is trained to learn a set of prototypes +P = {pm}M +m=1, where pm ∈ Rρ1×ρ2×D, with each of +the C classes containing M/C prototypes. Without loss +of generality, we assume ρ1 = ρ2 = 1, but the extension +to general values is trivial. A typical ProtoPNet comprises +four components: a CNN backbone, add-on layers, a pro- +totype layer, and a fully connected (FC) layer. An input +image x is fed to the CNN backbone fθ : X → F (pa- +rameterised by θ ∈ Θ, where F ⊂ R ¯ +H× ¯ +W × ¯ +D) and then +passed on to the add-on layers, denoted by fω : F → +V (parameterised by ω ∈ Ω), to produce a feature map +V ∈ V ⊂ R ¯ +H× ¯ +W ×D. +The prototype layer computes +the similarity between the feature map V and the M D- +dimensional prototypes {pm}M +m=1 to generate M similarity +maps S(i,j) +m += sim(V(i, j, :), pm), where i ∈ {1, ..., ¯H}, +j ∈ {1, ..., ¯W}, and sim(·, ·) represents a similarity mea- +sure, e.g., cosine similarity [10] and projection metric [48]. +The prototype layer outputs M similarity scores from max- +pooling S = +� +max +i∈{1,..., ¯ +H},j∈{1,..., ¯ +W } S(i,j) +m +�M +m=1, which are +fed to the FC layer fφ : S → ∆, parameterised by φ ∈ Φ, +to produce the classification prediction ˆy ∈ ∆ ⊂ [0, 1]C, +where ∆ denotes the probability space for C classes. +4. ST-ProtoPNet +An overview of our proposed ST-ProtoPNet method +is illustrated in Fig. 3, which comprises a shared CNN +backbone fθ(·), two interpretable ProtoPNet classification +branches, namely: 1) the support ProtoPNet represented by +add-on layers fω(s)(·), prototype layer with support proto- +types P(s), and FC layer fφ(s)(·) which outputs the classi- +fication probability distribution ˆy(s) ∈ ∆; and 2) the trivial +ProtoPNet branch with its add-on layers fω(t)(·), trivial pro- +totypes P(t), and FC layer fφ(t)(·) that generates probabil- +ity predictions ˆy(t) ∈ ∆. The final classification is obtained +by combining the classification logits from both the support +and trivial ProtoPNets. In our implementation, we construct +the support and trivial ProtoPNet mainly based on the orig- +inal ProtoPNet [4] and TesNet [48], as explained below. +4.1. Support ProtoPNet +The support ProtoPNet is designed to produce support +(i.e., hard-to-learn) prototypes for classification that are as +close as possible to the classification boundary, as shown in +Fig. 1 and Fig. 2. The loss function to optimise the support +ProtoPNet branch is defined as: +θ∗,ω(s)∗, P(s)∗, φ(s)∗ = +arg +min +θ,ω(s),P(s),φ(s) +� +(x,y)∈D +ℓspt(x, y, θ, ω(s), P(s), φ(s)). +(1) +The loss for each training sample (x, y) ∈ D in Eq. (1) +above is represented by: +ℓspt(x, y, θ, ω(s), P(s), φ(s)) = ℓce(x, y, θ, ω(s), P(s), φ(s)) +− λ1ℓct(x, y, θ, ω(s), P(s)) ++ λ2ℓsp(x, y, θ, ω(s), P(s)) +− λ3ℓcls(P(s)) ++ λ4ℓort(P(s)), +(2) +where λ1, λ2, λ3, and λ4 are hyper-parameters to balance +each term, ℓce(·) denotes the cross-entropy classification +loss, ℓct(·) and ℓsp(·) represent the clustering and separa- +tion losses, respectively, which are introduced to regularise +the ProtoPNet’s training, as follows: +ℓct(x, y, θ, ω(s), P(s)) = max +p∈P(s) +y +max +v∈V(s) sim(v, p), +(3) +ℓsp(x, y, θ, ω(s), P(s)) = max +p/∈P(s) +y +max +v∈V(s) sim(v, p), +(4) +where V(s) = fω(s)(fθ(x)) is the feature map extracted +from the input image x, v represents one of the ¯H × ¯W +feature vectors in V(s) obtained by matrix vectorisation, p +is a normalised prototype (i.e., unit vector) in P(s), sim(·, ·) +is one of the similarity functions defined in Section 3, and +P(s) +y +denotes the set of prototypes of class y. The clustering +loss in Eq. (3) and separation loss in Eq. (4) aim to learn +a meaningful feature space in which the image features of +a certain class are clustered around the prototypes of the +class, and also well separated from those of other classes. +As mentioned in Section 1, the effect of the clustering +and separation losses above tend to push the prototypes +of different classes as far as possible from the classifica- +tion boundary, resulting in trivial prototypes, as displayed +in Fig. 1(a) and Fig. 2(a). In order to learn the proposed +support prototypes, we introduce the following novel close- +ness loss ℓcls to explicitly enforce the prototypes of different +classes to be close to each other, which is formulated as: +ℓcls(P(s)) = +C−1 +� +c1=1 +C +� +c2=c1+1 +min +pm∈Pc1,pn∈Pc2 +p⊤ +mpn. +(5) +4 + +DIWGU2IOUX +X noiengmid +V +★ +B +B +V +口 +V +GSLUIua +1 +- +WGILC +■ +cb(²)ogeml bolodelnunogininT-e +lsdel28loInnigino +Wgbbru +CGIGLUIGg CJ2IGL2During training, the closeness loss ℓcls maximises the +pair-wise prototype similarity, in the form of dot product +p⊤ +mpn between different classes in Eq. (5), with the goal of +pulling the prototypes close to the classification boundary. +As the prototypes move gradually towards the classification +boundary, they are able to capture harder and harder visual +features from training samples. On the other hand, since +the prototypes are located near the classification boundary, +this can put pressure on the support ProtoPNet’s learning +and enforce it to learn highly discriminative feature repre- +sentations to achieve accurate classification, which is bene- +ficial to extract more meaningful semantic information from +training samples. +Ideally, each prototype of a class should focus on unique +object parts of the training images (e.g., head, tail, and claw +of birds), so that the prototypes can represent rich and di- +verse visual patterns. However, there is no particular con- +straints to guarantee such prototype diversity and the issue +of prototype duplication [10] often occurs in the ProtoPNet +family of models. To encourage the intra-class prototype di- +versity, we employ an orthonormality loss [48] so that pro- +totypes within a class can represent dissimilar visual pat- +terns of training samples, which is defined as: +ℓort(P(s)) = +C +� +c=1 +∥Pc +⊤Pc − IM/C∥2 +F , +(6) +where ∥·∥2 +F represents Frobenius norm, Pc ∈ D ×R(M/C) +stands for a matrix composed of the prototypes of class +c (prototypes in each column of Pc are normalised), and +IM/C is an identity matrix of size M/C × M/C. +4.2. Trivial ProtoPNet +As described in Section 4.1, the support ProtoPNet is +developed to learn support (i.e., hard-to-learn) prototypes +by forcing them to be close to the classification boundary. +Considering that training samples contain not only hard vi- +sual features but also important trivial ones that the support +prototypes cannot fully capture, we propose to also learn +trivial prototypes to provide complementary classification +information, and exploit both the support and trivial proto- +types for improved interpretable classification. +The loss objective to train the trivial ProtoPNet branch is +defined as follows: +θ∗,ω(t)∗, P(t)∗, φ(t)∗ = +arg +min +θ,ω(t),P(t),φ(t) +� +(x,y)∈D +ℓtrv(x, y, θ, ω(t), P(t), φ(t)). +(7) +The loss for each training image (x, y) ∈ D in Eq. (7) above +is represented by: +ℓtrv(x, y, θ, ω(t), P(t), φ(t)) = ℓce(x, y, θ, ω(t), P(t), φ(t)) +− λ1ℓct(x, y, θ, ω(t), P(t)) ++ λ2ℓsp(x, y, θ, ω(t), P(t)) ++ λ3ℓdsc(P(t)) ++ λ4ℓort(P(t)), +(8) +where λ1, λ2, λ3, and λ4 are hyper-parameters, ℓce(·) is the +cross-entropy loss, the clustering loss ℓct, separation loss +ℓsp, and orthonormality loss ℓort are the same as in the sup- +port ProtoPNet defined in Eq. (3), (4) and (6), respectively. +The trivial ProtoPNet targets the learning of easy proto- +types that are far from the classification boundary and have +a good discrimination ability. To help achieve this, we in- +troduce a new discrimination loss ℓdsc to facilitate the inter- +class separability between prototypes of different classes. +This is formulated by minimising the pair-wise prototype +similarities of different classes, as follows: +ℓdsc(P(t)) = +C−1 +� +c1=1 +C +� +c2=c1+1 +max +pm∈Pc1,pn∈Pc2 +p⊤ +mpn. +(9) +4.3. Training and Testing +Training. Following the training strategies in the Pro- +toPNet family [4, 10], the training procedure of our ST- +ProtoPNet consists of 3 stages: 1) stochastic gradient de- +scent (SGD) optimisation of the CNN backbone, add-on +layers, and prototype layer, using a fixed FC layer initialised +with +1.0 and -0.5 for correct and incorrect connection +weights, respectively; 2) prototype projection by updating +each prototype with its nearest latent training image patch; +and 3) optimisation of the FC layer, with an additional L1 +regularisation on the incorrect connection weights (initially +fixed at -0.5). In each stage, we alternate the optimisation +of each branch of the ST-ProtoPNet between mini-batches. +Testing. +To exploit the complementary results from +both branches of the ST-ProtoPNet, the final classification +is obtained from the summed logits predicted by the two +branches. It is worth noticing that this ensemble strategy +introduces no loss of interpretablity but improved accuracy. +5. Experiments +We perform experiments on three fine-grained classifi- +cation benchmark datasets: CUB-200-2011 [45], Stanford +Cars [25], and Stanford Dogs [19]. To achieve fair com- +parison, we follow previous studies [4, 48] by applying of- +fline data augmentations (e.g., random rotation, skew, shear, +and left-right flip) on the cropped CUB and cropped Cars +datasets (using bounding boxes provided). We also validate +5 + +our method on the full CUB and full Dogs datasets, and +employ the same online data augmentation methods (e.g., +random affine transformation and left-right flip) as used +in Deformable ProtoPNet [10]. All images are resized to +224 × 224 pixels as network input. +5.1. Experimental Settings +The proposed ST-ProtoPNet method is evaluated on the +following CNN architectures: VGG-16, VGG-19, ResNet- +34, ResNet-50, ResNet-152, DenseNet-121, and DenseNet- +161. All CNN backbones are pre-trained on ImageNet [7], +except for ResNet-50, which is pre-trained on iNatural- +ist [44] for the experiment on full CUB [10]. The add-on +layers include two 1 × 1 convolutional layers. For simplic- +ity, we utilise the same prototype dimension D = 64 for all +CNN backbones on the three datasets. For cropped CUB +and Cars datasets, following [48], we use 10 prototypes (5 +support and 5 trivial) per class and the projection metric +in the similarity function sim(·, ·). In full CUB and Dogs +datasets, to ensure comparison fairness with Deformable +ProtoPNet [10] that uses 10 2 × 2 (full CUB) and 10 3 × 3 +(full Dogs) deformable prototypes per class, we utilise the +same total number of prototypes, i.e., 40 1 × 1 (20 support +and 20 trivial) for full CUB and 90 1×1 (45 support and 45 +trivial) for full Dogs. Also, we employ the cosine similar- +ity in sim(·, ·) and obtain 14 × 14 ( ¯H = ¯W = 14) feature +maps by upsampling the original 7 × 7 feature maps via a +bi-linear interpolation step, as in [10]. In our implemen- +tation, we empirically set λ1 = 0.8, λ2 = 0.48 and 0.08 +for the support and trivial ProtoPNet branches respectively, +λ3 = 1.0, and λ4 = 0.001. More details about the experi- +mental setup can be found in the supplementary material. +5.2. Performance Comparison +Table 1 presents the classification accuracy (averaged +across 5 runs) of our proposed ST-ProtoPNet on cropped +CUB and cropped Cars datasets, where the Baseline is rep- +resented by non-interpretable black-box CNN models. As +can be seen, our ST-ProtoPNet outperforms other compet- +ing methods across all backbone architectures for the task +of bird species classification. Also, our method achieves the +best results for the car model identification task when using +VGG and DenseNet architectures as the CNN backbone. +In particular, our VGG19-based ST-ProtoPNet reaches an +average accuracy of 83.2% and 91.7% on CUB and Cars, +respectively, surpassing other methods with the most im- +provements across all backbones. Moreover, the support +ProtoPNet generally performs better than methods utilising +only trivial prototypes (e.g., ProtoPNet, TesNet, and Triv- +ial ProtoPNet), showing the importance of learning support +prototypes for the interpretable image classification. It is +worth noting that our proposed ST-ProtoPNet produces su- +perior performance over the support ProtoPNet method, in- +dicating that both support and trivial prototypes are useful +and can provide complementary information for achieving +accurate and interpretable classification. +Table 2 shows the classification results of different meth- +ods on full CUB and full Dogs datasets. In both datasets, +the classification accuracy of the original ProtoPNet method +is generally worse than the non-interpretable counterpart +(Baseline) under many CNN backbones. On the other hand, +the accuracy by the trivial ProtoPNet and support ProtoP- +Net are substantially better than those by Baseline, ProtoP- +Net, and Deformable ProtoPNet. However, the proposed +ST-ProtoPNet achieves more significant performance gains +that result in the best classification accuracy across most +CNN backbones, particularly when using a large number of +prototypes (i.e., 40 1 × 1 prototypes per class for CUB and +90 1×1 prototypes per class for Dogs), which demonstrates +the effectiveness of utilising both the trivial and support +prototypes for the interpretable image classification. Ad- +ditionally, when using a smaller number of prototypes, i.e, +10 1×1 prototypes per class, our ST-ProtoPNet method still +exhibits competitive classification accuracy across multiple +CNN backbones. +We further compare our ST-ProtoPNet with other deep- +learning methods that can provide different levels of inter- +pretability on CUB dataset, with results shown in Table 3, +where * and ** denote ensembled models trained with dif- +ferent CNN backbones or hyper-parameters. As can be ob- +served, an ensemble of three ST-ProtoPNets can achieve +high classification accuracy (87.9% on cropped images, +88.2% on full images), outperforming competing methods +that are also based on an ensemble of three models (e.g., +ProtoTree, TesNet, and ProtoPool). Moreover, the ensemble +of five ST-ProtoPNets exceeds all other competing methods +and obtains the best classification accuracy of 88.1% and +88.4% on cropped and full CUB images, respectively. Ex- +perimental results on Stanford Cars and Stanford Dogs are +presented in the supplementary material. +5.3. Visualisation Analysis +To better highlight the differences between the support +and trivial prototypes, we select 8 categories of birds with +visually similar features from cropped CUB-200-2011 to +form a subset, and show the learned prototypes of our +VGG19-based ST-ProtoPNet in Fig. 4, where each proto- +type is visualised by projecting onto its nearest training im- +age patch in the latent feature space [4]. We can see in +Fig. 4(a) that the support prototypes can capture subtle and +fine visual features of different classes. Importantly, notice +that the support prototypes only focus on relevant bird parts, +such as the head and belly. This is reasonable since our +learning algorithm is designed to produce prototypes that +are as close as possible to the classification boundary, where +the image prototypical parts should be discriminative but at +6 + +Method +CUB +Cars +VGG16 +VGG19 +ResNet34 ResNet152 Dense121 Dense161 +VGG16 +VGG19 +ResNet34 ResNet152 Dense121 Dense161 +Baseline +73.3 ± 0.2 74.7 ± 0.4 82.2 ± 0.3 80.8 ± 0.4 81.8 ± 0.1 82.1 ± 0.2 87.3 ± 0.4 88.5 ± 0.3 92.6 ± 0.3 92.8 ± 0.4 92.0 ± 0.3 92.5 ± 0.3 +ProtoPNet [4] +77.2 ± 0.2 77.6 ± 0.2 78.6 ± 0.1 79.2 ± 0.3 79.0 ± 0.2 80.8 ± 0.3 88.3 ± 0.2 89.4 ± 0.2 88.8 ± 0.1 88.5 ± 0.3 87.7 ± 0.1 89.5 ± 0.2 +TesNet [48] +81.3 ± 0.2 81.4 ± 0.1 82.8 ± 0.1 82.7 ± 0.2 84.8 ± 0.2 84.6 ± 0.3 90.3 ± 0.2 90.6 ± 0.2 90.9 ± 0.2 92.0 ± 0.2 91.9 ± 0.3 92.6 ± 0.3 +Trivial ProtoPNet +80.8 ± 0.2 81.2 ± 0.2 82.5 ± 0.2 83.1 ± 0.3 83.9 ± 0.3 84.6 ± 0.3 90.1 ± 0.2 90.7 ± 0.2 91.1 ± 0.2 91.5 ± 0.2 91.4 ± 0.3 92.4 ± 0.3 +Support ProtoPNet +81.7 ± 0.2 81.8 ± 0.3 83.0 ± 0.1 83.6 ± 0.2 84.7 ± 0.2 85.2 ± 0.3 90.9 ± 0.2 90.8 ± 0.2 91.0 ± 0.2 91.8 ± 0.2 91.7 ± 0.2 92.7 ± 0.3 +ST-ProtoPNet (ours) 82.9 ± 0.2 83.2 ± 0.2 83.5 ± 0.1 84.1 ± 0.2 85.4 ± 0.2 86.1 ± 0.2 91.1 ± 0.2 91.7 ± 0.2 91.4 ± 0.1 92.0 ± 0.2 92.3 ± 0.3 92.7 ± 0.2 +Table 1. Classification accuracy (%) on cropped CUB-200-2011 and Stanford Cars by competing methods using different CNN backbones. +Method +Prototype +CUB +Prototype +Dogs +VGG16 VGG19 ResNet34 ResNet50 ResNet152 Dense121 Dense161 +VGG16 VGG19 ResNet34 ResNet50 ResNet152 Dense121 Dense161 +Baseline +– +70.9 +71.3 +76.0 +78.7 +79.2 +78.2 +80.0 +– +75.6 +77.3 +81.1 +83.1 +85.2 +81.9 +84.1 +ProtoPNet [4] +1×1p, 10pc +70.3 +72.6 +72.4 +81.1 +74.3 +74.0 +75.4 +1×1p, 10pc +70.7 +73.6 +73.4 +76.4 +76.2 +72.0 +77.3 +ProtoPNet [4] +1×1p, 40pc +72.9 +74.2 +74.1 +84.8 +76.0 +76.6 +78.5 +1×1p, 90pc +73.9 +75.3 +76.1 +78.1 +79.7 +75.4 +78.8 +TesNet [48] +1×1p, 10pc +75.8 +77.5 +76.2 +86.5 +79.0 +80.2 +79.6 +1×1p, 10pc +74.3 +77.1 +80.1 +82.4 +83.8 +80.3 +83.8 +TesNet [48] +1×1p, 40pc +77.6 +79.2 +76.5 +87.3 +80.1 +80.9 +81.3 +1×1p, 90pc +78.5 +79.6 +81.2 +83.3 +84.5 +82.1 +85.2 +Deformable ProtoPNet [10] 2×2p, 10pc +75.7 +76.0 +76.8 +86.4 +79.6 +79.0 +81.2 +3×3p, 10pc +75.8 +77.9 +80.6 +82.2 +86.5 +80.7 +83.7 +Trivial ProtoPNet +1×1p, 40pc +80.0 +79.5 +77.5 +87.2 +80.8 +81.1 +82.1 +1×1p, 90pc +78.6 +80.4 +82.6 +85.0 +87.0 +82.3 +85.9 +Support ProtoPNet +1×1p, 40pc +80.4 +80.0 +78.4 +87.5 +80.2 +81.5 +82.4 +1×1p, 90pc +79.0 +80.6 +83.0 +85.1 +87.3 +82.6 +86.2 +ST-ProtoPNet (ours) +1×1p, 10pc +76.8 +77.6 +77.4 +86.6 +78.7 +78.6 +80.6 +1×1p, 10pc +74.2 +77.2 +80.8 +84.0 +85.3 +79.4 +84.4 +ST-ProtoPNet (ours) +1×1p, 40pc +81.0 +80.2 +78.2 +88.0 +81.2 +81.8 +82.7 +1×1p, 90pc +79.1 +80.9 +83.4 +85.7 +87.2 +82.9 +86.6 +Table 2. Classification accuracy (%) on full CUB-200-2011 and Stanford Dogs datasets by competing approaches using different CNN +backbones, where kpc represents k prototypes per class and ρ1×ρ2p denotes the spatial shape of the prototypes. +(a) Support prototypes +(b) Trivial prototypes +Laysan Albatross +Sooty Albatross +Crested Auklet +Crested Auklet +Parakeet Auklet +Rhinoceros Auklet +Brewer Blackbird +Redwinged Blackbird +Figure 4. Visual comparison between the support (a) and trivial (b) prototypes from cropped CUB-200-2011, where each row exhibits +prototypes of the same class. In each pair, the first column shows the original image with a prototype indicated in a yellow bounding box, +the second column demonstrates the prototype’s corresponding activation map. +the same time visually similar among different classes. By +contrast, it is observed in Fig. 4(b) that the trivial prototypes +tend to focus not only on the relevant bird parts but also the +background regions. For example, some trivial prototypes +of the Laysan Albatross and Sooty Albatross classes cap- +ture the sea surface as they often appear with the sea back- +ground. We argue that this is because the trivial ProtoPNet +may treat the background as the most easy-to-learn pattern, +focusing less on the target’s visual parts of the class. +Fig. 5 displays an example of the interpretable reasoning +process of our ST-ProtoPNet method in classifying a testing +bird image. As can be seen, each ProtoPNet branch calcu- +lates its own classification logits (weighted sum of similar- +ity scores), which is then combined to generate the final pre- +dictions. To be specific, when classifying a Brewer Black- +bird, the support prototypes are quite active on the belly or +lower surface of the bird. Meanwhile, the trivial prototypes +mostly have high activations on the bird’s head and tail. In +this case, the support ProtoPNet branch obtains a relatively +higher similarity score (22.913), compared with the trivial +branch (20.248). Note that our ST-ProtoPNet exploits both +the support and trivial prototypes to capture much richer +representations of the target object from different perspec- +tives, which enables the production of the final decision in +a complementary way. Notably, the ensemble classification +improves interpretability and accuracy using richer repre- +sentations for the target object. More examples of visual +prototypes and interpretable reasoning on Cars and Dogs +are provided in the supplementary material. +7 + +Interpretability +Method +Accuracy (%) +None +B-CNN [29] +85.1 (b) 84.1 (f) +Object-level attention +CAM [53] +70.5 (b) 63.0 (f) +CSG [28] +82.6 (b) 78.5 (f) +Part-level attention +PA-CNN [24] +82.8 (b) +– +MG-CNN [47] +83.0 (b) 81.7 (f) +MA-CNN [51] +– +86.5 (f) +RA-CNN [12] +– +85.3 (f) +TASN [52] +– +87.0 (f) +Part-level attention + Prototypes +Region [15] +81.5 (b) 80.2 (f) +ProtoPNet* [4] +84.8 (b) 81.1 (f) +ProtoTree* [32] +– +86.6 (f) +TesNet* [48] +86.2 (b) 83.5 (f) +ProtoPool* [36] +87.5 (b) +– +ST-ProtoPNet* (ours) +87.9 (b) 88.2 (f) +ProtoTree** [32] +– +87.2 (f) +Deformable ProtoPNet** [10] +– +87.8 (f) +ProtoPool** [36] +87.6 (b) +– +ST-ProtoPNet** (ours) +88.1 (b) 88.4 (f) +Table 3. Classification accuracy and interpretability of different +methods on CUB-200-2011. +“b” and “f” denote the model is +trained and tested on cropped and full images, respectively. *: +Ensembled of three models. **: Ensembled of five models. +Testing image +Prototype +Training image where +the prototype derives +Activation map +Similarity score +Connection weight +Individual logits +Combined logits +5.035 +0.981 +× +4.939 += +4.891 +0.964 +× +4.715 += +4.321 +0.979 +× +4.230 += +4.206 +0.972 +× +4.088 += +Support ProtoPNet +Trivial ProtoPNet +... +... +... +... +43.161 +... +... +... +... +... +... +... +... +... +... +22.913 +20.248 +Figure 5. An example of the interpretable reasoning of our ST- +ProtoPNet for classifying a testing Brewer Blackbird image. +5.4. Ablation Study +The Closeness and Discrimination Losses. In order to +validate the effectiveness of our proposed closeness loss in +Eq. (5) and discrimination loss in Eq. (9), we first conduct +ablation studies on full CUB and full Dogs datasets using +ResNet34 as the CNN backbone, with results provided in +Table 4. We can notice that both the closeness and discrim- +ination losses can improve the classification accuracy, com- +pared with the Baseline ProtoPNet trained with only clus- +tering, separation, and orthonormality losses. Importantly, +the closeness loss introduces a larger performance improve- +ment since it aims at learning support (i.e., hard-to-learn) +prototypes that lie near the classification boundary. +Combining Support and Trivial Prototypes. We also +investigate the importance of integrating the two comple- +mentary sets of support and trivial prototypes for achieving +improved classification. To achieve this, we first train a two- +branch model where both branches learn the same type of +Method +ℓct +ℓsp +ℓort +ℓdsc +ℓcls +Accuracy (%) +CUB +Dogs +Baseline ProtoPNet +✓ +✓ +✓ +76.7 +80.9 +Trivial ProtoPNet +✓ +✓ +✓ +✓ +77.5 +82.6 +Support ProtoPNet +✓ +✓ +✓ +✓ +78.4 +83.0 +Table 4. Ablation analysis of the closeness loss ℓcls in Eq. (5) to +learn support prototypes, and discrimination loss ℓdsc in Eq. (9) to +learn trivial prototypes on full CUB-200-2011 and Stanford Dogs. +Method +VGG16 +VGG19 +ResNet34 ResNet152 Dense121 Dense161 +Trivial Ensemble +81.4 ± 0.3 81.8 ± 0.2 82.7 ± 0.2 83.2 ± 0.3 84.4 ± 0.2 85.0 ± 0.3 +Support Ensemble +82.1 ± 0.2 82.4 ± 0.3 83.0 ± 0.2 83.7 ± 0.3 84.8 ± 0.2 85.5 ± 0.2 +Trivial Branch +81.0 ± 0.2 81.1 ± 0.3 82.4 ± 0.2 82.9 ± 0.3 84.1 ± 0.3 84.8 ± 0.3 +Support Branch +81.5 ± 0.3 81.8 ± 0.3 82.8 ± 0.2 83.4 ± 0.3 84.6 ± 0.2 85.4 ± 0.2 +ST-ProtoPNet (ours) 82.9 ± 0.2 83.2 ± 0.2 83.5 ± 0.1 84.1 ± 0.2 85.4 ± 0.2 86.1 ± 0.2 +Table 5. Ablation study of the combination of support and trivial +prototypes for improved classification on cropped CUB-200-2011. +prototypes and the final result is produced by the ensemble +of them (Trivial Ensemble and Support Ensemble). Besides, +for our ST-ProtoPNet, we also provide results of its individ- +ual branches (Trivial Branch and Support Branch). Table 5 +gives the experimental results on cropped CUB dataset. We +can notice that combining the two different types of pro- +totypes (ST-ProtoPNet) achieves superior performance over +combining only the same type of prototypes (Trivial Ensem- +ble and Support Ensemble), indicating that our performance +improvements are from not only the ensemble strategy but +also the two complementary sets of prototypes. Also, ST- +ProtoPNet indeed exhibits higher accuracy than its individ- +ual branches, further verifying that the results from the two +branches are complementary, and the combination of them +are effective to improve the final classification accuracy. +6. Conclusion and Future Work +In this paper, we propose the ST-ProtoPNet method to +exploit both support (i.e., hard-to-learn) and trivial (i.e., +easy-to-learn) prototypes for the interpretable image clas- +sification, where the two distinctive sets of prototypes +can provide complementary results. In addition, our ST- +ProtoPNet is a general approach that can be easily applied +to existing prototype-based interpretable models. One limi- +tation for our method is that we empirically adopt the same +number of support and trivial prototypes and the same to- +tal number of prototypes for each class. Considering the +different learning difficulties and imbalanced training sam- +ples among classes in other real-world datasets, e.g., Ima- +geNet [7], a better way to adaptively learn a flexible num- +ber of support and trivial prototypes is needed and deserves +to be further investigated in our future work. Furthermore, +given that we mimic the behaviour of the support vectors of +SVM classifier to obtain the support prototypes by forcing +them to be as close as possible to the classification bound- +ary, we plan to develop new methods to learn prototypes +with gradient-based kernel computations, e.g., neural tan- +gent kernel [16] and path kernel [8]. +8 + +References +[1] David Alvarez Melis and Tommi Jaakkola. Towards robust +interpretability with self-explaining neural networks. +Ad- +vances in Neural Information Processing Systems, 31, 2018. +1 +[2] Alina Jade Barnett, Fides Regina Schwartz, Chaofan Tao, +Chaofan Chen, Yinhao Ren, Joseph Y Lo, and Cynthia +Rudin. 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In Proceedings of the IEEE Conference +on Computer Vision and Pattern Recognition, pages 2921– +2929, 2016. 3, 8 +10 + diff --git a/B9E2T4oBgHgl3EQfnwhd/content/tmp_files/load_file.txt b/B9E2T4oBgHgl3EQfnwhd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9edf777da0261ef0977aeb460f19bcd6d2d69b64 --- /dev/null +++ b/B9E2T4oBgHgl3EQfnwhd/content/tmp_files/load_file.txt @@ -0,0 +1,1030 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf,len=1029 +page_content='Learning Support and Trivial Prototypes for Interpretable Image Classification Chong Wang1 Yuyuan Liu1 Yuanhong Chen 1 Fengbei Liu1 Yu Tian2 Davis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' McCarthy3 Helen Frazer4 Gustavo Carneiro5 1 Australian Institute for Machine Learning, University of Adelaide 2 Harvard University 3 St Vincent’s Institute of Medical Research 4 St Vincent’s Hospital Melbourne 5 CVSSP, University of Surrey Abstract Prototypical part network (ProtoPNet) methods have been designed to achieve interpretable classification by associating predictions with a set of training prototypes, which we refer to as trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', easy-to-learn) prototypes because they are trained to lie far from the classification boundary in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Note that it is possible to make an analogy between ProtoPNet and support vec- tor machine (SVM) given that the classification from both methods relies on computing similarity with a set of train- ing points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', trivial prototypes in ProtoPNet, and support vectors in SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' However, while trivial prototypes are lo- cated far from the classification boundary, support vectors are located close to this boundary, and we argue that this discrepancy with the well-established SVM theory can re- sult in ProtoPNet models with suboptimal classification ac- curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In this paper, we aim to improve the classification accuracy of ProtoPNet with a new method to learn support prototypes that lie near the classification boundary in the feature space, as suggested by the SVM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In addition, we target the improvement of classification interpretabil- ity with a new model, named ST-ProtoPNet, which exploits our support prototypes and the trivial prototypes to pro- vide complementary interpretability information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Experi- mental results on CUB-200-2011, Stanford Cars, and Stan- ford Dogs datasets demonstrate that the proposed method achieves state-of-the-art classification accuracy and pro- duces more visually meaningful and diverse prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Introduction Deep convolutional neural networks (CNN) [14, 26, 27] have made remarkable achievements in various visual tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', image recognition [14] and object detection [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' De- spite the excellent feature extraction and discrimination ability, CNNs are generally treated as black-box models due to their complex architectures, high-dimensional fea- Features lct lsp lct lsp dct dsp Classification boundary Classification boundary (a) The learning of trivial prototypes (b) (b) The learning of support prototypes (ours) (d) Prototypes Feature vectors Clustering Separation Our loss Classification boundary Classification boundary Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The difference between the learning of trivial and sup- port prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (a) Trivial prototypes: the separation loss pushes the prototypes of different classes as far as possible from the clas- sification boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (b) Support prototypes: our new closeness loss enforces the prototypes of different classes to be as close as possible to the classification boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ture spaces, and an enormous number of learnable param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Such lack of interpretability hinders their successful application in fields that require understandable and trans- parent decisions [35], such as disease diagnosis [42], finan- cial risk assessment [30], and autonomous driving [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Recently, increasing attention has been dedicated to the development of interpretable deep-learning models [1, 3, 4, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' A particularly interesting strategy is the prototype- based interpretable models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', prototypical part network (ProtoPNet) [4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' These methods are inherently inter- pretable since they can explain the model’s decisions by showing image classification activation maps associated with a set of class-specific image prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' These proto- types are automatically learned from training samples, with classification score being computed by comparing testing image parts to the learned training prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ProtoPNet [4] is trained to learn a classifier from a set of class-specific prototypes by minimising the cross-entropy classification loss and two additional regularisation losses, namely: 1) a clustering loss that pulls together training im- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='04011v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='CV] 8 Jan 2023 DIWGU2IOUX X noiengmid V ★ B B V 口 V GSLUIua 1 WGILC ■ cages to prototypes of their own class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' and 2) a separation loss that pushes apart training images from all prototypes of other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' More specifically, the clustering loss min- imises the distance of each image patch to at least one pro- totype of its own class, while the separation loss maximises the distance between all image patches and all other class prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The combination of these two losses pushes the prototypes as far as possible from the classification bound- ary, but still within the class distribution, so we call them trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', easy-to-learn) prototypes, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We also display these trivial prototypes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2(a), where we present the ProtoPNet learning results for the two-moon problem, depicting the training points (red and blue points) and prototypes (green and black stars) in both the data and feature spaces, learned with a feed-forward neural net- work1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Notice that the trivial prototypes are located as far as possible from the classification boundary in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Similar to ProtoPNet, the support vector machine (SVM) [6] classifier is trained by minimising a loss func- tion that learns a set of support vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Different from the ProtoPNet’s prototypes, these support vectors are located as close as possible to the classification boundary, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Given that the prototypes in ProtoPNet and support vectors in SVM play similar roles in classification problems, we argue that the ProtoPNet’s loss function may lead to suboptimal classification results because of the triv- ial prototypes being learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In this paper, we propose an alternative learning strat- egy of ProtoPNet from the SVM perspective, to force the learned prototypes to resemble support vectors of SVM and be located as close as possible to the classification bound- ary with the goal of increasing classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The strategy consists of a new closeness loss that minimises the distance between prototypes of different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 1(b), our new loss enforces the prototypes to move closer to the classification boundary, as also demonstrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2(b) that reveals the support prototypes produced by the introduction of our new closeness loss are indeed more similar to the support vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Furthermore, in order to im- prove interpretability, we propose a new ProtoPNet classi- fier that integrates the support and trivial prototypes (named ST-ProtoPNet), where the goal is to produce two distinc- tive and complementary sets of prototypes to obtain more meaningful classification explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Due to the different natures of the two sets of prototypes, they can also enable further improvements in terms of classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The major contributions of this work are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We provide the first study that makes an analogy be- tween the prototype learning from ProtoPNet methods and support vector learning from SVM, where we in- 1The network has an input layer of 2 nodes, a hidden layer of 256 nodes (activated by tanh), and an output layer of 2 nodes (activated by sigmoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Two-moon classification results from ProtoPNet and SVM classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (a) Trivial prototypes (stars) and training sam- ples (circles) in the feature (top) and data (bottom) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (b) Support prototypes (stars) and training samples (circles) in the fea- ture (top) and data (bottom) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Note that in (a) and (b), each learned prototype is projected onto the nearest training sample in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (c) Support vectors (stars) and training samples (circles) from a Radial Basis Function (RBF) kernel based SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' vestigate if by following SVM’s support vector learn- ing strategy and pulling the prototypes to be as close as possible to the classification boundary, it is possible to improve ProtoPNet’s classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We present a new ST-ProtoPNet method to exploit both support and trivial prototypes for the interpretable im- age classification, where the two sets of prototypes can provide complementary information to improve both interpretability and classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We conduct extensive experiments on three benchmark datasets which show that our ST-ProtoPNet method outperforms current state-of-the-art (SOTA) methods in terms of classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In our experiments, we also demonstrate that the combi- nation of the two types of prototypes contributes to richer interpretability, where trivial prototypes tend to focus on both local parts of the visual object of interest and the back- ground, while support prototypes mainly focus on 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 (a) Trivial prototypes (b) Support prototypes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 Class A: red, green Class B: blue, black 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 (c) Support vectors from SVM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Related Work In this section, we first review relevant studies on classi- fication interpretability where we focus on prototype-based methods, and then we briefly review support vector machine (SVM) classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Finally, we provide a short survey on ensemble classification for interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Classification Interpretability The interpretation of classification results produced by deep neural networks can be achieved by a variety of post-hoc explanation techniques, such as explanatory surro- gates [31,39,50], counterfactual examples [13,17,41], and saliency visualisation [38,40,49,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In comparison with post-hoc explanations, prototype- based interpretability is directly present in the model’s in- ner computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ProtoPNet [4] is the original work that uses class-specific prototypes for interpretable image clas- sification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Similar to ProtoPNet, TesNet [48] con- structs class-specific transparent basis concepts on Grass- mann manifold for the interpretable classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' De- rived from ProtoPNet, Deformable ProtoPNet [10] employs spatially-flexible and deformable prototypes to adaptively capture meaningful object features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In ProtoPShare [37], a data-dependent merge-pruning method is presented to share prototypes among classes, which can reduce the number of prototypes used for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Alternatively, Pro- toPool [36] introduces a fully differentiable prototype as- signment strategy to reduce the number of prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In Proto2Proto [18], a knowledge distillation method is de- signed to transfer interpretability from a teacher ProtoPNet to a shallow student ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ProtoTree [32] integrates the prototype learning into a binary neural decision tree that can explain its predictions by tracing a decision path throughout the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ViT-NeT [22] further establishes the prototype neural tree structure on visual transformers [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Because of the ability to self-explain classification re- sults, prototype-based interpretability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', ProtoPNet) has been widely utilised not only in the computer vision appli- cations above, but also in medical imaging [2, 20, 46] and face recognition [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' However, an open question faced by these methods is if the prototypes being learned are the op- timal ones in terms of classification and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' SVM vs Prototype-based Classification To better understand the optimality of prototypes, we consider the support vector machine (SVM) [6] classifier that finds support vectors to represent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' More specif- ically, SVM learns the maximum-margin classifier defined by a classification boundary that maximises the distance to the closest training samples, which are the support vectors for the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The testing of SVM classifiers consists of computing a weighted similarity between a testing sample and the support vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' It is interesting to note that the testing of prototype-based classifiers is also based on mea- suring the similarity between a testing image and a set of class-specific prototypes learned from the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Although the testing of SVM and prototype-based classi- fiers are similar, their training procedures are quite differ- ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' First, the training of a prototype-based classifier learns a fixed number of prototypes [4, 10], while the SVM clas- sifier learns to weight a variable number of support vectors from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Second, in prototype-based classi- fiers [4,10], the learned prototypes tend to be as far as pos- sible from the classification boundary, which is contrary to the SVM training objective mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The study of deep learning methods from an SVM the- oretical perspective is a rich area of research [5,8,33] with huge potential given the impact that both techniques have had in the whole society for the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' However, there are many practical questions that need to be addressed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', how to scale the kernel computation for large-scale datasets, how to shorten the training process [33], and how to integrate the learning of the SVM classifier with deep- learning features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In this paper, we focus on adapting the learning of ProtoPNet’s prototypes to make them similar to SVM’s support vectors, by forcing prototypes to be as close as possible to the classification boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Ensemble Classification Ensemble classification [9] is a traditional machine learning approach that combines the results from multiple classifiers, with the goals of improving learning generalisa- tion and classification calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The use of interpretable ensemble strategy to improve the classification accuracy has been explored in [4,10,48], which is achieved by summing the classification logits of multiple prototype-based clas- sifiers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', ProtoPNets trained with different CNN back- bones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In this work, we obtain the interpretable ensemble classification by combining the predictions of two ProtoP- Nets with highly distinctive prototypes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', support pro- totypes and trivial ones), which is different from previous studies [4,10,48] where the type of prototypes each individ- ual classifier produces is very similar given that the same objective function is employed for each classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' More specifically, the ensemble classification used in this paper targets the utilisation of two complementary sets of pro- totypes, particularly when the prototypes are learned from quite different objective functions, such as the ones for learning the support and trivial prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Preliminaries We assume to have a training set D = {(xn, yn)}|D| n=1, where x ∈ X ⊂ RH×W ×R is an image with R colour channels and y ∈ Y ⊂ {0, 1}C is a one-hot vector rep- resentation of the image class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The interpretable 3 Input image Ensemble Trivial ProtoPNet Add-on layers Add-on layers CNN Backbone fθ Non-cancer g FC layers k Support PPN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='96 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='31 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='58 Cancer Similarity score Logits Support ProtoPNet Prototypes Feature vectors FC layer FC layer Output logits Logits Logits fω (s) fω (t) fΦ (t) fΦ (s) P (s) P (t) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The architecture of our proposed ST-ProtoPNet method for the interpretable image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ProtoPNet [4, 10] is trained to learn a set of prototypes P = {pm}M m=1, where pm ∈ Rρ1×ρ2×D, with each of the C classes containing M/C prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Without loss of generality, we assume ρ1 = ρ2 = 1, but the extension to general values is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' A typical ProtoPNet comprises four components: a CNN backbone, add-on layers, a pro- totype layer, and a fully connected (FC) layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' An input image x is fed to the CNN backbone fθ : X → F (pa- rameterised by θ ∈ Θ, where F ⊂ R ¯ H× ¯ W × ¯ D) and then passed on to the add-on layers, denoted by fω : F → V (parameterised by ω ∈ Ω), to produce a feature map V ∈ V ⊂ R ¯ H× ¯ W ×D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The prototype layer computes the similarity between the feature map V and the M D- dimensional prototypes {pm}M m=1 to generate M similarity maps S(i,j) m = sim(V(i, j, :), pm), where i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', ¯H}, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', ¯W}, and sim(·, ·) represents a similarity mea- sure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', cosine similarity [10] and projection metric [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The prototype layer outputs M similarity scores from max- pooling S = � max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', ¯ H},j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', ¯ W } S(i,j) m �M m=1, which are fed to the FC layer fφ : S → ∆, parameterised by φ ∈ Φ, to produce the classification prediction ˆy ∈ ∆ ⊂ [0, 1]C, where ∆ denotes the probability space for C classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ST-ProtoPNet An overview of our proposed ST-ProtoPNet method is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 3, which comprises a shared CNN backbone fθ(·), two interpretable ProtoPNet classification branches, namely: 1) the support ProtoPNet represented by add-on layers fω(s)(·), prototype layer with support proto- types P(s), and FC layer fφ(s)(·) which outputs the classi- fication probability distribution ˆy(s) ∈ ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' and 2) the trivial ProtoPNet branch with its add-on layers fω(t)(·), trivial pro- totypes P(t), and FC layer fφ(t)(·) that generates probabil- ity predictions ˆy(t) ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The final classification is obtained by combining the classification logits from both the support and trivial ProtoPNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In our implementation, we construct the support and trivial ProtoPNet mainly based on the orig- inal ProtoPNet [4] and TesNet [48], as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Support ProtoPNet The support ProtoPNet is designed to produce support (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', hard-to-learn) prototypes for classification that are as close as possible to the classification boundary, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The loss function to optimise the support ProtoPNet branch is defined as: θ∗,ω(s)∗, P(s)∗, φ(s)∗ = arg min θ,ω(s),P(s),φ(s) � (x,y)∈D ℓspt(x, y, θ, ω(s), P(s), φ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (1) The loss for each training sample (x, y) ∈ D in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (1) above is represented by: ℓspt(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ω(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' P(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' φ(s)) = ℓce(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ω(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' P(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' φ(s)) − λ1ℓct(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ω(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' P(s)) + λ2ℓsp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ω(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' P(s)) − λ3ℓcls(P(s)) + λ4ℓort(P(s)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (2) where λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' λ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' λ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' and λ4 are hyper-parameters to balance each term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ℓce(·) denotes the cross-entropy classification loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ℓct(·) and ℓsp(·) represent the clustering and separa- tion losses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' which are introduced to regularise the ProtoPNet’s training,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' as follows: ℓct(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ω(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' P(s)) = max p∈P(s) y max v∈V(s) sim(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (3) ℓsp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' ω(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' P(s)) = max p/∈P(s) y max v∈V(s) sim(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (4) where V(s) = fω(s)(fθ(x)) is the feature map extracted from the input image x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' v represents one of the ¯H × ¯W feature vectors in V(s) obtained by matrix vectorisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' p is a normalised prototype (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', unit vector) in P(s), sim(·, ·) is one of the similarity functions defined in Section 3, and P(s) y denotes the set of prototypes of class y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The clustering loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (3) and separation loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (4) aim to learn a meaningful feature space in which the image features of a certain class are clustered around the prototypes of the class, and also well separated from those of other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' As mentioned in Section 1, the effect of the clustering and separation losses above tend to push the prototypes of different classes as far as possible from the classifica- tion boundary, resulting in trivial prototypes, as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 1(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In order to learn the proposed support prototypes, we introduce the following novel close- ness loss ℓcls to explicitly enforce the prototypes of different classes to be close to each other, which is formulated as: ℓcls(P(s)) = C−1 � c1=1 C � c2=c1+1 min pm∈Pc1,pn∈Pc2 p⊤ mpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (5) 4 DIWGU2IOUX X noiengmid V ★ B B V 口 V GSLUIua 1 WGILC ■ cb(²)ogeml bolodelnunogininT-e lsdel28loInnigino Wgbbru CGIGLUIGg CJ2IGL2During training, the closeness loss ℓcls maximises the pair-wise prototype similarity, in the form of dot product p⊤ mpn between different classes in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (5), with the goal of pulling the prototypes close to the classification boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' As the prototypes move gradually towards the classification boundary, they are able to capture harder and harder visual features from training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' On the other hand, since the prototypes are located near the classification boundary, this can put pressure on the support ProtoPNet’s learning and enforce it to learn highly discriminative feature repre- sentations to achieve accurate classification, which is bene- ficial to extract more meaningful semantic information from training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Ideally, each prototype of a class should focus on unique object parts of the training images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', head, tail, and claw of birds), so that the prototypes can represent rich and di- verse visual patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' However, there is no particular con- straints to guarantee such prototype diversity and the issue of prototype duplication [10] often occurs in the ProtoPNet family of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' To encourage the intra-class prototype di- versity, we employ an orthonormality loss [48] so that pro- totypes within a class can represent dissimilar visual pat- terns of training samples, which is defined as: ℓort(P(s)) = C � c=1 ∥Pc ⊤Pc − IM/C∥2 F , (6) where ∥·∥2 F represents Frobenius norm, Pc ∈ D ×R(M/C) stands for a matrix composed of the prototypes of class c (prototypes in each column of Pc are normalised), and IM/C is an identity matrix of size M/C × M/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Trivial ProtoPNet As described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1, the support ProtoPNet is developed to learn support (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', hard-to-learn) prototypes by forcing them to be close to the classification boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Considering that training samples contain not only hard vi- sual features but also important trivial ones that the support prototypes cannot fully capture, we propose to also learn trivial prototypes to provide complementary classification information, and exploit both the support and trivial proto- types for improved interpretable classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The loss objective to train the trivial ProtoPNet branch is defined as follows: θ∗,ω(t)∗, P(t)∗, φ(t)∗ = arg min θ,ω(t),P(t),φ(t) � (x,y)∈D ℓtrv(x, y, θ, ω(t), P(t), φ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (7) The loss for each training image (x, y) ∈ D in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (7) above is represented by: ℓtrv(x, y, θ, ω(t), P(t), φ(t)) = ℓce(x, y, θ, ω(t), P(t), φ(t)) − λ1ℓct(x, y, θ, ω(t), P(t)) + λ2ℓsp(x, y, θ, ω(t), P(t)) + λ3ℓdsc(P(t)) + λ4ℓort(P(t)), (8) where λ1, λ2, λ3, and λ4 are hyper-parameters, ℓce(·) is the cross-entropy loss, the clustering loss ℓct, separation loss ℓsp, and orthonormality loss ℓort are the same as in the sup- port ProtoPNet defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (3), (4) and (6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The trivial ProtoPNet targets the learning of easy proto- types that are far from the classification boundary and have a good discrimination ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' To help achieve this, we in- troduce a new discrimination loss ℓdsc to facilitate the inter- class separability between prototypes of different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' This is formulated by minimising the pair-wise prototype similarities of different classes, as follows: ℓdsc(P(t)) = C−1 � c1=1 C � c2=c1+1 max pm∈Pc1,pn∈Pc2 p⊤ mpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (9) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Training and Testing Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Following the training strategies in the Pro- toPNet family [4, 10], the training procedure of our ST- ProtoPNet consists of 3 stages: 1) stochastic gradient de- scent (SGD) optimisation of the CNN backbone, add-on layers, and prototype layer, using a fixed FC layer initialised with +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 for correct and incorrect connection weights, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 2) prototype projection by updating each prototype with its nearest latent training image patch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' and 3) optimisation of the FC layer, with an additional L1 regularisation on the incorrect connection weights (initially fixed at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In each stage, we alternate the optimisation of each branch of the ST-ProtoPNet between mini-batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' To exploit the complementary results from both branches of the ST-ProtoPNet, the final classification is obtained from the summed logits predicted by the two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' It is worth noticing that this ensemble strategy introduces no loss of interpretablity but improved accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Experiments We perform experiments on three fine-grained classifi- cation benchmark datasets: CUB-200-2011 [45], Stanford Cars [25], and Stanford Dogs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' To achieve fair com- parison, we follow previous studies [4, 48] by applying of- fline data augmentations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', random rotation, skew, shear, and left-right flip) on the cropped CUB and cropped Cars datasets (using bounding boxes provided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We also validate 5 our method on the full CUB and full Dogs datasets, and employ the same online data augmentation methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', random affine transformation and left-right flip) as used in Deformable ProtoPNet [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' All images are resized to 224 × 224 pixels as network input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Experimental Settings The proposed ST-ProtoPNet method is evaluated on the following CNN architectures: VGG-16, VGG-19, ResNet- 34, ResNet-50, ResNet-152, DenseNet-121, and DenseNet- 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' All CNN backbones are pre-trained on ImageNet [7], except for ResNet-50, which is pre-trained on iNatural- ist [44] for the experiment on full CUB [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' The add-on layers include two 1 × 1 convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' For simplic- ity, we utilise the same prototype dimension D = 64 for all CNN backbones on the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' For cropped CUB and Cars datasets, following [48], we use 10 prototypes (5 support and 5 trivial) per class and the projection metric in the similarity function sim(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In full CUB and Dogs datasets, to ensure comparison fairness with Deformable ProtoPNet [10] that uses 10 2 × 2 (full CUB) and 10 3 × 3 (full Dogs) deformable prototypes per class, we utilise the same total number of prototypes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', 40 1 × 1 (20 support and 20 trivial) for full CUB and 90 1×1 (45 support and 45 trivial) for full Dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Also, we employ the cosine similar- ity in sim(·, ·) and obtain 14 × 14 ( ¯H = ¯W = 14) feature maps by upsampling the original 7 × 7 feature maps via a bi-linear interpolation step, as in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In our implemen- tation, we empirically set λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8, λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='48 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='08 for the support and trivial ProtoPNet branches respectively, λ3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0, and λ4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' More details about the experi- mental setup can be found in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Performance Comparison Table 1 presents the classification accuracy (averaged across 5 runs) of our proposed ST-ProtoPNet on cropped CUB and cropped Cars datasets, where the Baseline is rep- resented by non-interpretable black-box CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' As can be seen, our ST-ProtoPNet outperforms other compet- ing methods across all backbone architectures for the task of bird species classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Also, our method achieves the best results for the car model identification task when using VGG and DenseNet architectures as the CNN backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In particular, our VGG19-based ST-ProtoPNet reaches an average accuracy of 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2% and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7% on CUB and Cars, respectively, surpassing other methods with the most im- provements across all backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Moreover, the support ProtoPNet generally performs better than methods utilising only trivial prototypes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', ProtoPNet, TesNet, and Triv- ial ProtoPNet), showing the importance of learning support prototypes for the interpretable image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' It is worth noting that our proposed ST-ProtoPNet produces su- perior performance over the support ProtoPNet method, in- dicating that both support and trivial prototypes are useful and can provide complementary information for achieving accurate and interpretable classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Table 2 shows the classification results of different meth- ods on full CUB and full Dogs datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In both datasets, the classification accuracy of the original ProtoPNet method is generally worse than the non-interpretable counterpart (Baseline) under many CNN backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' On the other hand, the accuracy by the trivial ProtoPNet and support ProtoP- Net are substantially better than those by Baseline, ProtoP- Net, and Deformable ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' However, the proposed ST-ProtoPNet achieves more significant performance gains that result in the best classification accuracy across most CNN backbones, particularly when using a large number of prototypes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', 40 1 × 1 prototypes per class for CUB and 90 1×1 prototypes per class for Dogs), which demonstrates the effectiveness of utilising both the trivial and support prototypes for the interpretable image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Ad- ditionally, when using a smaller number of prototypes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e, 10 1×1 prototypes per class, our ST-ProtoPNet method still exhibits competitive classification accuracy across multiple CNN backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We further compare our ST-ProtoPNet with other deep- learning methods that can provide different levels of inter- pretability on CUB dataset, with results shown in Table 3, where * and ** denote ensembled models trained with dif- ferent CNN backbones or hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' As can be ob- served, an ensemble of three ST-ProtoPNets can achieve high classification accuracy (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9% on cropped images, 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2% on full images), outperforming competing methods that are also based on an ensemble of three models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', ProtoTree, TesNet, and ProtoPool).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Moreover, the ensemble of five ST-ProtoPNets exceeds all other competing methods and obtains the best classification accuracy of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1% and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4% on cropped and full CUB images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Ex- perimental results on Stanford Cars and Stanford Dogs are presented in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Visualisation Analysis To better highlight the differences between the support and trivial prototypes, we select 8 categories of birds with visually similar features from cropped CUB-200-2011 to form a subset, and show the learned prototypes of our VGG19-based ST-ProtoPNet in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 4, where each proto- type is visualised by projecting onto its nearest training im- age patch in the latent feature space [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 4(a) that the support prototypes can capture subtle and fine visual features of different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Importantly, notice that the support prototypes only focus on relevant bird parts, such as the head and belly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' This is reasonable since our learning algorithm is designed to produce prototypes that are as close as possible to the classification boundary, where the image prototypical parts should be discriminative but at 6 Method CUB Cars VGG16 VGG19 ResNet34 ResNet152 Dense121 Dense161 VGG16 VGG19 ResNet34 ResNet152 Dense121 Dense161 Baseline 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 ProtoPNet [4] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 TesNet [48] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 Trivial ProtoPNet 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 Support ProtoPNet 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 ST-ProtoPNet (ours) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Classification accuracy (%) on cropped CUB-200-2011 and Stanford Cars by competing methods using different CNN backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Method Prototype CUB Prototype Dogs VGG16 VGG19 ResNet34 ResNet50 ResNet152 Dense121 Dense161 VGG16 VGG19 ResNet34 ResNet50 ResNet152 Dense121 Dense161 Baseline – 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 – 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ProtoPNet [4] 1×1p, 10pc 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 1×1p, 10pc 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 ProtoPNet [4] 1×1p, 40pc 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 1×1p, 90pc 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 TesNet [48] 1×1p, 10pc 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 1×1p, 10pc 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 TesNet [48] 1×1p, 40pc 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 1×1p, 90pc 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 Deformable ProtoPNet [10] 2×2p, 10pc 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 3×3p, 10pc 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 Trivial ProtoPNet 1×1p, 40pc 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 1×1p, 90pc 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 Support ProtoPNet 1×1p, 40pc 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 1×1p, 90pc 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ST-ProtoPNet (ours) 1×1p, 10pc 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 1×1p, 10pc 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ST-ProtoPNet (ours) 1×1p, 40pc 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 1×1p, 90pc 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Classification accuracy (%) on full CUB-200-2011 and Stanford Dogs datasets by competing approaches using different CNN backbones, where kpc represents k prototypes per class and ρ1×ρ2p denotes the spatial shape of the prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (a) Support prototypes (b) Trivial prototypes Laysan Albatross Sooty Albatross Crested Auklet Crested Auklet Parakeet Auklet Rhinoceros Auklet Brewer Blackbird Redwinged Blackbird Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Visual comparison between the support (a) and trivial (b) prototypes from cropped CUB-200-2011, where each row exhibits prototypes of the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In each pair, the first column shows the original image with a prototype indicated in a yellow bounding box, the second column demonstrates the prototype’s corresponding activation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' the same time visually similar among different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' By contrast, it is observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 4(b) that the trivial prototypes tend to focus not only on the relevant bird parts but also the background regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' For example, some trivial prototypes of the Laysan Albatross and Sooty Albatross classes cap- ture the sea surface as they often appear with the sea back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We argue that this is because the trivial ProtoPNet may treat the background as the most easy-to-learn pattern, focusing less on the target’s visual parts of the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 5 displays an example of the interpretable reasoning process of our ST-ProtoPNet method in classifying a testing bird image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' As can be seen, each ProtoPNet branch calcu- lates its own classification logits (weighted sum of similar- ity scores), which is then combined to generate the final pre- dictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' To be specific, when classifying a Brewer Black- bird, the support prototypes are quite active on the belly or lower surface of the bird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Meanwhile, the trivial prototypes mostly have high activations on the bird’s head and tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In this case, the support ProtoPNet branch obtains a relatively higher similarity score (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='913), compared with the trivial branch (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='248).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Note that our ST-ProtoPNet exploits both the support and trivial prototypes to capture much richer representations of the target object from different perspec- tives, which enables the production of the final decision in a complementary way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Notably, the ensemble classification improves interpretability and accuracy using richer repre- sentations for the target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' More examples of visual prototypes and interpretable reasoning on Cars and Dogs are provided in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 7 Interpretability Method Accuracy (%) None B-CNN [29] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 (b) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 (f) Object-level attention CAM [53] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 (b) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 (f) CSG [28] 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 (b) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 (f) Part-level attention PA-CNN [24] 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 (b) – MG-CNN [47] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 (b) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 (f) MA-CNN [51] – 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 (f) RA-CNN [12] – 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 (f) TASN [52] – 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 (f) Part-level attention + Prototypes Region [15] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 (b) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 (f) ProtoPNet* [4] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 (b) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 (f) ProtoTree* [32] – 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 (f) TesNet* [48] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 (b) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 (f) ProtoPool* [36] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 (b) – ST-ProtoPNet* (ours) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 (b) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 (f) ProtoTree** [32] – 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 (f) Deformable ProtoPNet** [10] – 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 (f) ProtoPool** [36] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 (b) – ST-ProtoPNet** (ours) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 (b) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 (f) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Classification accuracy and interpretability of different methods on CUB-200-2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' “b” and “f” denote the model is trained and tested on cropped and full images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' *: Ensembled of three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' **: Ensembled of five models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Testing image Prototype Training image where the prototype derives Activation map Similarity score Connection weight Individual logits Combined logits 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='981 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='939 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='964 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='715 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='979 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='230 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='972 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='088 = Support ProtoPNet Trivial ProtoPNet .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='161 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='913 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='248 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' An example of the interpretable reasoning of our ST- ProtoPNet for classifying a testing Brewer Blackbird image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Ablation Study The Closeness and Discrimination Losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In order to validate the effectiveness of our proposed closeness loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (5) and discrimination loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (9), we first conduct ablation studies on full CUB and full Dogs datasets using ResNet34 as the CNN backbone, with results provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We can notice that both the closeness and discrim- ination losses can improve the classification accuracy, com- pared with the Baseline ProtoPNet trained with only clus- tering, separation, and orthonormality losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Importantly, the closeness loss introduces a larger performance improve- ment since it aims at learning support (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', hard-to-learn) prototypes that lie near the classification boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Combining Support and Trivial Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We also investigate the importance of integrating the two comple- mentary sets of support and trivial prototypes for achieving improved classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' To achieve this, we first train a two- branch model where both branches learn the same type of Method ℓct ℓsp ℓort ℓdsc ℓcls Accuracy (%) CUB Dogs Baseline ProtoPNet ✓ ✓ ✓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 Trivial ProtoPNet ✓ ✓ ✓ ✓ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 Support ProtoPNet ✓ ✓ ✓ ✓ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Ablation analysis of the closeness loss ℓcls in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (5) to learn support prototypes, and discrimination loss ℓdsc in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' (9) to learn trivial prototypes on full CUB-200-2011 and Stanford Dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Method VGG16 VGG19 ResNet34 ResNet152 Dense121 Dense161 Trivial Ensemble 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 Support Ensemble 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 Trivial Branch 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 Support Branch 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ST-ProtoPNet (ours) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='2 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Ablation study of the combination of support and trivial prototypes for improved classification on cropped CUB-200-2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' prototypes and the final result is produced by the ensemble of them (Trivial Ensemble and Support Ensemble).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Besides, for our ST-ProtoPNet, we also provide results of its individ- ual branches (Trivial Branch and Support Branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Table 5 gives the experimental results on cropped CUB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' We can notice that combining the two different types of pro- totypes (ST-ProtoPNet) achieves superior performance over combining only the same type of prototypes (Trivial Ensem- ble and Support Ensemble), indicating that our performance improvements are from not only the ensemble strategy but also the two complementary sets of prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Also, ST- ProtoPNet indeed exhibits higher accuracy than its individ- ual branches, further verifying that the results from the two branches are complementary, and the combination of them are effective to improve the final classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Conclusion and Future Work In this paper, we propose the ST-ProtoPNet method to exploit both support (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', hard-to-learn) and trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', easy-to-learn) prototypes for the interpretable image clas- sification, where the two distinctive sets of prototypes can provide complementary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In addition, our ST- ProtoPNet is a general approach that can be easily applied to existing prototype-based interpretable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' One limi- tation for our method is that we empirically adopt the same number of support and trivial prototypes and the same to- tal number of prototypes for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Considering the different learning difficulties and imbalanced training sam- ples among classes in other real-world datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', Ima- geNet [7], a better way to adaptively learn a flexible num- ber of support and trivial prototypes is needed and deserves to be further investigated in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Furthermore, given that we mimic the behaviour of the support vectors of SVM classifier to obtain the support prototypes by forcing them to be as close as possible to the classification bound- ary, we plan to develop new methods to learn prototypes with gradient-based kernel computations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=', neural tan- gent kernel [16] and path kernel [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 8 References [1] David Alvarez Melis and Tommi Jaakkola.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vi- sion and Pattern Recognition, pages 5012–5021, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 8 [53] Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' Learning deep features for discrimi- native localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2921– 2929, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} +page_content=' 3, 8 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfnwhd/content/2301.04011v1.pdf'} diff --git a/CNFQT4oBgHgl3EQfODbS/content/tmp_files/2301.13274v1.pdf.txt b/CNFQT4oBgHgl3EQfODbS/content/tmp_files/2301.13274v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..36577b8b53f4e5ebfea62595f1f0822352a61ca4 --- /dev/null +++ b/CNFQT4oBgHgl3EQfODbS/content/tmp_files/2301.13274v1.pdf.txt @@ -0,0 +1,6356 @@ +THE OPEN DIHYPERGRAPH DICHOTOMY FOR GENERALIZED +BAIRE SPACES AND ITS APPLICATIONS +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Abstract. The open graph dichotomy for a subset X of the Baire space ωω states that +any open graph on X either admits a coloring in countably many colors or contains a +perfect complete subgraph. It is a strong version of the open coloring axiom for X that +was introduced by Todorˇcevi´c and Feng to study definable sets of reals. We first show +that its recent infinite dimensional generalization by Carroy, Miller and Soukup holds +for all subsets of the Baire space in Solovay’s model, extending a theorem of Feng from +dimension 2. Our main theorem lifts this result to generalized Baire spaces κκ in two +ways. +(1) For any regular infinite cardinal κ, the following holds after a L´evy collapse of an +inaccessible cardinal λ > κ to κ+. +Suppose that H is a κ-dimensional box-open directed hypergraph +on a subset of κκ such that H is definable from a κ-sequence of +ordinals. Then either H admits a coloring in κ many colors or there +exists a continuous homomorphism from a canonical large directed +hypergraph to H. +(2) If λ is a Mahlo cardinal, then the previous extends to all relatively box-open +directed hypergraphs on any subset of κκ that is definable from a κ-sequence of +ordinals. +We derive several applications to definable subsets of generalized Baire spaces, among +them variants of the Hurewicz dichotomy that characterizes subsets of Kσ sets, an +asymmetric version of the Baire property, an analogue of the Kechris-Louveau-Woodin +dichotomy that characterizes when two disjoint sets can be separated by an Fσ set, the +determinacy of V¨a¨an¨anen’s perfect set game for all subsets of κκ, and an analogue of +the Jayne-Rogers theorem that characterizes the functions which are σ-continuous with +closed pieces. Some of these applications lift results of Carroy, Miller and Soukup from +the countable setting and extend results of V¨a¨an¨anen, L¨ucke, Motto Ros and the authors +in the uncountable setting. +2020 Mathematics Subject Classification. (Primary) 03E15, (Secondary) 03E35, 03E55. +Key words and phrases. Generalized Baire space, directed hypergraph, dichotomy, graph coloring, open +graph dichotomy, large cardinal. +We are grateful to Stevo Todorˇcevi´c for comments and discussions. We further thank Rapha¨el Carroy +and Ben Miller for answering a query regarding their work. +The first-listed author was partially supported by EPSRC grant number EP/V009001/1 and FWF +grant number I4039. This project has received funding from the European Union’s Horizon 2020 research +and innovation programme under the Marie Sk�lodowska-Curie grant agreement No 794020 of the first- +listed author. The second-listed author was partially supported by NKFIH grant number K129211. For +the purpose of open access, the authors have applied a ‘Creative Commons Attribution’ (CC BY) public +copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission. +1 +arXiv:2301.13274v1 [math.LO] 30 Jan 2023 + +2 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Contents +1 +Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2 +Preliminaries +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2.1 +Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2.2 +Dihypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2.3 +Trees and order preserving maps . . . . . . . . . . . . . . . . . . . . . . +11 +2.4 +Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +3 +Dihypergraphs and homomorphisms +. . . . . . . . . . . . . . . . . . . +17 +3.1 +Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +3.2 +Order homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +4 +The open dihypergraph dichotomy for definable sets . . . . . . . . . +22 +4.1 +Three important steps +. . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +4.1.1 +Reflecting to intermediate models . . . . . . . . . . . . . . . . . . . +23 +4.1.2 +Names and witnessing functions . . . . . . . . . . . . . . . . . . . . +26 +4.1.3 +Constructing continuous homomorphisms +. . . . . . . . . . . . . . +30 +4.2 +The countable case +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +4.3 +The uncountable case . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +4.3.1 +κ-analytic sets +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +4.3.2 +Definable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +4.3.3 +The Quotient Lemma for Add(µ, 1) . . . . . . . . . . . . . . . . . . +36 +4.4 +A global version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +48 +5 +Variants +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +5.1 +Closure properties of maps . . . . . . . . . . . . . . . . . . . . . . . . . +50 +5.2 +The open graph dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . +54 +5.3 +Counterexamples +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +56 +5.4 +♦i +κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +5.5 +Strong variants +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +62 +6 +Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +72 +6.1 +The Hurewicz dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +6.2 +The Kechris-Louveau-Woodin dichotomy . . . . . . . . . . . . . . . . . +76 +6.3 +Applications of ODDκ +κ(κκ) +. . . . . . . . . . . . . . . . . . . . . . . . . +81 +6.3.1 +The perfect set property for closed sets . . . . . . . . . . . . . . . . +81 +6.3.2 +ODDκ +κ(X) for κ-analytic sets +. . . . . . . . . . . . . . . . . . . . . +82 +6.4 +The determinacy of V¨a¨an¨anen’s perfect set game +. . . . . . . . . . . . +86 +6.5 +The asymmetric Baire property +. . . . . . . . . . . . . . . . . . . . . . +92 +6.6 +The Jayne-Rogers theorem . . . . . . . . . . . . . . . . . . . . . . . . . +95 +7 +Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 +References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +3 +1. Introduction +It is natural to wonder whether Ramsey’s theorem for n-tuples of natural numbers +[Ram30] can be extended to the set of real numbers. Sierpinski’s counterexample is a +partition of pairs of reals into two pieces such that no uncountable homogeneous set +exists [Sie33]. Since his example is not constructive, one can ask if a version of Ramsey’s +theorem holds for simple partitions. Galvin gave a partial answer to this by proving that +for any partition of pairs of reals into two open pieces there exists an uncountable (in fact, +perfect) homogeneous set [Gal68]. Blass generalized this to partitions of n-tuples into +finitely many Borel pieces by weakening the conclusion [Bla81]: instead of homogeneity, +one requires that all pairs in the set belong to at most (n − 1)! pieces. +The next step was to generalize these results to spaces other than the reals. Abraham, +Rubin and Shelah formulated a strengthening of Galvin’s theorem for countably based +metric spaces of size ℵ1 [ARS85]. Abstracting from many applications, Todorˇcevi´c in- +troduced the open coloring axiom OGA [Tod89].1 For a topological space X, OGA(X) +states that any open graph G on X either admits a coloring2 in countably many colors or +else contains an uncountable complete subgraph. Furthermore, OGA states that OGA(X) +holds for all countably based metric spaces X. OGA follows from the proper forcing ax- +iom PFA [Tod89] and has many applications [Bek91,TF95,Vel92]. Feng and Todorˇcevi´c +studied a stronger version of OGA(X), the open graph dichotomy OGD(X), that is useful +for applications to definable sets of reals [Fen93, TF95]. Here the uncountable complete +subgraph has to be perfect. OGD(X) is consistent for all definable sets X of reals and in +fact, it is provable for all analytic sets [Fen93].3 Note that the previous statements only +mention open graphs because they fail for closed graphs. +The verbatim analogues of the open graph axiom and dichotomy fail in dimension 3 +and higher: an open hypergraph on the Cantor space might neither admit a countable +coloring nor have an uncountable complete subhypergraph. However, one can replace the +latter by a continuous homomorphic image of a canonical large hypergraph. Carroy, Miller +and Soukup proved an infinite dimensional version of OGD for directed hypergraphs4 on +analytic sets and extended this to all sets of reals assuming the axiom of determinacy +AD [CMS20].5 A similar idea is implicit in work of Aviles and Todorˇcevi´c [AT11].6 The +main motivation of this dichotomy is to provide new proofs and strong versions of several +1It is also often called the open coloring axiom OCA. Note that a graph G on a set X can be identified +with a partition or coloring of pairs in X, for example by coloring edges in G red and edges in the +complement of G blue. Galvin’s theorem is then equivalent to the statement that for any clopen graph G +on the reals, there exists either a perfect independent set or a perfect complete subgraph. +2Recall that a coloring of a graph G assigns different colors to adjacent vertices. +3See [Fen93, p. 676-7] for an alternative approach due to Todorˇcevi´c. +4A directed hypergraph, or dihypergraph, of dimension d is a set of nonconstant functions with do- +main d. +5It follows that this dichotomy is consistent with ZFC for all directed hypergraphs which are definable +from a countable sequence of ordinals. To see this, take a model of AD + DC and add a well-ordering of +the reals by a σ-closed homogeneous forcing. +6We would like to thank Stevo Todorˇcevi´c for pointing out that the idea for an infinite dimensional +version of OGD(X) is implicit in the proof of [AT11, Theorem 7]. + +4 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +seemingly unrelated results in descriptive set theory [CMS20]. For instance, the Hurewicz +dichotomy characterizes the circumstances in which an analytic sets is contained in a Kσ +set [Hur28,SR75,Kec77]. This dichotomy has also been studied by theoretical computer +scientists [dB18]. A dichotomy due to Kechris, Louveau and Woodin describes when an +analytic set can be separated from another set by an Fσ set [KLW87]. +A celebrated +theorem of Jayne and Rogers characterizes functions that can be obtained as a countable +union of continuous functions on closed sets [JR82]. +The topological properties of the generalized Baire space κκ of functions on a regular +uncountable cardinal κ resemble that of the Baire space ωω. In fact, larger classes of spaces +including κκ have been identified as analogues to countably based complete metric spaces +at uncountable cardinals [CS16, ARS21]. The study of descriptive set theory for these +spaces is motivated, inter alia, by connections with model theory and classification theory +[MV93, FHK14, Mor22a, Mor22b].7 There is ample background literature on generalized +Baire spaces, beginning with results on the structure of closed, analytic and coanalytic +subsets [V¨a¨a91,MV93,L¨uc12,LS15]. It is known that some properties of definable subsets +of generalized Baire spaces are closely linked to statements in combinatorial set theory, for +instance to the existence of Kurepa trees [LS]. Extensions of several classical dichotomies +to the uncountable setting pave the way for a structure theory of definable sets. For +instance, the Hurewicz dichotomy for analytic sets is consistent by joint work of the +first-listed author with L¨ucke and Motto Ros [LMS16]. Assuming the existence of an +inaccessible cardinal, it is also consistent for all definable sets [LMS18], and so is the +perfect set property PSP for definable sets [Sch17]. Moreover, the analogue of OGD for +analytic sets is consistent relative to an inaccessible cardinal by a result of the second- +listed author [Szir18]. While we do not study analogues of the original OGA here, there +is a search for principles at cardinals beyond ω1 that can take its place [MV21]. +Our aim is to extend the above results about the open graph dichotomy OGD and +its infinite dimensional variant to uncountable cardinals and use this to provide several +applications to definable subsets of generalized Baire spaces. +We next introduce the +relevant variants of the open dihypergraph dichotomy. We assume that κ is a regular +infinite cardinal with κ<κ = κ throughout this paper, unless otherwise mentioned. The +κ-Baire space is the set κκ of functions from κ to κ equipped with the bounded topology +(or <κ-box topology), i.e., the topology generated by the base {Ns : s ∈ <κκ}, where +Ns = {x ∈ κκ : s ⊆ x} for each s ∈ <κκ.8 The κ-Cantor space is κ2 with the subspace +topology. A graph is a symmetric irreflexive binary relation. A graph on a topological +space X is open if it is an open subset of the product space X × X. +Consider the following analogue of Todorˇcevi´c’s [Tod89] open graph axiom for regular +uncountable cardinals κ. For any class C, OGAκ(C) states that the following holds for all +subsets X ∈ C of κκ: +OGAκ(X): If G is an open graph on X, then either G admits a κ-coloring +or G has a complete subgraph of size κ+. +7See the survey [V¨a¨a95] for early developements in this area. +8The ω-Baire space is the Baire space ωω, since the bounded topology equals the product topology for +κ = ω. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +5 +The following analogue of Feng’s open graph dichotomy [Fen93] is a stronger version of +this axiom. +Definition 1.1. For any class C, OGDκ(C) states that the following holds for all subsets +X ∈ C of κκ: +OGDκ(X): If G is an open graph on X, then either G admits a κ-coloring +or G has a κ-perfect complete subgraph.9 +Note that G has a κ-perfect complete subgraph if and only if there exists a continuous +homomorphism from the complete graph Kκ2 to G. Thus OGDκ(X) can be formulated +for arbitrary topological spaces as well. +In the countable case,10 Carroy, Miller and +Soukup introduced the box-open d-dimensional dihypergraph dichotomy as an analogue of +OGDω(X) in higher dimensions 2 ≤ d ≤ ω [CMS20]. We now consider its analogue for +regular infinite cardinals κ with κ<κ = κ. +Let X be any set and d a set of size at least 2. A d-dihypergraph11 on X is a subset +H of dX consisting of non-constant sequences. Recall that a homomorphism is a map +that takes hyperedges to hyperedges.12 A subset of X is H-independent if it contains no +hyperedges, and a κ-coloring is a partition of X into κ many H-independent sets. Let +Hκd denote the following d-dihypergraph on κd: +Hκd := +� +t∈<κd +� +i∈d +Nt⌢⟨i⟩ = {x ∈ d(κd) : ∃t ∈ <κd ∀i ∈ d t⌢⟨i⟩ ⊆ xi}. +.. +.. +t0 t1 +tα +.. +x0 +x1 +xα +<κd +tα = t⌢⟨α⟩ +Figure 1. Hκd +We will understand d as a discrete topological space and equip κd with the <κ-box +topology. +Ordinals are equipped with the discrete topology. +If X is a subset of the +κ-Baire space, then dX is equipped with the box-topology unless otherwise mentioned. +9I.e., G has a complete subgraph whose domain is a κ-perfect subset of κκ. See Definitions 2.5 and 2.6. +10I.e., for κ = ω. +11I.e., a d-dimensional directed hypergraph. +12See Subsection 2.2 for the precise definitions of the concepts discussed in this paragraph. + +6 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Definition 1.2. For any d-dihypergraph H on κκ,13 let ODDH +κ denote the following +statement: +ODDH +κ : Either H admits a κ-coloring, or there exists a continuous homo- +morphism f : κd → κκ from Hκd to H. +Definition 1.3. For classes C and D, let ODDd +κ(C, D) denote statement that ODDH↾X +κ +holds whenever +(a) X is a subset of κκ in C and +(b) H is a box-open14 d-dihypergraph on κκ in D. +If D is the class of all sets or all d-dihypergraphs on κκ, then we omit it from the notation +and write ODDd +κ(C) for short. If C has a unique element X, then we write X instead of +C, and similarly for D. +For example, ODDd +κ(X) states that ODDH↾X +κ +holds for all box-open d-dihypergraphs H +on κκ. It is equivalent that ODDH +κ holds for all d-dihypergraphs H on X that are box-open +on X.15 Following [CMS20], we call ODDd +κ(X) the box-open d-dimensional dihypergraph +dichotomy for X, or open dihypergraph dichotomy for short. The open graph dichotomy +OGDκ(X) is equivalent to ODD2 +κ(X), and therefore it follows from ODDd +κ(X) for any d.16 +The preliminary Sections 2 and 3 provide definitions and basic results on directed +hypergraphs, trees, forcing, homomorphisms and order preserving maps. +The following theorem, which is the main result of this paper, is proved in Section 4. +Let Dκ denote the class of all sets that are definable from a κ-sequence of ordinals. +Theorem 1.4. Suppose κ < λ are regular infinite cardinals and 2 ≤ d ≤ κ. For any +Col(κ, <λ)-generic filter G over V , the following statements hold in V [G]:17 +(1) ODDd +κ(Dκ, Dκ) if λ is inaccessible in V . +(2) ODDd +κ(Dκ) if λ is Mahlo in V .18 +A crucial idea in the proof of the uncountable case is to construct a forcing which adds +a perfect tree of generic filters over an intermediate model that induces a homomorphism +of dihypergraphs. This technical part of the proof is done in Subsection 4.3. +Note that ODDd +κ(Dκ, Dκ) ⇒ ODDd +κ(Dκ) holds for any d < κ, since any box-open d- +dihypergraph on κκ is in Dκ by virtue of the topology having a base of size κ.19 It thus +suffices to assume in (2) that λ is inaccessible in V if d < κ. It is open whether the Mahlo +cardinal is neccessary for (2) if d = κ. However, the inaccessible cardinal is necessary +13The definition of ODDH +κ also makes sense for dihypergraphs H on arbitrary topological spaces. +14I.e, H is an open subset of d(κκ) in the box topology. +15In general, we have to work with box-open dihypergraphs H on κκ instead of box-open dihypergraphs +on X. For instance, ODDH↾X +κ +might hold only for definable box-open dihypergraphs H, but H↾X is not +necessarily definable. This is relevant for several applications. +16See Subsection 5.2 and Lemma 3.7. +17(1) is equivalent to (1) in the abstract by Lemma 3.3, and (2) is equivalent to (2) in the abstract by +Lemma 3.4. +18Recall that a cardinal λ is Mahlo if the set of inaccessible cardinals ν < λ is stationary in λ. +19See Lemma 3.2. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +7 +for (1), since the statement implies the perfect set property PSPκ(X) for all closed sets +X ⊆ κκ and thus the existence of an inaccessible cardinal above κ in L.20 +We also obtain a global version of Theorem 1.4 (1) for all regular infinite cardinals in +Subsection 4.4. It remains open whether (2) admits a global version. +In Section 5, we investigate alternatives to the open dihypergraph dichotomy. +We +observe that ODDd +κ(X) is equivalent to a dichotomy for d-hypergraphs with the open +graph dichotomy OGDκ(X) as a special case for d = 2. We then affirm the optimality +of the formulation of ODDH +κ in various aspects. While Hκd shows that for any d ≥ 3, +the existence of a continuous homomorphism cannot be replaced by the existence of a +large complete subhypergraph,21 it is natural to ask for homomorphisms with additional +properties such as injectivity. While this cannot be done for d = κ = ω,22 it is possible +in the uncountable setting assuming a weak version of ♦κ. The next result follows from +Theorem 5.35. +Theorem 1.5. Assume ♦i +κ.23 If H is a box-open d-dihypergraph on a subset of κκ where +2 ≤ d ≤ κ, then ODDH +κ is equivalent to the following statement: +ODDIH +κ : Either H admits a κ-coloring or there exists an injective contin- +uous homomorphism from Hκd to H. +For d < κ, or alternatively with additional assumptions on H, the continuous homo- +morphism in ODDH +κ can be chosen to be a homeomorphism onto a closed subset even +without assuming ♦i +κ. This is demonstrated in Subsection 5.5.24 It is open whether the +seemingly weak assumption ♦i +κ in the previous theorem can be removed for d = κ.25 It +holds for all inaccessible cardinals κ and all successor cardinals κ ≥ ω2 with κ<κ = κ by a +result of Shelah [She10].26 Moreover, if κ is a regular uncountable cardinal, then ♦i +κ holds +in Col(κ, <λ)-generic extensions V [G] where λ > κ is inaccessible. Therefore ODDκ +κ can +be replaced by ODDIκ +κ in Theorem 1.4.27 +Section 6 studies a number of applications of the open dihypergraph dichotomy. We +now describe the main results of the section. The missing notation will be introduced in +the respective subsections. The Hurewicz dichotomy characterizes analytic subsets of Kσ +sets as those that do not contain a topological copy of the Baire space [Hur28]. Kechris +and Saint Raymond proved this independently for more complex sets under appropri- +ate determinacy assumptions [SR75, Kec77]. In the uncountable setting, the Hurewicz +dichotomy has a topological version that is more similar to the classical one [LMS16], +20See [Jec71, Sections 3&4] and [L¨uc12, Section 7]. The inaccessible cardinal is also neccessary in the +countable case by [Jec03, Theorem 25.38]. +21See Subsection 5.3 for stronger counterexamples. +22The dihypergraph in Definition 5.29 is a counterexample by Lemma 5.30 and Proposition 5.32. +23♦i +κ and its relationship with some other versions of ♦κ is discussed in Subsection 5.4. Note that ♦i +κ +implies that κ is uncountable by Remark 5.21. +24See Definitions 5.24, 5.37 and Theorems 5.26, 5.40. +25See Problem 7.7 +26See Lemma 5.20. +27See Corollary 5.36. + +8 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +as well as a combinatorial version using superperfect trees [LMS18]. The next result is +proved in Theorems 6.3 and 6.8 using a variant of a dihypergraph from [CMS20]. +Theorem 1.6. ODDκ +κ(X, Dκ) implies the following analogues of the Hurewicz dichotomy: +(1) THDκ(X): Either X contains a closed homeomorphic copy of κκ, or X can be +covered by κ many κ-compact subsets of κκ. +(2) HDκ(X): Either X contains a κ-superperfect subset, or X can be covered by the +sets of branches of at most κ many <κ-splitting subtrees of <κκ. +Kechris, Louveau and Woodin proved a dichotomy that characterizes when an analytic +set of reals can be separated from a disjoint set by an Fσ set [KLW87, Theorem 4]. Their +theorem strengthens the Hurewicz dichotomy for analytic sets in the sense that one can +derive the latter by a short argument using compactifications.28 Let Rκ denote the set of +elements of κκ that take the value 0 unboundedly often and Qκ the set of those that do +not. Based on an argument in [CMS20], we prove a stronger version of the next theorem +where the sets X and Y are not necessarily disjoint in Theorem 6.19. +Theorem 1.7. ODDκ +κ(X) implies the following analogue of the Kechris-Louveau-Woodin +dichotomy for all subsets Y of κκ disjoint from X: +KLWκ(X, Y ): Either there is a Σ0 +2(κ) set A separating X from Y ,29 or +there is a homeomorphism f between κ2 and a closed subset of κκ that +reduces (Rκ, Qκ) to (X, Y ).30 +We study dihypergraphs with domain κκ in Subsection 6.3. There is no immediate +reason why the restriction ODDκ +κ(κκ) to these dihypergraphs should be useful. While it +implies ODDκ +κ(X) for subsets X of κκ that are continuous images of κκ by Lemma 3.5, for +uncountable κ not all closed subsets of κκ are of this form [LS15, Theorem 1.5]. However, +an argument resembling the proof of Theorem 1.7 shows the next implications. It follows +that ODDκ +κ(κκ, Dκ) has the same consistency strength as an inaccessible cardinal. We will +further see that ODDκ +κ(κκ) suffices for some other applications. +Theorem 1.8. +(1) ODDκ +κ(κκ) ⇒ ODDκ +κ(Σ1 +1(κ)). +(2) ODDκ +κ(κκ, Dκ) ⇒ ODDκ +κ(Σ1 +1(κ), Dκ). +By Cantor-Bendixson analysis, any set of reals can be decomposed as the disjoint union +of a crowded set and a countable scattered set.31 V¨a¨an¨anen showed that an analogue of +this statement for all closed subsets of κκ is consistent32 but not provable [V¨a¨a91]. To +28See [Kec95, Theorem 21.22 & Corollary 21.23] and note that this argument does not generalize to +higher cardinals. +29I.e., X ⊆ A and A ∩ Y = ∅. +30I.e., f(Rκ) ⊆ X and f(Qκ) ⊆ Y . +31Recall that a set of reals X is crowded if it has no isolated points, and if and only if its closure is +perfect. X is scattered if each nonempty subspace contains an isolated point. +32V¨a¨an¨anen used a measurable cardinal and Galgon reduced this to an inaccessible cardinal [Gal16]. +The second-listed author showed that in fact, the restriction to closed sets is equivalent to the perfect set +property PSPκ(X) for closed sets [Szir18]. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +9 +this end, he generalized the notions of scattered and crowded sets to subsets of κκ via a +game Vκ(X) of length κ.33 We extend V¨a¨an¨anen’s result to all subsets of κκ in the next +theorem. This is proved in Theorems 6.31 and 6.35. +Theorem 1.9. ODDκ +κ(κκ) implies the following version of the Cantor-Bendixson decom- +position for all subsets X of κκ: +CB2 +κ(X): X equals the disjoint union of a κ-scattered set of size κ and a +κ-crowded set. +In particular, Vκ(X) is determined for all subsets X of κκ. +Moreover, ODDκ +κ(κκ, Dκ) implies the previous statements for all definable subsets of +κκ. +While there is a verbatim analogue of the Baire property at higher cardinals, it is not +as useful as in the countable setting since even simple sets such as the club filter do not +satisfy it [HS01]. The asymmetric Baire property ABPκ(X) introduced in [Sch17] is a +more general condition that is equivalent to the determinacy of the Banach-Mazur game +of length κ for X. In the countable setting, ABPω(Dω) is thus equivalent to the Baire +property for sets in Dω. The next result is proved in Theorem 6.45. +Theorem 1.10. ODDκ +κ(X, Dκ) implies the asymmetric Baire property ABPκ(X). +The proof of this theorem relies on a characterization of the Baire property via ho- +momorphisms of dihypergraphs. +A new consequence in the countable setting is that +ODDω +ω(X, Dω) implies the Baire Property for X. +A celebrated of Jayne and Rogers characterizes ∆0 +2-measurable functions on the reals +as those that are σ-continuous with closed pieces [JR82, Theorem 5]. A simpler proof of +this theorem was subsequently discovered by Semmes and Motto Ros [RS10]. Its possible +generalizations are a subject of intense study [Ros13, GKN21]. Recently, Carroy, Miller +and Soukup found a new proof that derives the theorem from the open dihypergraph +dihotomy [CMS20]. Their proof allows us to generalize the Jayne-Rogers theorem to the +uncountable setting. +Theorem 1.11. ODDκ +κ(Dκ, Dκ) implies the following analogue of the Jayne-Rogers the- +orem for all X ∈ Dκ: +JRκ(Dκ): Let X ∈ Dκ be a subset of κκ and let f : X → κκ. Then f +is ∆0 +2(κ)-measurable if and only if it is a union of κ many continuous +functions on closed sets. +The above two versions of the Hurewicz dichotomy, the asymmetric Baire property +and the Jayne-Rogers theorem for definable subsets of κκ are consistent relative to an +inaccessible cardinal by Theorem 1.4. The first two statements in particular provide new +proofs of the main results of [Sch17, LMS18]. The Kechris-Louveau-Woodin dichotomy +KLWκ(Dκ),34 the determinacy of V¨a¨an¨anen’s perfect set game and in fact CB2 +κ(X) for +arbitrary subsets X of κκ are consistent relative to a Mahlo cardinal. The latter extends +33More exactly, V¨a¨an¨anen worked with local versions Vξ(X, x) for x ∈ κκ of this game. +34I.e., KLWκ(X, Y ) for all definable X and arbitrary subsets Y of κκ. + +10 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +a result of V¨a¨an¨anen [V¨a¨a91] from closed sets to arbitrary subsets of κκ. Moreover, it +lowers the upper bound for the consistency strength of a dichotomy studied in [SzV17] +from a measurable to a Mahlo cardinal. +2. Preliminaries +2.1. Notation. α, β, γ, δ denote ordinals and κ, λ, µ, ν cardinals. κ always denotes an +infinite cardinal with κ<κ = κ. Lim and Succ denote the classes of limit and successor +ordinals, respectively. If α = β + 1, then α − 1 denotes β. An α-sequence is a function +f : α → V . ⟨c⟩α denotes the constant sequence of length α with value c. For sequences sα +for α < β, � +α<β sα denotes their concatenation. For any set d, α < κ and any sequence +x = ⟨xi : i ∈ d⟩ ∈ d(κκ), let x↾α := ⟨xi↾α : i ∈ d⟩. Given a function f : X → Y , write f(U) +for the pointwise image of a subset U of X under f, and write f−1(W) for the preimage +of a subset W of Y . For any set d, let fd : dX → dY denote the function defined by +letting fd(⟨xi : i ∈ d⟩) := ⟨f(xi) : i ∈ d⟩ for all ⟨xi : i ∈ d⟩ ∈ dX. idX denotes the identity +function on X. +A class X is definable from a set y if X = {x : ϕ(x, y)} for some first order formula +ϕ(v0, v1) with two free variables. Let Dκ denote the class of those sets which are definable +from a κ-sequence of ordinals. By a definable set, we always mean an element of Dκ when +κ is clear from the context. We use the following notation for definable subsets of the +κ-Baire space: If ϕ(v0, v1) is a first order formula with two free variables and y is a set, +write +Xκ +ϕ,y := {x ∈ κκ : ϕ(x, y)}. +We further write Xϕ,y for Xκ +ϕ,y if κ is clear from the context. +Basic open subsets of κd are denoted Nt := {x ∈ κd: t ⊆ x} for any d ≤ κ and t ∈ <κd. +The closure of a subset X of κd is denoted X. The κ-Borel subsets of κd are defined by +closing the set of basic open sets under unions of length κ and complements. The κ-Borel +hierarchy begins with Σ0 +1(κ) (i.e., open) sets. For any γ with 1 < γ < κ+, Σ0 +γ(κ) sets +are of the form � +α<κ Aα, where each Aα is in Π0 +β(κ) for some β < γ. For any γ with +0 < γ < κ+, Π0 +γ(κ) sets are complements of Σ0 +γ(κ) sets and ∆0 +γ(κ) sets are both Σ0 +γ(κ) +and Π0 +γ(κ). κ-analytic or Σ1 +1(κ) subsets of κd are of the form f(X), where f : κκ → κd +is continuous and X is a closed subset of κκ. A Π1 +1(κ) set is a complement of a Σ1 +1(κ) +set. A partial function f : κκ ⇀ κκ is called C-measurable with respect to a collection C +of subsets of κκ if f−1(U) ∈ C for every open subset U of κκ. +2.2. Dihypergraphs. In this subsection, X denotes a set and d a set of size at least 2. +∆d +X denotes the diagonal, i.e., the set of constant sequences in dX. +Definition 2.1. A d-dihypergraph35 on X is a subset H of dX \ ∆d +X. +Let Sym(d) denote the set of all permutations of d. We write xπ := ⟨xπ(α) : α < d⟩ for +any sequence x = ⟨xα : α < d⟩ and π ∈ Sym(d). +Definition 2.2. A d-hypergraph on X is a d-dihypergraph H on X that is closed under +permutation of hyperedges, i.e., xπ ∈ H for all x ∈ H and π ∈ Sym(d). +35This is short for d-dimensional directed hypergraph. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +11 +A digraph on X is then a 2-dihypergraph on X, while a graph on X is a 2-hypergraph +on X, i.e., a symmetric irreflexive binary relation on X. +Definition 2.3. +(a) Kd +X := dX − ∆d +X is the complete d-hypergraph on X. +(b) KX = K2 +X is the complete graph on X. +Definition 2.4. Let H and I be d-dihypergraphs on X and Y , respectively. +(a) A homomorphism from H to I is a function f : X → Y such that fd(H) ⊆ I. +(b) A subset Z of X is H-independent if H ∩ dZ = ∅. +(c) Let λ be a cardinal. A function c : X → λ is a λ-coloring of H if c−1({α}) is +H-independent for all α < λ.36 +Given a family of topological spaces ⟨Xi : i ∈ d⟩, the box-topology on � +i∈d Xi is the +topology generated by sets of the form � +i∈d Ui, where Ui is an open subset of Xi for each +i ∈ d. If X is a subset of κκ, then we always work with the box topology on dX unless +otherwise mentioned. +In the following, let H be a d-dihypergraph on a topological space X. We call H box- +open if it is open as a subset of dX with the box-topology.37 For finite d, we will simply +say that H is open on X. +Definition 2.5. +(a) The domain domH of H is the set of all x ∈ X such that x is an element of some +hyperedge of H.38 +(b) H is relatively box-open if it is a box-open dihypergraph on its domain dom H. +Note that H is box-open on X if and only if H is relatively open and domH is an open +subset of X. +2.3. Trees and order preserving maps. Suppose s, t ∈ ≤κκ and A ⊆ ≤κκ. Let lh(t) := +dom(t) and lh(A) := sup{lh(u) : u ∈ A}. Let t↓ := {s ∈ <κκ : s ⊊ t} and succ(t) := {u ∈ +<κκ : t ⊊ u}. Let s ∧ t denote the maximal r ∈ ≤κκ with r ⊆ s and r ⊆ t. s and t are +called compatible, denoted s∥t, if s ⊆ t or t ⊆ s.39 s and t are called incompatible, denoted +s⊥t, if they are not compatible. Note that s ∧ t is the node where s and t split if s ⊥ t, +and s ∧ t = s if s ⊆ t. For incompatible s and t, let ∆(s, t) := min{α < κ : s(α) ̸= t(α)}. +Then ∆(s, t) = lh(s ∧ t) and s ∧ t = s↾∆(s, t) = t↾∆(s, t). +A subtree of <κκ is a subset that is closed under forming initial segments (i.e., it is +downwards closed). If A is a subset of ≤κκ, let T(A) := {t ∈ <κκ : ∃a ∈ A t ⊆ a} denote +the tree of initial segments t ∈ <κκ of elements of A. If T is a subtree of <κκ, then +[T] := {x ∈ κκ: ∀α < κ x↾α ∈ T} denotes the set of its branches. Note that [T(X)] is the +closure of any subset X of κκ. +36Equivalently, if c is a homomorphism from H to Kd +λ +37We say H is box-closed if it is closed as a subset of Kd +X with the box-topology. Note that H is +box-closed if and only if H ∪ ∆d +X is a closed subset of the space dX. +38More precisely, domH = � +i∈d pi(H), where pi denotes projection onto the ith coordinate. +39For general posets P, this is called comparable. Conditions p, q ∈ P are called compatible if there +exists a common extension r ≤ p, q. + +12 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Definition 2.6. Suppose T is a subtree of <κκ and t ∈ T. +(a) t is splitting in T is there exist u ⊥ v in T with t ⊆ u, v. +(b) T is cofinally splitting if for each t ∈ T, there exists a splitting node u ∈ T with +t ⊆ u. +(c) T is <κ-closed if for any strictly increasing sequence t = ⟨ti : i < α⟩ in T with +α < κ, there exists an upper bound t ∈ T for t. +(d) T is κ-perfect if it is cofinally splitting and <κ-closed. +(e) T is pruned if every node t ∈ T extends to a branch x ∈ [T]. +(f) A closed subset X of κκ is κ-perfect if T(X) is κ-perfect. +The κ-perfect set property (κ-PSP) holds for a subset X of κκ if either |X| ≤ κ or X +contains a κ-perfect subset. PSPκ(C) states that the κ-PSP holds for all subsets X ∈ C +of κκ. +Suppose that P and Q are posets. A function ι: P → Q is called strict order preserving +if p < q implies ι(p) < ι(q) for all p, q ∈ P and strict order reversing if p < q implies +ι(p) > ι(q) for all p, q ∈ P. +Definition 2.7. Let ι be a partial function from <κκ to <κκ. +(a) ι is ⊥-preserving if s ⊥ t implies ι(s) ⊥ ι(t) for all s, t ∈ dom(ι). +(b) Suppose that ι is strict order preserving. ι is called continuous if ι(t) = � +s⊊t ι(s) +for all t ∈ dom(ι) with lh(t) ∈ Lim and t↓ ⊆ dom(ι). +Definition 2.8. Suppose that ι is a strict order preserving partial function from <κκ to +<κκ. +(a) Let [ι] denote the partial function from κκ to κκ where dom([ι]) consists of those +x ∈ κκ with x↾α ∈ dom(ι) for unboundedly many α < κ, and for all x ∈ dom([ι]) +[ι](x) = � +α<κ +ι(x↾α). +(b) Let T(ι) := T(ran(ι)) denote the tree of initial segments of elements of ran(ι). +We will usually assume that dom(ι) is a subtree of <κκ, in which case dom([ι]) = +[dom(ι)]. +It is clear from the definitions that T(ι) = T(ran([ι])). +Thus, [T(ι)] is the +closure of ran([ι]) in κκ. +Lemma 2.9. Let ι be a strict order preserving partial function from <κκ to <κκ. +(1) [ι] is a continuous function from dom([ι]) to κκ. +(2) If ι is ⊥-preserving, then [ι] is a homeomorphism between dom([ι]) and ran([ι]). +Proof. Since [ι] is strict order preserving, we have [ι] +� +Nt ∩ dom([ι]) +� +⊆ Nι(t) for all t ∈ +dom(ι). Item (1) follows easily from this observation. To show (2), Suppose that ι is +⊥-preserving. Then Nι(s) ∩ Nι(t) = ∅ for all s, t ∈ dom(ι) such that s ⊥ t. Therefore [ι] is +injective, and [ι] +� +Nt ∩ dom([ι]) +� += Nι(t) ∩ ran([ι]) for all t ∈ dom(ι). This implies that [ι] +is a homeomoprhism between dom([ι]) and ran([ι]). +□ + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +13 +2.4. Forcing. A forcing P = (P, ≤P, 1P) is a triple such that ≤P is a pre-order (i.e., a +reflexive transitive relation) on P and 1P is a largest element of P with respect to ≤P. +We confuse P with its domain P, and we usually write ≤, ⊥, and ⊩ instead of ≤P, ⊥P, +1P and ⊩P, respectively. For all p, q ∈ P, we let p ∥ q (or sometimes p ∥P q) denote the +statement that p and q are compatible (i.e., that p ̸⊥ q). We let B(P) denote the Boolean +completion of P.40 If P is separative, we assume that P is a dense subset of B(P). +Definition 2.10. +(a) An atom in a forcing P is a condition p ∈ P with no incompatible extensions. +Moreover, a forcing P is non-atomic if it has no atoms. +(b) A forcing P is homogeneous if for all p, q ∈ P, there is an automorphism π: P → P +such that π(p) and q are compatible. +Definition 2.11. +Suppose P, Q are forcings. +(a) A dense embedding ι : P → Q is a homomorphism with respect to ≤, ⊥ and 1 +such that ι(P) is a dense subset of Q. +(b) Two forcings P and Q are equivalent (P ≃ Q) if there exist dense embeddings +ι : P → R and ν : Q → R into some forcing R. +(c) Let ι : P → Q be a dense embedding. We define a P-name σι for each Q-name σ +by recursion on the rank as +σι := +� +(τ ι, p) : p ∈ P, ∃q ∈ Q +� +ι(p) ≤ q ∧ (σ, q) ∈ τ +�� +. +It is easy to check that in Definition 2.11 (c), if G is a P-generic filter over V and H is +the upwards closure of ι(G) in Q, then we have (σι)G = σH. +We recall the following standard facts (see e.g. [Kun13]). Two forcings are equivalent +if and only if they have isomorphic Boolean completions. Suppose ι : P → Q is a dense +embedding between forcings P and Q. Given a P-generic filter G over V , the upwards +closure H of ι(G) in Q is a Q-generic filter over V , and G = ι−1(H). Conversely, suppose +H is a Q-generic filter over V . Then G = ι−1(H) is a P-generic filter over V , and H is +equal to the upwards closure of G in Q. In both of the above cases, we have V [G] = V [H]. +The following definition of the forcing for adding Cohen subsets is non-standard, but +is essential in several arguments below. +Definition 2.12. Suppose that κ is a regular uncountable cardinal. +(a) Add(κ, 1) is defined as the forcing +Add(κ, 1) := {p : α → κ | α < κ}, +ordered by reverse inclusion. +(b) Add(κ, ξ) is defined as the <κ-support product � +i<ξ Add(κ, 1) for any ordinal ξ. +We use the following convention. For a given y ∈ κκ, we sometimes confuse y with +the set y↓ = {t ∈ <κκ : t ⊊ y}. For example, we say that y is Add(κ, 1)-generic if and +only if y↓ is an Add(κ, 1)-generic filter. Furthermore, if y is Add(κ, 1)-generic and σ is an +40Note that B(P) is unique up to isomorphism. + +14 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Add(κ, 1)-name, let σy := σy↓. We also let � +i<ξ yi denote � +i<ξ yi↓ whenever ⟨yi : i < ξ⟩ +is a sequence of elements of κκ. +We will often use the standard facts about adding Cohen subsets and collapse forcings +which are found in Lemmas 2.13 and 2.14 below. +Lemma 2.13. +Suppose that κ is an uncountable cardinal such that κ<κ = κ. If P is a +non-atomic <κ-closed forcing of size κ, then P has a dense subset which is isomorphic to +the dense subforcing +Add∗(κ, 1) = {p ∈ Add(κ, 1) : dom p ∈ Succ} +of Add(κ, 1). In particular, P is equivalent to Add(κ, 1). +We omit the proof of Lemma 2.13, since is straightforward and well-known. +Lemma 2.14 ([Fuc08, Lemma 2.2]). Let κ be a regular cardinal. Suppose that ν > κ is +a cardinal with ν<κ = ν. Let P be a separative <κ-closed forcing of size ν which forces +that ν has size κ. Then P has a dense subset which is isomorphic to the dense subforcing +Col∗(κ, ν) = {p ∈ Col(κ, ν) : dom p ∈ Succ} +of Col(κ, ν). In particular, P is equivalent to Col(κ, ν). +Lemma 2.14 can be proven using an adaptation of the proof of [Jec03, Lemma 26.7] +which can be found in the proof of [Fuc08, Lemma 2.2]. +We will use the following notation for subforcings of the L´evy collapse Col(κ, < λ). +Suppose that κ < λ are cardinals, α ≤ λ and I ⊆ λ is not an ordinal (to avoid a conflict +with the notation for the standard collapse). Then let Pα := Col(κ, <α) and +PI = Col(κ, I) = {p ∈ Col(κ, < λ) : dom p ⊆ I × κ}. +The notation PI = Col(κ, I) will be used for intervals I, for which we use the standard +notation +(α, γ) = {β ∈ Ord : α < β < γ} +[α, γ) = {β ∈ Ord : α ≤ β < γ}. +In particular, let Pα := P[α,λ) for all α < λ. If G is a Pλ-generic filter over V , we write +Gα := G ∩ Pα, GI := G ∩ PI and Gα := G ∩ Pα. We will also often use the following +consequence of Lemma 2.14. +Corollary 2.15. Let λ > κ be an inaccessible cardinal and γ < λ. If P is a <κ-closed +forcing of size <λ, then the forcings P × Pλ and P[γ,λ) are equivalent. +Corollary 2.15 is a variant of [Fuc08, Corollary 2.4]. (More specifically, [Fuc08, Corol- +lary 2.4] asserts that the conclusion of Corollary 2.15 holds when γ = 0, under the weaker +assumption that λ > κ is a cardinal such that for all cardinals µ < λ, we have µ<κ < λ.) +Corollary 2.15 follows easily from Lemma 2.14 using a straightforward analogue of the +proof of [Fuc08, Corollary 2.4] (see also [Fuc08, Corollary 2.3]). + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +15 +Definition 2.16. Suppose that P, Q are forcings. +(a) A complete embedding i: P → Q is a homomorphism with respect to ≤, ⊥ and +1 with the property that for every q ∈ Q, there is a condition p ∈ P (called a +reduction of q to P) such that for every r ≤ p in P, i(r) is compatible with q. +(b) P is a complete subforcing of Q (P ⋖ Q) if P is a subforcing of Q and the inclusion +map idP : P → Q is a complete embedding. +(c) Suppose that i: P → Q is a complete embedding and G is P-generic over V . The +quotient forcing Q/G for G in Q is defined as the subforcing +Q/G := {q ∈ Q : ∀p ∈ G i(p) ∥ q} +of Q. Moreover, we fix a P-name Q/P for for the quotient forcing for ˙G in P, +where ˙G is the canonical P-name for the P-generic filter. We also refer to Q/P as +(a name for) the quotient forcing for P in Q. +We recall the following standard facts (see e.g. [Kun13] and Exercises (C7), (C8), (D4) +and (D5) in [Kun80, Chapter VII]). +Fact 2.17. Suppose P and Q are forcings and i : P → Q. +(i) If i is a homomorphism with respect to ≤, ⊥ and 1, then i is a complete embedding +if and only if for all maximal antichains A of P, i(A) is a maximal antichain of Q. +If P and Q are complete Boolean algebras, then i is a complete embedding (in the +sense of Definition 2.16) if and only if i is an injective complete homomorphism +of Boolean algebras. +(ii) Any complete embedding i : P → Q defines a complete embedding j : B(P) → +B(Q). If P and Q are separative, we may assume that i ⊆ j. Specifically, if P is a +complete subforcing of Q and Q is separative, we may assume that B(P) ⊆ B(Q). +(iii) Suppose i is a complete embedding. Let G be a P-generic filter over V . If D +is a dense subset of Q, then D ∩ Q/G is a dense subset of Q/G. Therefore all +Q/G-generic filters H over V [G] are also Q-generic filters over V , and G = i−1(H) +holds for all such H. Conversely, if H is Q-generic over V and G = i−1(H), then G +is a P-generic filter over V and H is Q/G-generic over V [G]. Furthermore, under +either of the above assumptions, we have V [H]Q = V [G][H]Q/G. +Definition 2.18. Given complete Boolean algebras P and Q and a complete embedding +i : P → Q, the retraction associated to i is the map πi : Q → P defined by letting +πi(q) := �P{p ∈ P : i(p) ≥ q} +for all q ∈ Q. +The following lemma is standard, but we include a proof for the reader. +Lemma 2.19. Suppose i : P → Q is a complete embedding between the complete Boolean +algebras P and Q. +(1) For all q ∈ Q, πi(q) is the largest reduction of q to P. +(2) If G is a P-generic filter over V , then q ∈ Q/G if and only if πi(q) ∈ G. + +16 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Proof. Let q ∈ Q. First, observe that i(p) ⊥ q if and only if p ⊥ πi(q) holds for all p ∈ P. +Indeed, given p ∈ P, we have i(p) ⊥ q iff i(p) ∧ q = 0 iff ¬i(p) ≥ q. By the definition of πi +and since i is a Boolean homomorphism, the last statement is equivalent to ¬p ≥ πi(q), +and is therefore also equivalent to p ∧ πi(q) = 0 and to p ⊥ πi(q). +The above observation easily implies that πi(q) is a reduction of q to P. To see that πi(q) +is the largest reduction of q to P, suppose that p ∈ P and p ̸≤ πi(q). Let r := p ∧ ¬πi(q). +Then p ≥ r ≥ 0P and r ⊥ πi(q), and therefore i(r) ⊥ q. Thus, r witnesses that p is not a +reduction of q to P. +The above observation also implies that if G is a P-generic filter over V and q ∈ Q, +then q ∈ Q/G holds if and only if p ⊥ πi(q) holds for all p ∈ P, and therefore if and only +if πi(q) ∈ G. +□ +As usual, by a name σ for a subset of a set x, we mean a name σ such that ⊩ σ ⊆ x. +By a name σ for an element of κκ, we mean a name σ such that ⊩ σ ∈ κκ. By a name σ +for a new object, we mean a name σ such that ⊩ σ /∈ ˇV . In general, if ϕ(v) is any formula +with one variable v, then by a name σ for an object with property ϕ, we mean a name σ +such that ⊩ ϕ(σ). +We will use the following notation when working with quotient forcings induced by +names. +Definition 2.20. Suppose Q is a complete Boolean algebra and σ is a Q-name for a +subset of a set x. Let B(σ) := BQ(σ) denote the complete Boolean subalgebra of Q that +is completely generated by the set of Boolean values {�y ∈ σ�Q : y ∈ x}. +The following lemma is standard, but we include a proof for the reader. +Lemma 2.21. Suppose that Q is a complete Boolean algebra and σ is a Q-name for a +subset of a set x. If H is a Q-generic filter over V , then V +� +σH� += V +� +BQ(σ) ∩ H +� +. +Proof. Let A := {�y ∈ σ�Q : y ∈ x}. By [Jec03, Lemma 15.40], we have V +� +BQ(σ) ∩ H +� += +V +� +A ∩ H +� +. The fact that V +� +A ∩ H +� += V +� +σH� +follows from the observation that for all +y ∈ x, we have y ∈ σH iff �y ∈ σ�Q ∈ H iff �y ∈ σ�Q ∈ A ∩ H. +□ +In the next definition and lemmas, suppose that Q ∈ V is a forcing, q ∈ Q and σ ∈ V +is a Q-name for an element of κκ. +Definition 2.22. +(a) σ[q] := �{t ∈ <κκ : q ⊩ t ⊆ σ}. +(b) σ(q) := {(α, β) : q ⊩ (α, β) ∈ σ}. +(c) The tree T σ,q of possible values for σ below q is defined as follows: +T σ,q := {t ∈ <κκ : ∃r ≤ q t ⊆ σ[r]}. +For clarity, we at times write σ[q,Q], σ(q,Q) and T σ,q +Q +instead of σ[q], σ(q) and T σ,q, respec- +tively. +The next two lemmas list basic properties of σ[q] and T σ,q. The first one follows imme- +diately from the definitions. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +17 +Lemma 2.23. +(1) σ[q] ⊆ σq, and σ[q] = σ(q) if and only if dom(σ(q)) is an ordinal. +(2) σ[q]↾α = σ(q)↾α for all ordinals α with α ⊆ dom(σ(q)). +Lemma 2.24. +(1) If q ̸⊩Q σ ∈ ˇV , then σ[q] ∈ <κκ equals the stem of T σ,q. +(2) q ⊩V +Q σ ∈ [T σ,q]. +(3) If S is a subtree of <κκ with q ⊩V +Q σ ∈ [S], then T σ,q ⊆ S. +(4) If M ⊇ V is a transitive model of ZFC with (<κκ)V = (<κκ)M, then +(T σ,q)V = (T σ,q)M. +Proof. (1)-(3) are immediate. (4) holds since the formula “q ⊩ t ⊆ σ” is absolute between +M and V , as it can be defined by a recursion which uses only absolute concepts [Kun13, +Theorem II.4.15]. +□ +3. Dihypergraphs and homomorphisms +Subsection 3.1 contains several claims on the dichotomy ODDd +κ(X, H) which will be +used in some later arguments. In Subsection 3.2, we give an equivalent characterization +of the existence continuous homomorphisms from Hκd to box-open dihypergraphs H via +certain strict order preserving maps. This characterization will be an important ingredient +throughout the rest of the paper. We also characterize H-independence at the level of +subtrees of <κκ. +3.1. Basic facts. Throughout this subsection, we assume that 2 ≤ d ≤ κ. +We first +observe that the two options in the definition of ODDH +κ are mutually exclusive. +Lemma 3.1. Let H be a d-dihypergraph on a topological space X. Suppose that there is +a homomorphism f : κd → X from Hκd to H. +(1) H does not have a κ-coloring. +(2) H↾f(Nt) does not have a κ-coloring for any t ∈ <κd. +Note that (1) implies that |X| > κ and (2) implies that |f(Nt)| > κ for all t ∈ <κd. +Proof. The proof of (1) is a straightforward analogue of the proof for the κ = ω case in +[CMS20, Theorem 1]. We give the details for completeness. Suppose c′ : X → κ is a +κ-coloring of H, and let c := c′ ◦ f. Since f is a homomorphism from Hκd to H, c is a +κ-coloring of Hκd. We recursively define a continuous increasing sequence ⟨tα : α < κ⟩ +such that tα ∈ αd and c−1({α})∩Ntα+1 = ∅ for each α < κ. We use the Hκd-independence +of c−1({α}) at stage α + 1 of the construction. Then � +α<κ tα is an element of κd which +is not in c−1({α}) for any α < d, a contradiction. +(2) follows from (1), since the map ft : κd → f(Nt); x �→ f(t⌢x) is a homomorphism +from Hκd to H↾f(Nt) for any given t ∈ κd. +□ +We did not need to assume that f is continuous or that H is box-open. Therefore, +if ODDH +κ holds and there exists any homomorphism from Hκd to H, then there already +exists a continuous one. This is analogous to the fact that for any subset X of κκ that + +18 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +satisfies the perfect set property, the existence of an injective function from κ2 to X +already implies the existence of a continuous injective function. +Lemma 3.2. Let 2 ≤ d < κ and X ⊆ κκ. +(1) If U is a box-open subset of d(κκ) then U ∈ Dκ. +(2) ODDd +κ(X) is equivalent to ODDd +κ(X, Dκ). +Proof. For (1), note that U is given by a sequence x: κ → d(<κκ) since the base of the +topology has size |d(<κκ)| = κ. Since x can be coded by an element of κ2, we have U ∈ Dκ. +Moreover, (2) follows from (1). +□ +The next two lemmas give reformulations of ODDd +κ(Dκ) and ODDd +κ(Dκ, Dκ). Recall +that H is a relatively box-open dihypergraph if it is box-open on its domain domH. +Lemma 3.3. ODDd +κ(Dκ) is equivalent to the statement that ODDH +κ holds for all relatively +box open d-dihypergraphs H with domH ∈ Dκ. +Proof. This follows from the observation that a dihypergraph H is relatively box-open +with domH ∈ Dκ if and only if H = H′↾X for some subset X ∈ Dκ of κκ and some +box-open d-dihypergraph H′ on κκ. To see the direction from right to left, note that +domH′↾X is a relatively open subset of X. +□ +Lemma 3.4. +ODDd +κ(Dκ, Dκ) is equivalent to the statement that ODDH +κ holds for all +relatively box-open d-dihypergraphs H ∈ Dκ. +Proof. This follows from the equivalence of the following statements for any d ≤ κ and +any d-dihypergraph H on κκ: +(a) H ∈ Dκ is relatively box-open. +(b) H = H′↾X for some subset X ∈ Dκ of κκ and some box-open d-dihypergraph +H′ ∈ Dκ on κκ. +For the implication (a)⇒(b), note that X := domH ∈ Dκ and +H′ := �{ � +α κ is any cardinal that +is not Mahlo. We claim that there exists a function F : κκ → κκ in V [G] such that +F↾V [Gα] /∈ V [Gα] for all α with κ ≤ α < λ. +First suppose λ is singular in V . Then λ remains singular in V [G]. Hence λ ̸= (κ+)V [G] +and λ is collapsed in V [G], so there exists some x ∈ (κκ)V [G] with x /∈ V [Gα] for all α < λ. +Now suppose λ is regular in V . Let C be a club in λ whose elements are not inaccessible. +Moreover, if λ is a successor cardinal then we assume C does not contain cardinals, while +if λ is a limit cardinal, then C only contains cardinals. For any α < λ, let α′ denote the +least ordinal above α in C. Take any function F : κκ → κκ in V [G] with the following +property: if α < λ is least with x ∈ V [Gα], then F(x) /∈ V [Gα′]. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +25 +We show that F↾V [Gα] /∈ V [Gα] for all α with κ ≤ α < λ. To see this, fix any such α. +We distinguish two cases: +Case 1. α is not a limit point of C. +Then there is some γ ∈ C with γ < α ≤ γ′. Take any x ∈ V [Gα] \ V [Gγ]. Let β > γ +be least with x ∈ V [Gβ]. Then F(x) /∈ V [Gβ′] by the definition of F. Since α ≤ γ′ ≤ β′, +we have F(x) /∈ V [Gα] as required. +Case 2. α is a limit point of C. +We claim that there is some x ∈ κκ such that α is least with x ∈ V [Gα]. First suppose +that λ is successor cardinal in V .50 Let µ := |α|V , so µ < α < µ+V . Since |α| = κ +in V [Gµ+1], G(µ+1,α) can be coded as a subset x of κ via a bijection f : κ → α with +f ∈ V [Gµ+1]. Then α is least with x ∈ V [Gα]. Now suppose λ is a limit cardinal in V . +Since C contains only cardinals and the GCH holds, α is singular in V and the claim +follows as in the case of singular λ. Thus α ̸= (κ+)V [Gα] and α is collapsed in V [Gα], so +the claim follows. +Then F(x) /∈ V [Gα′] by the definition of F. Thus, F(x) /∈ V [Gα] as required. +Remark 4.4. The proof of Lemma 4.1 (2) can be adapted to show that if λ > κ is +inaccessible, then the following strengthening of ♦κ+ holds in Pλ-generic extensions V [G]: +♦+ +κ+: there exists a sequence ⟨Aα : α < κ+⟩ of sets Aα ⊆ P(α) such that +(a) |Aα| ≤ κ for all α < κ+, and +(b) for all X ⊆ κ+, there exists a club C in κ+ such that for all α ∈ C, +X ∩ α ∈ Aα and C ∩ α ∈ Aα. +We show that in V [G], ⟨Aα = P(α)V [Gα] : α < κ⟩ is a ♦+ +κ+-sequence. It is clear that (a) +holds in V [G]. +To show (b), we use an argument similar to the proof of Lemma 4.1 (2). We note that +this version of the argument works assuming only that λ is inaccessible. (In particular, +we do not need to assume that λ is Mahlo.) +Let X be a subset of κ+ = λ in V [G], and let ˙X ∈ V be a Pλ-name with ˙XG = X. +Arguing in V , we first define a function f : λ → λ as follows. For each α < λ, let A be +a maximal antichain in Pλ consisting of conditions which decide “α ∈ ˙X”. Since Pλ has +the λ-c.c., there is some µ < λ such that A ⊆ Pξ. Let f(α) := µ. +Let C be a club of limit ordinals ν < λ that are closure points of f. We will show that +C satisfies (b). Since C ∈ V , it suffices to show that X ∩ ν ∈ Aν = P(ν)V [Gν] for any +given ν ∈ C. +Claim. For any α < ν, exactly one of the following holds: +(i) ∃p ∈ Gν p ⊩V +Pλ α ∈ ˙X. +(ii) ∃p ∈ Gν p ⊩V +Pλ α /∈ ˙X. +Proof. By the definition of f, there is A ⊆ Pf(γ) such that A is a maximal antichain in +Pλ of conditions deciding α ∈ ˙X. Since ν is a closure point of f, we have f(γ) < ν. Thus +A is a maximal antichain in Pν. Let p ∈ Gν ∩ A. +□ +50For successor cardinals λ, the proof does not use that α is a limit point of C, so Case 1 is not needed. + +26 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +As in the proof of Lemma 4.1 (2), it now follows that for all α < ν, +α ∈ X ∩ ν ⇐⇒ ∃p ∈ Gν p ⊩V +Pλ α ∈ ˙X. +Thus, X ∩ ν is an element of Aν = P(ν)V [Gν]. +4.1.2. Names and witnessing functions. +Assumptions 4.5. The following assumptions will be used throughout the rest of this +section, unless explicitly stated otherwise. In V : +• κ is a regular infinite cardinal, 2 ≤ d ≤ κ and λ > κ is an inaccessible cardinal. +G is a Pλ = Col(κ, <λ)-generic filter over V . +• κ<κ = κ holds in V . This may be assumed without loss of generality, because P2 +forces κ<κ = κ and P2 is equivalent to Pλ. +In V [G]: +• X is a subset of κκ in Dκ. More specifically +X = Xϕ,a = {x ∈ κκ : ϕ(x, a)}, +where ϕ(x, a) is a first order formula with a parameter a ∈ κOrd. +• H is a box-open d-dihypergraph on κκ such that H↾X has no κ-coloring. +• λ is a Mahlo cardinal in V , or H ∈ Dκ in V [G].51,52 +To prove Theorem 1.4, it is sufficient to prove that ODDH↾X +κ +holds under these assump- +tions. Since we assume that H↾X has no κ-coloring, it is enough to show that there is a +continuous homomorphism from Hκd to H↾X. +We work in V [G] unless mentioned otherwise. Let T be a subtree of <κκ. Recall from +Definition 3.12 that T is an H-independent tree if for all sequences ⟨tα ∈ T : α < d⟩, we +have � +α ω). +We assume κ > ω throughout the section. +61For κ = ω, Pλ equals Col(ω, <λ), where λ is inaccessible. +62Here, we identify x with x↓ = {t ∈ Add(ω, 1) : t ⊊ x}. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +33 +4.3.1. κ-analytic sets. We begin with the special case of Theorem 1.4 for κ-analytic +sets. The proof is much less complicated than the general case, since we will not need the +Quotient Lemma. It adapts an argument from [Szir18, Section 3.1] for the open graph +dichotomy OGDκ +� +Σ1 +1(κ) +� +. We first give a proof for closed subsets of κκ. It resembles the +proof of the countable case of Theorem 1.4. +Lemma 4.22. In V [G], suppose X is a closed subset of κκ. Then ODDH↾X +κ +holds. +Proof. Recall our Assumptions 4.5. In particular, we assume that H↾X has no κ-coloring. +It suffices to show that there is a continuous homomorphism f : κd → X from Hκd to +H↾X. +In V [G], let S be a subtree of <κκ such that [S] = X = Xϕ,a. There is some α < λ such +that S ∈ V [Gα], since Pλ has the λ-c.c. Now let γ > α and let τ be an Add(κ, 1)-name +as in Lemma 4.11. In V [Gγ], 1Add(κ,1) forces that 1Pλ ⊩ ˇτ ∈ [S] by Lemma 4.11 (4.4). +Since the formula “x ∈ [S]” is absolute between transitive models of ZFC, we have +(4.7) +1Add(κ,1) ⊩V [Gγ] τ ∈ [S]. +Work in V [G]. We construct functions h : <κd → Add(κ, 1) and t : <κd → <κκ with +the following properties for all u, v ∈ <κd: +(i) If u ⊊ v, then t(u) ⊊ t(v) and h(v) ≤ h(u). +(ii) h(u) ⊩V [Gγ] t(u) ⊆ τ. +(iii) � +α κ and +ordinals d with 2 ≤ d ≤ κ, ODDId +κ(Dκ, Dκ) holds in all Pλ-generic extensions of V . If +additionally λ is Mahlo, then we obtain ODDId +κ(Dκ). +To prove this, recall that after the Quotient Lemma 4.16, we defined a continuous +homomorphism f from Hκd to H↾X by letting f(x) := τ [h](x), where γ and τ are as in +Lemma 4.11. In particular, note that τ is a name in V [Gγ] for a new element of κκ. It +remains to show that f is injective. Let x, y be distinct elements of κd, let x′ := [h](x) and +y′ := [h](y). Then f(x) = τ x′ ∈ V [Gγ][x′] − V [Gγ] and f(y) = τ y′ ∈ V [Gγ][y′] − V [Gγ]. +Since x′ and y′ are mutually generic over V [Gγ], we have V [Gγ][x′] ∩ V [Gγ][y′] = V [Gγ]. +Therefore f(x) ̸= f(y). +Theorem 5.35 and Corollary 5.36 below provide an alternative proof using a weak ♦ +principle. +Remark 4.29. Note that the Quotient Lemmas 4.27 and 4.26 both fail in the countable +case, since ODDIω +ω(Dω, Dω) fails by Proposition 5.32 below. While Lemma 4.27 does hold +without the last condition on mutual genericity by Lemma 4.20, we do not know if this +is the case for Lemma 4.26. +65See Corollary 2.15. +66See Corollary 2.15. +67See Definition 5.24 below. + +36 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +4.3.3. The Quotient Lemma for Add(µ, 1). This subsection is dedicated to the proof +of the Quotient Lemma 4.26 for Add(µ, 1). Our proof uses ideas from [Sch17]. +Throughout this subsection, we make the same assumptions as in Lemma 4.26. In +particular, we assume that M is a transitive class model of ZFC. In M, we let µ be +an uncountable cardinal with µ<µ = µ, we assume that 2 ≤ δ ≤ µ, and we let s : +Add(µ, 1) × δ → Add(µ, 1) be a step function for Add(µ, 1).68 +We first give a short description of our proof of Lemma 4.26. +Working in M, we +define a forcing Q whose domain contains partial functions of size <µ approximating an +s-function.69 We will first prove that Q is equivalent to Add(µ, 1). We then show that if h +is the s-function added in the natural way by a given Q-generic filter, then h satisfies the +requirements in Lemma 4.26. A significant portion of the arguments in this subsection +are aimed at proving the second requirement that elements of ran([h]) have well-behaved +quotients. +We now present the detailed proof. We argue either in M or in a Q-generic extension +of M throughout this subsection. +Recall that an s-function for Add(µ, 1) is a strict order preserving map h : <µδ → <µµ 70 +which is “built along” s in the sense of Definition 4.15. In the next definition, we consider +partial functions with the same property. Our forcing Q will consist of such partial s- +functions of size <µ which satisfy the additional technical requirement of “strictness” +(also defined below). Strictness will be needed for the separativity of Q. +Definition 4.30. +(a) A partial s-function is a strict order preserving partial function p : <µδ ⇀ <µµ +such that dom(p) is a subtree of <µδ, and if u, u⌢⟨α⟩ ∈ dom(p), then +p +� +u⌢⟨α⟩ +� +⊋ s +� +p(u), α +� +. +(b) A partial s-function p is strict if for all t ∈ dom(p) with lh(t) ∈ Lim, we have +p(t) ⊋ � +u⊊t +p(u). +We now define the forcing Q which will be used in our proof of the Quotient Lemma 4.26 +for Add(µ, 1). +We also define a larger partial order Q, since it will be useful in our +argument. +Definition 4.31. +(a) Let the domain of Q consist of those strict partial s-functions q such that |dom(q)| < +µ. +(b) Let the domain of Q consist of all strict partial s-functions. +(c) We equip both Q and Q with the following the partial order ≤: Let p ≤ q if the +following conditions hold: +(i) dom(q) ⊆ dom(p). +(ii) q(t) = p(t) for every non-terminal node t of dom(q). +68See Definition 4.13 +69See Definition 4.15. +70Note that <µµ is Add(µ, 1) with the ordering reversed. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +37 +(iii) q(t) ⊆ p(t) for every terminal node t of dom(q). +Thus, a condition q is extended by moving its topmost values upwards and end- +extending the tree dom(q). Note that the definition of Q is absolute to transitive ZFC- +models N ⊇ M with (<µµ)M = (<µµ)N. Specifically, since Q is <µ-closed (by Lemma 4.36 +below), the definition of Q is absolute between M and Q-generic extensions of M. How- +ever, Q +N may contain total s-functions h : <µδ → <µµ which are not in M. +We now discuss some properties of Q which will be useful in this subsection. The next +lemma gives an equivalent reformulation of the definition of strict partial s-functions (i.e., +of the definition of Q). +Lemma 4.32. Let q : <µδ ⇀ <µµ be a partial function whose domain is a subtree of <µδ. +Then q is a strict partial s-function (i.e., q ∈ Q) if and only if the following statement +holds for all t ∈ dom(q): +(4.8) +q(t) ⊋ +� +α ht(dom pn), and +(iv) lh(ran pn+1) > lh(ran pn). +Let p := �{pn : n < ω}. By Lemma 4.34 (4), we have p ∈ Q and p ≤ q. To prove that +p ∈ Q∗, let i < ξ. We show that p ∈ Q∗(ϑi, σi). The fact that ht(dom p) ∈ Lim and +lh(ran p) ∈ Lim follows from (iii), (iv) and Lemma 4.42. Furthermore, (i) implies that +ht(dom pn) ⊆ dom (ϑi)(pn+1) and lh(ran pn) ⊆ dom (σi)(pn+1) hold for all n < ω. From +this and Lemma 4.42, we obtain: +ht(dom p) = � +n<ω +ht(dom pn) ⊆ � +n<ω +dom (ϑi)(pn+1) ⊆ dom (ϑi)(p), +lh(ran p) = � +n<ω +lh(ran pn) ⊆ � +n<ω +dom (σi)(pn+1) ⊆ dom (σi)(p). +Thus, (a) and (b) in Definition 4.41 hold for ϑi, σi and p. To show that (c) also holds, +let γ < ht(dom p). +Then γ ≤ ht(dom pn) for some n < ω. +This implies by (i) that +γ ⊆ dom (ϑi)(pn+1) or equivalently, that γ ≤ lh +� +(ϑi)[pn+1] +� +. Let +t = (ϑi)(pn+1)↾γ = (ϑi)[pn+1]↾γ. +Then t ∈ dom(pn+2) ⊆ dom(p) by (ii) and because p ≤ pn+2. Furthermore, we have +t ⊆ (ϑi)(pn+1) ⊆ (ϑi)(p). Therefore (ϑi)(p)↾γ = t ∈ dom(p), as required. +□ +Let p ∈ Q∗(ϑ, σ), where ϑ and σ are Q-names such that 1Q ⊩ ϑ ∈ µδ ∧ [˙h](ϑ) = σ. +Lemma 4.44 below is a stronger form of the following observation: Suppose that t is the +minimal initial segment of ϑ[p] such that t /∈ dom(p).80 Then p can be extended as follows +to an element r = p ∪ {(t, u)} of Q by choosing r(t) = u in such a way that r remains +a strict partial s-function. Since p ∈ Q∗(ϑ, σ), we have lh(t) = ht(dom p) ∈ Lim. By +Lemma 4.32 it suffices to have u ⊋ v, where +v = � � +p(t↾α) : α < ht(dom p) +� +. +In this way, we may obtain an extension r ∈ Q of p with σ(r) ⊇ u. Corollary 4.45 will +show that v = σ(p). +We state Lemma 4.44 for <µ many pairs of names (ϑi, σi), since this version is used in +some arguments below. +Lemma 4.44. Using the same assumptions as in Lemma 4.43, suppose that p ∈ Q∗. +For all i < ξ, let ti = (ϑi)(p)↾ht(dom p) and let ui be an arbitrary node of <µµ such that +ui ⊋ � � +p(ti↾α) : α < ht(dom p) +� +. Let +r = p ∪ {(ti, ui) : i < ξ}, +80t exists, since ϑ[p] /∈ dom(p) by requirement (a) in the Definition 4.41 of Q∗(ϑ, σ). For example, if ϑ +is a canonical name for an element x of (µµ)V , then ϑ[p] = x. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +43 +If ti ̸= tj for all i < j < ξ, then the following hold: +(1) r ∈ Q and r ≤ p. +(2) r ⊩ ti ⊆ ϑi for all i < ξ.81 +(3) r ⊩ ui ⊆ σi for all i < ξ. +Proof. We first show (1). Since p ∈ Q∗, we have for all i < ξ that ti is a minimal element +of <µδ which satisfies ti /∈ dom(p). This fact and the assumptions that ξ < µ and that +ti ̸= tj for all i < j < ξ imply that r is a partial function from <µδ to <µµ whose domain is +a subtree of <µδ of size <µ. Because p ∈ Q∗, we also have that lh(ti) = ht(dom p) ∈ Lim. +Thus, our assumption about the values ui = r(ti) (where i < ξ) guarantees that r is a +strict partial s-function. Therefore r ∈ Q, and it is clear that r ≤ p. +(2) holds because r ≤ p and p ⊩ ti ⊆ ϑi as ti ⊆ (ϑi)(p) for all i < ξ. +To show (3), let i < ξ. We have r ⊩ ˙h(ti) ⊆ [˙h](ϑi) = σi by (2) and the definition of +[˙h]. Furthermore, since r ⊩ ˙h ≤ r holds by Lemma 4.39, r also forces that ui ⊆ ˙h(ti). +Therefore r ⊩ ui ⊆ σi. +□ +The next corollary states that for p ∈ Q∗, σ(p) can be calculated from p and ϑ. +Corollary 4.45. Suppose that Q∗ = Q∗(ϑ, σ), where ϑ and σ are Q-names such that +1Q ⊩ ϑ ∈ µδ ∧ σ = [˙h](ϑ). For all p ∈ Q∗, we have +σ(p) = � � +p(ϑ(p)↾α) : α < ht(dom p) +� +. +In particular, dom(σ(p)) = lh(ran p). +Proof. Let p ∈ Q∗, and let v := � � +p(ϑ(p)↾α) : α < ht(dom p) +� +. We first show that +v = σ(p). By Lemma 4.39, we have p ⊩ ˙h ≤ p. Since p ⊩ ϑ(p) ⊆ ϑ, this implies that for +all α < ht(dom p), we have +p ⊩ p(ϑ(p)↾α) ⊆ ˙h(ϑ(p)↾α) ⊆ [˙h](ϑ) = σ. +Therefore we have p ⊩ v ⊆ σ, or equivalently, v ⊆ σ(p). Assume, seeking a contradiction, +that v ⊊ σ(p). Let u ⊋ v be such that u ⊥ σ(p), t := (ϑ(p))↾ht(dom p) and r := p∪{(t, u)}. +By Lemma 4.44, we have r ∈ Q and r ⊩ u ⊆ σ. However, we also have r ≤ p and therefore +r ⊩ σ(p) ⊆ σ, a contradiction. +To show the second part of the statement, observe that σ(p) = v implies that +dom(σ(p)) = lh(v) = �{lh(p(ϑ(p)↾α) : α < ht(dom p)} ⊆ lh(ran p). +Conversely, lh(ran p) ⊆ dom(σ(p)) follows from our assumption that p ∈ Q∗. +□ +The next lemma will be used to prove <µ-closure of the quotient forcing for elements +of ran([hK]). +Corollary 4.46. We make the same assumptions as in Corollary 4.45. Suppose that +⟨pα : α < ξ⟩ is a strictly decreasing chain of elements of Q∗ of length ξ < µ. Let +p := �{pα : α < ξ}. +Then p ∈ Q∗ and σ(p) = � +α<ξ σ(pα). +81In fact, this holds for p instead of r. + +44 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Proof. We first show that p ∈ Q∗. Since | dom(p)| < µ, we have p ∈ Q by Lemma 4.34. +The fact that ht(dom p) and lh(ran p) are limit ordinals follows from Lemma 4.42 in the +case that ξ ∈ Lim, and from the fact that pα ∈ Q∗ in the case that ξ = α+1. Furthermore, +ht(dom p) = � +α<ξ +ht(dom pα) ⊆ � +α<ξ +dom(ϑ(pα)) ⊆ dom(ϑ(p)), +lh(ran p) = � +α<ξ +lh(ran pα) ⊆ � +α<ξ +dom(σ(pα)) ⊆ dom(σ(p)) +by Lemma 4.42 using that pα ∈ Q∗ and p ≤ pα holds for all α < ξ. Thus, (a) and (b) in +Definition 4.41 hold. To show that (c) also holds, let γ < ht(dom p). Then γ < ht(dom pα) +for some α < ξ. Because p ≤ pα, we have ϑ(pα) ⊆ ϑ(p). This fact and pα ∈ Q∗ imply that +ϑ(p)↾γ = ϑ(pα)↾γ ∈ dom(pα) ⊆ dom(p). +To show the second part of the statement, let v := � +α<ξ σ(pα). Then v ⊆ σ(p), since +p ≤ pα holds for all α < ξ. Furthermore, the fact that p ∈ Q∗ and pα ∈ Q∗ for all α < ξ +imply by Corollary 4.45 and Lemma 4.42 that +dom(σ(p)) = lh(ran p) = � +α<ξ +lh(ran pα) = � +α<ξ +dom(σ(pα)) = dom v. +Thus, we have σ(p) = v. +□ +The next lemma proves the first part of the Quotient Lemma 4.26 for Add(µ, 1). +Lemma 4.47. Suppose K is Q-generic over M and ⟨xi : i < ξ⟩ is a sequence of distinct +elements of (µδ)M[K] of length ξ < µ. Then � +i<ξ[hK](xi) is Add(µ, ξ)-generic over M.82 +Proof. Let i < ξ. By basic facts about forcing, there exist Q-names ϑi and σi such that +(ϑi)K = xi and 1Q ⊩ ϑi ∈ µδ ∧ [˙h](ϑi) = σi. In more detail, let ϑ′ be a Q-name such that +(ϑ′)K = x and let +ϑ := +�� +(α, β), q +� +: (α, β) ∈ µ × δ and q ⊩ ϑ′ ∈ µδ ∧ (α, β) ∈ ϑ′� +, +σ := +�� +(α, β), q +� +: (α, β) ∈ µ × µ and q ⊩ (α, β) ∈ [˙h](ϑ) +� +. +Since xi ̸= xj for all i < j < ξ, there exists an ordinal α < µ and a condition q ∈ K +such that q ⊩ ϑi↾α ̸= ϑj↾α for all i < j < ξ. We may also assume that ht(dom q) ≥ α, +since the set {p ∈ Q : ht(dom p) ≥ α} is dense in Q by Lemma 4.37. +Claim. Suppose that D is a dense subset of Add(µ, ξ). Then the subset +D′ = +� +r ∈ Q : ∃u ∈ D ∀i < ξ r ⊩ u(i) ⊆ σi +� +of Q is dense below q. +Proof. Assume that q′ ∈ Q and q′ ≤ q. There is a condition p ≤ q′ in � +i<ξ Q∗(ϑi, σi) by +Lemma 4.43. For all i < ξ, we let ti ∈ <µδ and ui ∈ <µµ be nodes which are chosen as in +Lemma 4.44; that is, we let ti = (ϑi)(p)↾ht(dom p), and we let ui be such that +ui ⊋ � � +p(ti↾α) : α < ht(dom p) +� +. +In addition, we can assume ⟨ui : i < ξ⟩ ∈ D. Since p ≤ q, we have ti ̸= tj for all i < j < ξ. +Therefore r = p ∪ {(ti, ui) : i < ξ} is a condition in Q below p which forces that ui ⊆ σi +82[hK](xi) is identified with the filter of its initial segments in Add(µ, 1). + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +45 +for all i < ξ by Lemma 4.44. Thus, r ≤ q′ and r ∈ D′. This completes the proof of the +claim. +□ +To see that G′ = � +i<ξ[hK](xi) is Add(µ, ξ)-generic, suppose that D is a dense subset +of Add(µ, ξ). By the previous claim, D′ ∩ K ̸= ∅. Thus, there exists u ∈ D such that +for all i < ξ, we have u(i) ⊆ (σi)K = [hK](xi) and therefore u ∈ G′. This completes the +proof of Lemma 4.47. +□ +Remark 4.48. The previous lemma shows that the Quotient Lemma 4.26 for Add(µ, 1) +holds with the following stronger ξ-dimensional variant of statement (1): +If ⟨xi : i < ξ⟩ is a sequence of distinct elements of (µδ)M[K] of length ξ < µ, +then � +i<ξ[hK](xi) is Add(µ, ξ)-generic over M. +In the rest of this subsection, we will prove the second part of the Quotient Lemma +Lemma 4.26 for Add(µ, 1). We need to show that the quotient forcing in a Q-generic +extension M[K] of M for an element σK of ran[hK] is equivalent to Add(µ, 1). +The +precise statement is given in Lemma 4.56 below. +For this purpose, it will be more convenient to work with complete Boolean algebras +instead of arbitrary forcings. We therefore need the next lemma. +Lemma 4.49. Q is separative. +Proof. Assume that p, q ∈ Q are such that p ̸≤ q. We aim to find r ∈ Q such that r ≤ p +and r ⊥ q. We can assume p ∥ q. +First, suppose that q(t) ̸⊆ p(t) holds for some t ∈ dom(p) ∩ dom(q). Then p(t) ⊊ q(t) +and t is a terminal node of dom(p) by Lemma 4.35 (1) and (2). Let v ∈ <µµ be a node +extending p(t) with v ⊥ q(t). Let r be the element of Q obtained from p by extending +the value of t to v. That is, let dom(r) := dom(p), and for all u ∈ dom(r), let +r(u) := +� +� +� +v +if u ̸= t, +p(u) +if u = t. +Then r ≤ p, and r ⊥ q by Lemma 4.35 (1). +Now, suppose that q(t) ⊆ p(t) holds for all t ∈ dom(p) ∩ dom(q). Then q(t) = p(t) for +all t ∈ dom(p) ∩ dom(q) such that t is non-terminal in dom(q) by Lemma 4.35 (3). Thus, +since p ̸≤ q, we must have dom(q) ̸⊆ dom(p). Let t ∈ dom(q) be a node of minimal length +such that t /∈ dom(p), and let +u := +� +αB �v ⊆ σ�, then u ⊊ v. +Proof. Part (1) is immediate. For (2), suppose �u ⊆ σ� >B �v ⊆ σ�. We have u ∥ v, since +otherwise �u ⊆ σ� ⊥B �v ⊆ σ�. If v ⊆ u, then �v ⊆ σ� ≤B �u ⊆ σ� by (1). Therefore +u ⊊ v. +□ +Recall from Definition 2.20 that B(σ) = BQ(σ) denotes the complete Boolean subalge- +bra of B that is completely generated by {�(α, β) ∈ σ� : α, β < µ}. Note that σ(p) ∈ <µµ +for all p ∈ Q∗, by Corollary 4.45. Define the map π∗ : Q∗ → B(σ) by letting +π∗(p) := �σ(p) ⊆ σ� +for all p ∈ Q∗. Let R∗ := ran(π∗) = {π∗(q) : q ∈ Q∗}. +Lemma 4.52. For all p, q ∈ Q∗: +(1) q ≤B π∗(q). +(2) If p ≤B q, then π∗(p) ≤B π∗(q). +Proof. The first statement holds because q ⊩B σ(q) ⊆ σ. To show statement (2), suppose +p ≤B q. Then σ(p) ⊇ σ(q), and therefore π∗(p) ≤B π∗(q) by Lemma 4.51 (1). +□ +Lemma 4.53. +(1) If q ∈ Q∗, then π∗(q) is a reduction83 of q to R∗ with respect to the inclusion +id: R∗ → B. +(2) R∗ is a complete subforcing of B. +Proof. To show (1), suppose r ∈ R∗ and r ≤B π∗(q). We need to show q ∥ r. We can +assume r 0. We first claim that γ is a successor ordinal. Fix some δβ < κ such that ⟨xα↾β : +δβ ≤ α < κ⟩ is constant for each β < γ and let δ := supβ<γ δβ < κ. +If γ were a +limit ordinal, then ⟨xα↾γ : δ ≤ α < κ⟩ would be constant. Let γ = β + 1. We may +assume that ⟨xα↾β : α < κ⟩ has a constant value t ∈ <κκ by replacing x by a tail. Then +xα↾γ = t⌢⟨xα(β)⟩ for all α < κ. +Next, we claim that ⟨xα(β) : α < κ⟩ is unbounded in κ; in particular d = κ. Towards +a contradiction, suppose that there is a strict upper bound δ < κ with xα(β) < δ for +all α < κ. Since ⟨xα(β) : α < κ⟩ is not eventually constant, there exist i < j < κ and +disjoint unbounded subsets I, J of κ with xα(β) = i for all α ∈ I and xα(β) = j for all +α ∈ J. Then [ι](xα) ⊇ ι(t⌢⟨i⟩) for all α ∈ I and [ι](xα) ⊇ ι(t⌢⟨j⟩) for all α ∈ J. Since ι +is ⊥-preserving, ι(t⌢⟨i⟩) ⊥ ι(t⌢⟨j⟩). But then ⟨[ι](xα) : α < κ⟩ cannot be convergent. +To show that y ∈ Limι +t(X) as required, we may assume that ⟨xα(β) : α < κ⟩ is +injective by replacing x with a subsequence. Let z = ⟨zα : α < κ⟩ be a sequence with +zα ∈ Nt⌢⟨α⟩ ∩ X and zα = xδ if α = xδ(β). Since ι is ⊥-preserving, [ι] is injective. Thus +y is a limit point of the set {[ι](zα) : α < κ} and hence y ∈ Limι +t(X). +□ +87Recall that y is a limit point of a set X if and only if it is in the closure of X \ {y}. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +51 +The next corollary follows from the previous lemma and Lemmas 2.9 and 3.11. +Corollary 5.2. Suppose ι : <κd → <κκ is strict order- and ⊥-preserving. If d < κ or +if d = κ and Limι +t = ∅ for all t ∈ <κκ, then [ι] is a homeomorphism of κd onto a closed +subset of κκ. +The next lemma shows that the assumption that ι is ⊥-preserving can be omitted from +the previous lemma if κ is weakly compact. Recall that a topological space is κ-compact +(or κ-Lindel¨of ) if every open cover has a subcover of size strictly less than κ. A subset X +of κκ is bounded by b ∈ κκ if x(α) < b(α) for all x ∈ X and α < κ. Recall that κ > ω is +a weakly compact cardinal if and only if κ2 is κ-compact and if and only if every closed +and bounded subset of κκ is κ-compact88 [LMS16, Lemma 2.6]. +Lemma 5.3. Suppose that κ = ω or κ is weakly compact. Let f : κd → κκ be a continuous +map. +(1) If 2 ≤ d < κ, then f is a closed map. +(2) If d = κ, then +f(X) = f(X) ∪ +� +t∈T(X) +Limf +t (X). +for all subsets X of κκ. In particular, f is a closed map if Limf +t = ∅ for all +t ∈ <κκ. +Proof. (1) holds since the latter is preserved under continuous images and κ-compact +subsets of κκ are closed. +For (2), note that clearly f(X) ⊆ f(X) and that Limf +t (X) ⊆ f(X) for all t ∈ <κκ. +It suffices to show y ∈ f(X) ∪ � +t∈T(X) Limf +t (X) for any y ∈ f(X). +Fix a sequence +⟨f(xα) : α < κ⟩ converging to y with xα ∈ X for all α < κ. We may assume f(xα) ̸= y +for all α < κ. +If x := ⟨xα : α < κ⟩ is bounded, then it is contained in a κ-compact subset C of X. +Then f(C) is also κ-compact and thus closed, so that y ∈ f(C) ⊆ f(X). Now suppose +that x is unbounded. Let β < κ be least such that ⟨xα(β) : α < κ⟩ is unbounded in +κ. We may assume that ⟨xα(β) : α < κ⟩ is injective by replacing x with a subsequence. +There are only |κβ| many possibilities for xα↾β and |κβ| < κ, since κ is inaccessible. We +may thus assume that ⟨xα↾β : α < κ⟩ has a constant value t ∈ <κκ by replacing x with a +further subsequence. Then xα↾(β + 1) = t⌢⟨xα(β)⟩ for all α < κ. Let z = ⟨zα : α < κ⟩ +be a sequence with zα ∈ Nt⌢⟨α⟩ ∩ X and zα = xγ if α = xγ(β). Since y ̸= f(xγ) for all +γ < κ, y is a limit point of the set {f(zα) : α < κ} and thus y ∈ Limf +t (X). +□ +The next remark shows that the following assumptions cannot be omitted: ι is ⊥- +preserving in Lemma 5.1 and κ is weakly compact in Lemma 5.3. +Remark 5.4. Suppose that κ is not weakly compact. +(1) There exists a strict order preserving map ι: <κ2 → <κ2 such that ran([ι]) is not +closed, and thus [ι] is not a closed map. To see this, note that there exists a barrier +88Conversely, it is easy to see that κ-compact subsets of κκ are always closed and bounded for all +cardinals κ with κ<κ = κ. + +52 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +A in <κ2 of size κ<κ = κ,89 i.e. a maximal antichain in <κ2 of size κ<κ = κ with +the following equivalent properties: (a) every x ∈ κκ has an initial segment in A +and (b) T(A) has no κ-branches. If κ is inaccessible, let A be the boundary of a +κ-Aronszajn subtree of <κ2. If κ = µ+, let A := µ2. Fix an injective enumeration +⟨tα : α < κ⟩ of A. Further fix a strictly increasing sequence ⟨γα : α < κ⟩ with +γα ≥ lh(tα) and let ι(tα) := ⟨0⟩γα⌢⟨1⟩ for all α < κ. We now extend ι to a strict +order preserving map from <κ2 to <κ2. If t ⊊ u for some u ∈ A, let ι(t) := ⟨0⟩lh(t). +We further extend ι above A arbitrarily to a strict order preserving map. Then +⟨0⟩κ witnesses that ran([ι]) is not closed. Note that we can also guarantee that [ι] +is injective by extending ι in a ⊥-preserving way above A.90 +(2) There exists a strict order preserving map θ : <κκ → <κκ such that Limθ +t = ∅ for +all t ∈ <κκ and ran([θ]) is not closed. To see this, define θ(s⌢t) := ι(s)⌢t for all +s ∈ <κ2 and t ∈ <κκ with t(0) ≥ 2, where ι is as in (1). Note that if [ι] is injective, +then so is [θ]. +If µ is a regular cardinal, let cofµ denote the class of limit ordinals α with cof(α) = +µ, and for any ordinal γ, let cofγ +µ := cofµ ∩ γ. We define cof≤µ, cof<µ, cofγ +>µ, cofγ +>µ +etc. similarly for arbitrary ordinals µ. Given a class A of limit ordinals and a subtree T +of <κκ, we say that T is A-closed if every strictly increasing sequence in T with length in +A has an upper bound in T. +The next lemma shows that the relevant closure properties of T(ι) are determined by +the following sets. For ι: <κd → <κκ and u ∈ <κd, we write T ι,u for the downwards +closure of {ι(u⌢⟨α⟩ : α < κ} in <κd. +Lemma 5.5. If 2 ≤ d ≤ κ, ι is strict order and ⊥-preserving and µ < κ is a regular +infinite cardinal, then the following conditions are equivalent: +(1) T(ι) is cofκ +µ-closed. +(2) For all u ∈ <κd, T ι,u is cofκ +µ-closed. +Proof. (1) ⇒ (2): Suppose that w is a limit of nodes in T ι,u with w ∈ T(ι) \ T ι,u. There +exists some η < κ with ι(u⌢⟨η⟩) ⊆ w, since otherwise w properly extends ι(u) and is +incompatible with ι(t) for all t ⊋ u, but this would entail w /∈ T(ι). Then ι(u⌢⟨ξ⟩) ⊥ w +for all ξ ̸= η, so w cannot be a limit of nodes in T ι,u. +(2) ⇒ (1): Suppose that w is a limit of nodes in T(ι) with lh(w) ∈ cofκ +µ. We shall show +that w ∈ T(ι). +For all γ < lh(w), pick some sγ such that w↾γ ⊆ ι(sγ) and lh(sγ) > lh(u). +Let +A := {t ∈ <κd : ι(t) ⊆ w}. Note that the elements of A are pairwise comparable, since ι +is ⊥-preserving. Hence u := � A and v := �{ι(t) : t ∈ A} are elements of <κd. +We can assume that v ⊊ w. Then u ⊊ sγ for all γ with lh(v) < γ < lh(w). To see +this, note that for any t with ι(t) ⊆ w, we have ι(t) ⊆ v ⊆ w↾γ ⊆ ι(sγ) and hence t ⊆ sγ. +Since this holds for all such t, we have u ⊆ sγ. Since lh(sγ) > lh(u), we have u ⊊ sγ. +89A similar argument also works without the assumption that κ<κ = κ. +90I.e., so that for all u ∈ A and all incompatible v, w ⊋ u, we have ι(v) ⊥ ι(w)). E.g. let ι(u⌢v) := +ι(u)⌢v for all u ∈ A and v ∈ <κκ. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +53 +For each γ as above, we can thus pick some ηγ with u⌢⟨ηγ⟩ ⊆ sγ. +It suffices to +show w↾γ ⊆ ι(u⌢⟨ηγ⟩), since this implies w ∈ T ι,u ⊆ T(ι). +To see this, note that +w↾γ ∥ ι(u⌢⟨ηγ⟩), since both are initial segments of ι(sγ). But ι(u⌢⟨ηγ⟩) ⊆ w↾γ ⊆ w +would entail u⌢⟨ηγ⟩ ⊆ u by the definition of u. +□ +The next corollary is immediate: +Corollary 5.6. Suppose that ι : <κd → <κκ is strict order and ⊥-preserving. +(1) If 2 ≤ d < κ, then T(ι) is cofκ +>d-closed. +(2) If the nodes ι(t⌢⟨α⟩) split at the same node for each t ∈ <κd,91 then T(ι) is +<κ-closed. +The next remark shows that the restriction to cofκ +>d is necessary in the previous corol- +lary. +Remark 5.7. Suppose d ≤ κ and η ∈ cofκ +≤d. Then there exists a strict order and ⊥- +preserving function ι : <κd → <κκ such that T(ι) is not {η}-closed. To see this, choose a +strictly increasing sequence ⟨γα : α < cof(η)⟩ cofinal in η, and let ι = ιd,η be any ⊥- and +strict order preserving map from <κd to <κκ such that +ι (⟨α⟩) = +� +� +� +⟨0⟩γα⌢⟨1⟩ +if α < cof(ν), +⟨α⟩ +if α ∈ [cof(ν), d). +Then the node ⟨0⟩η witnesses that T(ι) is not {η}-closed. When d = κ, it is also easy to +make sure that Limι +t = ∅ for all t ∈ <κκ.92 +Suppose that d = κ. Then there exists a strict order and ⊥-preserving function ψ : +<κd → <κκ such that T(ψ) is not {η}-closed for all limit ordinals η < κ, and at the same +time Limψ +t = ∅ for all t ∈ <κκ. To see this, let ⟨ηα : α < κ⟩ be an enumeration of all +the limit ordinals below κ. For all α < κ, let ια be the map ικ,ηα defined in the previous +paragraph. Define ψ : <κκ → <κκ where ψ(∅) := ∅ and ψ(⟨α⟩⌢t) := ⟨α⟩⌢ια(t) for all +α < κ and t ∈ <κκ. Then T(ψ) is not {η}-closed for any limit ordinal η < κ, and we may +also ensure that Limψ +t = ∅ for all t ∈ <κκ. +Lemma 5.8. Suppose 2 ≤ d < ω. +(1) If ι : <κd → <κκ is ⊥- and strict order preserving, then ran([ι]) is a κ-perfect set. +(2) Conversely, if X ⊆ κκ has a κ-perfect subset, then there exists a ⊥- and strict +order preserving map ι : <κd → <κκ with ran([ι]) ⊆ X. +Proof. For (1), let ι : <κd → <κκ be ⊥- and strict order preserving. By Lemma 5.1, +ran([ι]) is a closed set, so ran([ι]) = [T(ι)]. It thus suffices to show that T(ι) is a κ-perfect +tree. T(ι) is <κ-closed by Corollary 5.6. T(ι) is also cofinally splitting because ι preserves +⊥. For (2), let T be a κ-perfect tree T with [T] ⊆ X. It is easy to construct a ⊥- and +strict order preserving map ι : <κd → T by recursion. Then ran([ι]) ⊆ [T] ⊆ X. +□ +91I.e., the ι(t⌢⟨α⟩)’s extend pairwise different immediate successors of the some st ∈ <κκ, for each +t ∈ <κd. Equivalently, ι is an order homomorphism for the dihypergraph Hsuper +κ +defined in Definition 6.6. +92Note that Limι +∅ = ∅ is guaranteed by the choice of the ι(⟨α⟩)’s. Let, for example, ι(⟨α⟩⌢t) = ι(⟨α⟩)⌢t +for all α < κ and t ∈ <κκ. + +54 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +5.2. The open graph dichotomy. Throughout this subsection, we assume that X is +a subset of κκ. We say that a graph G on X has a κ-perfect complete subgraph if there +exists a κ-perfect subset Y ⊆ X such that the complete graph KY is a subset of G. Recall +that the open graph dichotomy OGDκ(X) is the following statement: +OGDκ(X): If G is an open graph on X, then either G admits a κ-coloring +or G has a κ-perfect complete subgraph. +In this subsection, we show that the open graph dichotomy OGDκ(X) is equivalent to +ODD2 +κ(X).93 This is a special case of a result for dimension 2 ≤ d ≤ κ. +Recall that a d-hypergraph is a d-dihypergraph that is closed under permutation of hy- +peredges. For any d-dihypergraph I,94 let Isym denote the smallest hypergraph containing +I, i.e., +Isym := �{xπ : x ∈ I, π ∈ Sym(d)}. +Then Hsym +κd += �{� +α∈d Nt⌢⟨π(α)⟩ : t ∈ <κd, π ∈ Sym(d)} and Hsym +κ2 +equals the complete +graph Kκ2. +Definition 5.9. OGDd +κ(X) states that the following holds for all box-open d-hypergraphs +H on X: +OGDH +κ : Either H admits a κ-coloring, or there exists a continuous homo- +morphism from Hsym +κd +to H. +We first show that OGD2 +κ(X) is equivalent to the open graph dichotomy OGDκ(X). +Since Hsym +κ2 +is the complete graph Kκ2 in the two-dimensional case, it suffices to show that +a graph G on X has a κ-perfect complete subgraph if and only if there is a continuous +homomorphism from Kκ2 to G. The next lemma is a more general version of this for +dihypergraphs. +We say a d-dihypergraph I has a κ-perfect complete subhypergraph if +there exists a κ-perfect subset Y ⊆ X such that Kd +Y ⊆ I. +Lemma 5.10. Suppose that 2 ≤ d ≤ κ and I is a d-dihypergraph on X. +(1) If f is a continuous homomorphism from Hκd to I, then ran(f) has a κ-perfect +subset. +(2) I has a κ-perfect complete subhypergraph if and only if there is a continuous ho- +momorphism from Kd +κd to I. +Proof. For (1), note first that f is also a homomorphism from Hκd to Kd +ran(f) on ran(f). +Since Kd +ran(f) is relatively box-open, there exists a continuous order homomorphism ι : +<κd → <κκ for Kd +ran(f) by Lemma 3.11. +For all t ∈ <κd, there exists u, v ⊋ t with +ι(u) ⊥ ι(v) since otherwise, every sequence in � +α ∆ +� +x1, x2� +for all x ∈ 3Y . +We claim that the reverse lexicographic order on Y is a wellorder with order type at most +κ + 1. As before, all splitting nodes of T(Y ) lie on a branch z. If s ⊊ z is a splitting +node, then exactly one element of Y splits off at s. Therefore the order type of Y is at +most κ + 1. +We have not studied whether there exists a d-hypergraph with the properties of the +previous example for any d ≥ 4. +He [He05, Theorem 4.3], Di Prisco and Todorˇcevi´c +[DT98, Sections 3 & 4] showed for dimensions 3 ≤ n < ω that open n-hypergraphs with +additional structural properties either admit a countable coloring or contain a perfect +complete subhypergraph. +The next example shows that ODDH +κ may fail for some closed hypergraph H in any +dimension d ≥ 2. +Example 5.15. For any d ≥ 2, there exists a a product-closed98 d-hypergraph H ∈ Dκ +on κκ with the properties: +(a) H does not admit a κ-coloring. +(b) +(i) H does not have a complete subhypergraph of size κ+. +(ii) There is no continuous homomorphism from Hκd to H. +We next provide an example for d = 2. +An example for κ = ω can be found in +[TF95, Proposition 10.1], so suppose κ is uncountable. We follow [Jec03, Exercise 29.9] +and [Szir18, Example 3.3]. The example is based on the notion of a strongly κ-dense +linear order. This is a linear order with no (<κ, <κ)-gaps. A (<κ, <κ)-gap in a linear +order (L, ≤) is a pair (A, B) of subsets of L with the properties: +(1) a < b for all a ∈ A and b ∈ B. +(2) There is no x ∈ L with a < x < b for all a ∈ A and b ∈ B. +(3) |A|, |B| < κ. +It is easy to see that a strongly κ-dense linear order L of size κ exists for any κ with κ<κ = +κ.99 For instance, let Qκ denote the set of all x ∈ κ{−1, 0, 1} that take values in {−1, 1} +up to some α < κ and are constant with value 0 from α. Qκ with the lexicographical +order is strongly κ-dense. +Let (L, <) be a strongly dense linear order of size κ, and let +X := {y ∈ P(L) : ≤↾y is a well-order}. +Since |L| = κ, P(L) can be identified with κ2. Since κ is uncountable, X is a closed sub- +set.100 Therefore it suffices to define a closed graph H on X with the required properties. +For x, y ∈ X, let +(x, y) ∈ H ⇐⇒ x ⊊ y or y ⊊ x. +98I.e., H is a relatively closed subset of Kdκκ, where d(κκ) has the product topology. Equivalently, +H ∪ ∆dκκ is a closed subset of d(κκ) with the product topology. +99It is also easy to show using a back-and-forth argument that any two strongly κ-dense linear orders +of size κ are isomorphic. +100For κ = ω, X would be a Π1 +1 subset. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +59 +It is clear that H is a closed graph on X which has no complete subgraph of size κ+. +Thus there is no continuous homomorphism from Hκ2 to H by Corollary 5.12. +We now show that H has no κ-coloring. Towards a contradiction, suppose that X = +� +α<κ Xα, where Xα is H-independent for each α < κ. Construct a strictly ⊆-increasing +chain ⟨xα : α < κ⟩ in X, a strictly ≤-increasing chain ⟨qα : α < κ⟩ in L and a strictly +≤-decreasing chain ⟨rα : α < κ⟩ in L such that for all α < κ, we have qα < rα and p < qα +for all p ∈ xα. Whenever possible, choose xα, qα, rα such that xα ∈ Xα. Finally, let +x := � +α<κ xα and suppose that x ∈ Xβ. We have xβ ∈ Xβ by the construction and +(xβ, x) ∈ H, so Xβ cannot be H-independent. +Given the example for d = 2, the case d > 2 follows from the next claim. +Claim. Suppose d > 2 and G is a graph on κκ. Let HG be the d-hypergraph consisting of +those sequences x such that ran(x) = {x, y} for some (x, y) ∈ G. +(1) +(i) If G is closed on κκ, then HG is product-closed on κκ. +(ii) If G ∈ Dκ, then HG ∈ Dκ. +(2) A function c : κκ → κ is a κ-coloring of HG if and only if it is a κ-coloring of G. +(3) +(i) HG does not have a complete subgraph of size 3. +(ii) If there is a continuous homomorphism from Hκd to HG, then there is a +continuous homomorphism from Hκ2 to G. +Proof. (1) is clear. (2) holds since a subset of κκ is HG-independent if and only if it is +G-independent. +In (3), (i) is immediate. For (ii), suppose that f is a continuous homomorphism from +Hκd to HG. We construct a continuous ⊥- and strict order preserving map ι : <κ2 → <κd +such that +f(Nι(t⌢⟨0⟩)) × f(Nι(t⌢⟨1⟩)) ⊆ G +holds for all t ∈ <κd. We construct ι(t) by recursion on lh(t). Let ι(∅) := e(∅) = ∅. +If lh(t) ∈ Lim, and ι(s) has been defined for all s ⊊ t, then let ι(t) := � +s⊊t ι(s). We +now assume ι(t) has been defined and construct ι(t⌢⟨0⟩) and ι(t⌢⟨1⟩). For each i < d, +take xi extending ι(t)⌢⟨i⟩. Since x := ⟨xi : i < d⟩ is a hyperedge of Hκd and f is a +homomorphism from Hκd to Kdκκ, there exist j < k < d with f(xj) ̸= f(xk). Using the +continuity of f, take α < κ such that α > lh(ι(t)) and f(Nxj↾α) ∩ f(Nxk↾α) = ∅. Since f +is a homomorphism from Hκd to HG and xi↾α extends ι(t)⌢⟨i⟩ for each i < d, we have +� +i 2, let +IG denote the d-dihypergraph that consists of all ⟨x⟩⌢⟨y : 1 ≤ i < d⟩ with (x, y) ∈ G. + +60 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +It is again clear that (1), (2) and (3)(i) hold for IG. (3)(ii) holds as well, since for any +homomorphism f from Hκd to IG, f↾κ2 is a homomorphism from Hκ2 to G. +Remark 5.17. Farah and Todorˇcevi´c [TF95, Proposition 10.1] show that there exists a +graph on ωω as in the previous example. It is open whether this is possible for κκ when +κ is uncountable. Note that the closed set X in the above example is not homeomorphic +to κκ. In fact, one can show as in [LS15, Section 2] that X is not a continuous image of +κκ. To see this, one only has to replace the relation Rx coded by a subset x of κ by the +relation sup{ran(a) : a ∈ An} ≥ sup (ran(x↾n)) ≥ x(n − 1), +a contradiction. +102This step uses that κ is regular. + +62 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Remark 5.22. +(i) When κ = λ+ ≥ ℵ1, requirement (a) in Definition 5.18 (the definition of ♦i +κ) +is equivalent to the requirement that |Aα| ≤ λ for all α < κ. Therefore ♦i +κ is +equivalent to ♦κ, by a result of Kunen’s [JK69] (see also [Kun13, Section III.7]). +(ii) If κ = λ+ ≥ ℵ2, then κ<κ = κ is equivalent to ♦κ [She10], and is therefore also +equivalent to ♦i +κ. +(iii) The failure of ♦κ at the least inaccessible cardinal is consistent [Gol22] and there- +fore ♦κ and ♦i +κ are not equivalent for inaccessible cardinals. +To the best knowledge of the authors, it is not known whether any of the assumptions +κ<κ = κ, ♦i +κ and ♦κ are equivalent when κ is weakly inaccessible but not inaccessible. +The following lemma contains a δ-dimensional equivalent version of ♦i +κ, for ordinals +2 ≤ δ < κ, which we will use in some of our later arguments (for δ = 2). Recall that ∆δ +X +denotes the set of constant sequences in δX and that Kδ +X = δX − ∆δ +X. +Lemma 5.23. The following statements are equivalent whenever κ is a regular uncount- +able cardinal and 2 ≤ δ < κ: +(1) ♦i +κ. +(2) There exists a sequence ⟨Bα : α < κ⟩ of sets Bα ⊆ Kδακ such that +(a) |Bα| < κ for all α < κ, and +(b) for all x ∈ Kδκκ, there exists α < κ with x↾α ∈ Bα.103 +Proof. We first show that (2) implies (1). Suppose that ⟨Bα : α < κ⟩ is a sequence which +satisfies the requirements of (2). For all α < κ, let Aα := {s0 : ⟨sβ : β < δ⟩ ∈ Bα}. We +claim that ⟨Aα : α < κ⟩ is a ♦i +κ-sequence. To see this, suppose x ∈ κκ and γ < κ. It is +enough to find α < κ such that α > γ and x↾α ∈ Aα. Let x = ⟨xβ : β < δ⟩ be an element +of Kδκκ such that x0 = x and x↾γ ∈ ∆δγκ. By (2)(b), there exists α < κ such that x↾α ∈ Bα. +Then x = x0 ∈ Aα. We also have α > γ, because x ∈ Bα ⊆ Kδακ and x↾γ ∈ ∆δγκ. +To show that (1) implies (2), let ⟨Aα : α < κ⟩ be a ♦i +κ-sequence. For every γ ≤ κ, +let gγ : δ·γκ → δ(γκ) be the bijection which maps each s ∈ δ·γκ to the unique element +gγ(s) = ⟨tβ : β < δ⟩ of δ(γκ) such that tβ(α) = s(δ · α + β) for all α < γ and β < δ. For +each γ < κ, let +Bγ = � +β<δ +� +gγ +� +s↾(δ · γ) +� +: s ∈ Aδ·γ+β +� +. +We claim that the sequence ⟨Bγ : γ < κ⟩ satisfies the requirements in (2). Since κ is +regular, |Bγ| < κ holds for all γ < κ. To see that (2)(b) also holds, suppose that x ∈ Kδκκ. +Let z := g−1 +κ (x). Since z ∈ κκ and ⟨Aα : α < κ⟩ is a ♦i +κ-sequence, there exist ordinals +γ < κ and β < δ such that z↾(δ · γ + β) ∈ Aδ·γ+β. Then, by the definitions of Bα, gγ and +gκ, we have Bγ ∋ gγ(z↾(δ · γ)) = gκ(z)↾γ = x↾γ. +□ +5.5. Strong variants. Throughout the rest of this section, we assume that 2 ≤ d ≤ κ +and H is a d-dihypergraph on κκ. We further assume H is relatively box-open104 unless +it is explicitly stated otherwise. We will consider the following strengthenings of ODDH +κ . +103Recall that we let x↾α := ⟨xi↾α : i < δ⟩ for any sequence x = ⟨xi : i < δ⟩ in κκ and α < κ. +104I.e., H is box-open on its domain domH. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +63 +Definition 5.24. +ODDIH +κ : Either H admits a κ-coloring, or there exists an injective continuous +homomorphism from Hκd to H. +ODDHH +κ : Either H admits a κ-coloring, or there exists a homomorphism +from Hκd to H which is a homeomorphism between κd and its image. +ODDCH +κ : Either H admits a κ-coloring, or there exists a homomorphism +from Hκd to H which is a homeomorphism between κd and a closed subset +of κκ. +Note that these are ordered in increasing strength. +The next lemma characterizes the existence of a homomorphism as in ODDHH +κ via +order homomorphisms for any d ≤ κ. +Lemma 5.25. +(1) If ι is a a ⊥-preserving order homomorphism for H,105 then [ι] is a homomorphism +from Hκd to H which is a homeomorphism between κd and its image.106 +(2) Conversely, if there exists a homomorphism f from Hκd to H which is a homeo- +morphism between κd and its image, then there exists a continuous ⊥-preserving +order homomorphism ι for H with ran([ι]) ⊆ ran(f). +Proof. First, suppose ι is a ⊥-preserving order homomorphism for H. +Then [ι] is a +homomorphism from Hκd to H by Lemma 3.11 (1), and [ι] and is a homeomorphism +between κd and its image by Lemma 2.9 (2). +The argument for (2) is similar as the proof of Lemma 3.11 (2). We construct continuous +strict order preserving maps ι : <κκ → <κκ and e : <κd → <κd with the following +properties for all t ∈ <κd and α, β < d: +(i) f(Ne(t)) ⊆ Nι(t). +(ii) ι(tα) ⊥ ι(tβ). +(iii) � +α κ for all α < d, by Lemma 3.1 (2), we may construct by +recursion on α a sequence ⟨zα ∈ κd : α < d⟩ such that e(u)⌢⟨α⟩ ⊆ zα and [ι](zα) ̸= [ι](zβ) +for all α, β < d with α ̸= β. Since d < κ, there exists γ < κ such that ι(zα↾γ) ⊥ ι(zβ↾γ) +and e(u)⌢⟨α⟩ ⊆ zα↾γ for all α, β < d with α ̸= β. +Let e(u⌢⟨α⟩) := zα↾γ and let +i(u⌢⟨α⟩) := ι(zα↾γ) for all α < d. +By the construction, i is ⊥-preserving, strict order preserving and continuous. To see +that i is an order homomorphism for H, observe first that [i] = [ι] ◦ [e]. This is because +for all z ∈ κd, +[i](z) = � +t⊊z +i(t) = � +t⊊z +ι(e(t)) = [ι] ◦ [e](z) = [ι] ◦ [e](z) + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +65 +by (iv). Therefore ran([i]) ⊆ ran[ι] ⊆ domH. Suppose that t ∈ <κd. By (ii) and (iv). +i(t⌢⟨α⟩) ⊇ ι +� +e(t)⌢⟨α⟩ +� +for all α < d, and hence +� +α lh(u) = ξ. +□ +We claim that there is no continuous homomorphism f : κκ → κκ from Hκκ to Hπ with +closed range. Towards a contradiction, suppose that f is such a homomorphism. Let +⟨xα : α < κ⟩ ∈ Hκκ be an arbitrary hyperedge of Hκκ. Since ⟨f(xα) : α < κ⟩ ∈ Hπ, there +is some v ∈ <κκ such that v⌢⟨0⟩α⌢⟨1⟩ ⊆ f(xα) for all α < κ. If ran(f) is closed, then +v⌢⟨0⟩κ ∈ ran(f). But this contradicts the previous claim. +□ +We next show that ODDH +κ does not always imply ODDHH +κ and that for κ = ω, ODDH +ω +may not even imply ODDIH +ω . The following dihypergraph will provide a counterexample +to both implications. +Definition 5.29. Let Dκ denote the κ-dihypergraph on κκ consisting of all somewhere +dense sequences ⟨xα : α < κ⟩.109 +Observe that Dκ is in Dκ and it is in fact a κ-Borel subset of the space κ(κκ) with +the <κ-box topology. We first show that Dκ is a box-open dihypergraph which satisfies +ODDDκ +κ . +Proposition 5.30. +(1) Dκ is box-open on κκ. +(2) There is a continuous homomorphism from Hκκ to Dκ. +107If we choose π to be continuous, we obtain the map defined at the beginning of Subsection 6.2. +108In fact, Hπ is the smallest dihypergraph H on κκ such that π is an order homomorphism for (κκ, H). +Hπ is the same as the dihypergraph Hπ,κκ defined in Remark 3.10. +109I.e., {xα : α < κ} ∩ Nt is a dense subset of Nt for some t ∈ <κκ. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +67 +Proof. To show (1), let x = ⟨xα : α < κ⟩ ∈ Dκ. Then U = � +α<κ Nxα↾α is a box-open +neighborhood of x. We claim that U ⊆ Dκ. Let y = ⟨yα : α < κ⟩ ∈ U. Take t ∈ <κκ such +that x is dense in Nt. Then for any u ∈ <κκ with u ⊇ t, there is some α > lh(u) with +xα ∈ Nu. Since xα↾α = yα↾α, we have yα ∈ Nu. Thus, y is also dense in Nt. +To show (2), let ι : <κκ → <κκ be any strict order preserving map such that for all +t ∈ <κκ, ⟨ι(t⌢⟨α⟩) : α < κ⟩ enumerates succ(ι(t)) = {u ∈ <κκ : ι(t) ⊊ u}. Then ι is an +order homomorphism for Dκ, because any sequence y ∈ � +α<κ Nι(t⌢⟨α⟩) is dense in Nι(t). +By Lemma 3.11 (1), [ι] : κκ → κκ is a continuous homomorphism from Hκκ to Dκ. +□ +The next proposition shows that ODDHDκ +κ +fails, and thus ODDHκ +κ(κκ, Dκ) fails.110 +Proposition 5.31. There is no homomorphism from Hκκ to Dκ which is a homeomor- +phism between κκ and its image. +Proof. By Lemma 5.25 (2), it suffices to show that no order homomorphism ι for Dκ +preserves ⊥. Suppose that ι is an order homomorphism for Dκ. +Claim. For any given u ∈ <κκ, there exists α < β < κ such that ι(u⌢⟨α⟩) ∥ ι(u⌢⟨β⟩). +Proof. Suppose that ι(u⌢⟨α⟩) ⊥ ι(u⌢⟨β⟩) for all α < β < κ. Take a hyperedge ⟨xα : +α < κ⟩ of Hκκ with u⌢⟨α⟩ ⊆ xα for all α < κ and let Y = {[ι](xα) : α < κ}. Then +Y ∩ Nι(u⌢⟨α⟩) = {[ι](xα)} for all α < κ. On the other hand Y is dense in Nt for some +t ∈ <κκ, since [ι] is a homomorphism from Hκκ to Dκ by Lemma 3.11 (1). In particular, +there exist α < β < κ such that [ι](xα), [ι](xβ) ∈ Nt. Hence t is compatible with both +ι(u⌢⟨α⟩) and ι(u⌢⟨β⟩). Since these are incompatible, we have t ⊆ ι(u⌢⟨α⟩). Therefore +Y is also dense in Nι(u⌢⟨α⟩), contradicting the fact that |Y ∩ Nι(u⌢⟨α⟩)| = 1. +□ +This completes the proof of the proposition. +□ +The next proposition shows that ODDIDω +ω +fails, and thus ODDIω +ω(ωω, Dω) fails. +Proposition 5.32. There is no injective continuous homomorphism from Hωω to Dω. +Proof. Suppose that f : κκ → κκ is a continuous homomorphism from Hωω to Dω. We +shall construct x ̸= y with f(x) = f(y) using the next claim. +Claim. Suppose that u, t ∈ <κκ and f(Nu) ∩ Nt ̸= ∅. Then there exist u′, t′ ∈ <κκ such +that u′ ⊋ u, t′ ⊋ t and f(Nu′) is dense in Nt′. +Proof. Since f(Nu) ∩ Nt ̸= ∅ and f is continuous, there exists some u′ ⊋ u with f(Nu′) ⊆ +Nt. The set f(Nu′) = � +α<κ f(Nu′⌢⟨α⟩) is dense in Nw for some w ∈ <κκ, since f is a +homomorphism from Hωω to Dω. We have t ∥ w, since Nw ∩ Nt ̸= ∅. Let t′ ⊋ t ∪ w. Then +f(Nu′) is dense in Nt′. +□ +Using the previous claim, we now construct strictly increasing sequences ⟨un : n < ω⟩, +⟨vn : n < ω⟩ and ⟨tn : n < ω⟩ of elements of <κκ such that for all n < ω: +(i) un ⊥ vn. +(ii) f(Nun) and f(Nvn) are both dense in Ntn. +110The definition of ODDHκ +κ(κκ, Dκ) is analogous to Definition 1.3. + +68 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +We first define u0, v0 and t0. Fix a hyperedge ⟨xα : α < κ⟩ of Hωω with ⟨α⟩ ⊆ xα for all +α < κ. Since f is a homomorphism from Hωω to Dω, x is dense in Nt for some t ∈ <κκ. +Hence there is some α < κ with f(N⟨α⟩) ∩ Nt ̸= ∅. By the previous claim for ⟨α⟩ and t, +there is some u0 ⊋ ⟨α⟩ and some t′ +0 ⊋ t such that f(Nu0) is dense in Nt0. Since x is dense +in Nt, there is some β < κ with β ̸= α and f(N⟨β⟩) ∩ Nt′ +0 ̸= ∅. By the previous claim for +⟨β⟩ and t′ +0, there is some v0 ⊋ ⟨β⟩ and some t0 ⊋ t′ +0 such that f(Nv0) is dense in Nt0. We +have u0 ⊥ v0, since u0 ⊇ ⟨α⟩ and v0 ⊇ ⟨β⟩. +Suppose that un, vn and tn have been constructed. By the previous claim for un and +tn, there is some un+1 ⊋ un and some t′ +n+1 ⊋ tn such that f(Nun+1) is dense in Nt′ +n+1. By +the previous claim for vn and t′ +n+1, there is some vn+1 ⊋ vn and some tn+1 ⊋ t′ +n+1 such +that f(Nvn+1) is dense in Ntn+1. This completes the construction. +Finally, let x := � +n<ω un, y := � +n<ω vn and z := � +n<ω tn. We claim that f(x) = z. +Otherwise f(x) /∈ Ntn for some n < ω. Since f is continuous, there is some k ≥ n with +f(Nuk)∩Ntn = ∅ and thus f(Nuk)∩Ntk = ∅. But this contradicts (ii). Similarly, f(y) = z. +Thus x ̸= y and f(x) = f(y) = z. +□ +Remark 5.33. We will see in the next theorem that the previous proposition fails for +uncountable κ. Note that the construction in its proof cannot necessarily be continued +at limit stages: the inductive hypothesis need not be preserved, since the intersection of +less than κ many dense sets may be empty. +Remark 5.34. Let D− +κ denote the dihypergraph on κκ of all non-constant sequences x +such that for some t ∈ <κκ, x is contained in Nt and is dense in Nt. Note that D− +κ is +box-open on κκ and there is a continuous homomorphism from Hκκ to D− +κ (see the proof +of Proposition 5.30). Thus, Propositions 5.30–5.32 hold for any box-open dihypergraph +H on κκ with D− +κ ⊆ H ⊆ Dκ. +Recall the principle ♦i +κ from Subsection 5.4. Moreover, recall that ♦i +κ implies κ<κ = +κ > ω by Remark 5.21. Recall that H is assumed to be relatively box-open. +Theorem 5.35. Suppose that ♦i +κ holds. If there exists a continuous homomorphism from +Hκκ to H, then there exists an injective continuous homomorphism from Hκκ to H. +Thus, ♦i +κ implies ODDH +κ ⇔ ODDIH +κ . +Proof. We use the following the following (equivalent) 2-dimensional version of ♦i +κ, which +was obtained in Lemma 5.23: +There exists a sequence ⟨Bα : α < κ⟩ of sets Bα ⊆ Kακ of size |Bα| < κ +such that for all (x, y) ∈ Kκκ, there exists α < κ with (x↾α, y↾α) ∈ Bα. +Claim. There exists an order homomorphism i for H such that for all α < κ and all +(s, t) ∈ Bα, we have i(s) ⊥ i(t). +Proof. By Lemma 3.11 (2), let ι be an order homomorphism for H. We construct maps +i : <κκ → <κκ and e : <κκ → <κκ such that the following hold for all s, t ∈ <κκ and +α, β < κ: +(i) If (s, t) ∈ Bα, then i(s) ⊥ i(t). + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +69 +(ii) e(t)⌢⟨β⟩ ⊆ e(t⌢⟨β⟩). +(iii) If lh(t) ∈ Lim, then e(t) ⊇ � +u⊊t e(u). +(iv) i(t) = ι +� +e(t) +� +. +Note that (ii), (iii), (iv) and the fact that ι is a strict order preserving map imply that +the maps e and i are strict order preserving. +Let α < κ, and suppose that i(u) and e(u) have been constructed for all u ∈ <ακ. For +each t ∈ ακ, let e′(t) be defined as follows: +e′(t) := +� +� +� +� +� +� +� +� +� +∅ +if t = ∅, +e(u)⌢⟨β⟩ +if α ∈ Succ and t = u⌢⟨β⟩, +� +u⊊t +e(u) +if α ∈ Lim. +Let ⟨zt : t ∈ ακ⟩ be a sequence of elements of κκ such that e′(t) ⊊ zt holds for all t ∈ ακ +and [ι](zs) ̸= [ι](zt) holds for all s, t ∈ ακ with s ̸= t. Such a sequence can be defined +using recursion with respect to some well-ordering of ακ, since we have +��f +� +Ne′(t) +��� > κ +for all t ∈ ακ by Lemma 3.1 (2). Since |Bα| < κ and [ι](zt) = � +γ<κ ι(zt↾γ) for all t ∈ ακ, +there exists γ < κ such that ι(zs↾γ) ⊥ ι(zt↾γ) for all (s, t) ∈ Bα. For each t ∈ ακ, let e(t) +be an initial segment of zt such that e′(t) ⊆ e(t) and lh (e(t)) ≥ γ, and let i(t) := ι(e(t)). +This choice of e(t) and i(t) clearly ensures that (ii), (iii) and (iv) hold. Item (i) also +holds, since ι is strict order preserving, and therefore i(t) = ι(e(t)) ⊇ ι(zt↾γ) holds for +each t ∈ ακ. +Once the maps i and e have been constructed, the same argument as in the proof of +Theorem 5.26 shows that i is an order homomorphism for H. This completes the proof +of the claim. +□ +Let i be as in the claim. By Lemma 3.11 (1), [i] is a continuous homomorphism from +Hκκ to H. To see that [i] is injective, take x, y ∈ κκ with x ̸= y. Choose α < κ such that +(x↾α, y↾α) ∈ Bα. Then i(x↾α) ⊥ i(y↾α), and thus [i](x) ̸= [i](y). +□ +We do not know whether the assumption of ♦i +κ in the previous theorem can be removed. +The assumption is very weak: recall that ♦i +κ holds if κ is inaccessible or κ ≥ ω2 is a +successor cardinal with κ<κ = κ.111 It is open whether Theorem 5.35 holds for ω1 and for +all weakly inaccessible cardinals.112 +Note that ♦i +κ holds in all Col(κ, <λ)-generic extensions V [G], where λ > κ is inacces- +sible. We thus immediately obtain a stronger version of Theorem 1.4. The definitions +of the the principles in the next corollary are analogous to Definition 1.3, and moreover, +they can be reformulated similarly to Lemmas 3.3 and 3.4. +Corollary 5.36. Suppose κ is a regular infinite cardinal and 2 ≤ d ≤ κ. If λ > κ is +inaccessible and G is a Col(κ, <λ)-generic filter over V , then the following statements +hold in V [G]: +(1) ODDId +κ(Dκ, Dκ) if κ is uncountable. +(2) ODDId +κ(Dκ) if κ is uncountable and λ is Mahlo. +111See Lemma 5.20 and Remark 5.22. +112See Problem 7.7. + +70 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +If d < κ, then in fact ODDCd +κ(Dκ) holds in V [G]. +Proof. (1) and (2) for d = κ follow from Theorems 1.4 and 5.35. The last statement +follows from Lemma 3.2 and Corollary 5.27. (1) and (2) for d < κ follow. +□ +In the rest of this section, we show that some of the dichotomies in Figure 4 are +equivalent for dihypergraphs with additional properties. +Definition 5.37. +(a) For 2 ≤ d ≤ κ, let Id +κ denote the d-dihypergraph on κκ which consists of all +injective sequences ⟨xα : α < d⟩. Let Iκ := Iκ +κ. +(b) Let Nκ denote the κ-dihypergraph on κκ which consists of all non-constant se- +quences ⟨xα : α < κ⟩ with no injective convergent subsequences. +(c) Let Htop +κ +:= Iκ ∩ Nκ.113,114 +That is, Htop +κ +consists of all injective sequences in κκ which have no convergent subse- +quences. +Note that all of these dihypergraphs are in Dκ. Id +κ is box-open for d < κ, while Iκ is +not box-open. To see the second statement, take any injective sequence which converges +to one of its elements. Moreover, Nκ is not box-open, as witnessed by any non-constant +sequence which is eventually constant. To see that Htop +κ += Iκ ∩ Nκ is box-open, take +any x = ⟨xα : α < κ⟩ ∈ Htop +κ . For each xα, pick an open neighborhood Nxα↾γα which +does not contain any other elements of x. Then the sets Nxα↾γα are pairwise disjoint, so +� +α<κ(Nxα↾γα ∩ domH) ⊆ Htop +κ . +We shall prove the equivalences displayed in Figure 5. Recall that H is assumed to be +relatively box-open. +ODDCH +κ +ODDHH +κ +ODDIH +κ +ODDH +κ +♦i +κ +H ∩ Iκ is rela- +tively box-open +H ⊆ Iκ +H ⊆ Htop +κ +H ⊆ Nκ +Figure 5. Equivalences in some special cases +Lemma 5.38. Suppose that 2 ≤ d ≤ κ. +(1) A map f : κd → κκ is injective if and only if it is a homomorphism from Hκd to +Id +κ. +113The superscript in Htop +κ +indicates that this dihypergraph is important for the topological Hurewicz +dichotomy for dihypergraphs on κκ. +114Htop +κ +was defined in [CMS20, Section 2] for κ = ω. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +71 +(2) A strict order preserving map ι : <κd → <κκ is an order homomorphism for Id +κ if +and only if ι preserves ⊥. +Proof. For (1), suppose first that f is injective. Since any sequence in Hκd is injective, its +f-image is also injective, i.e., it is in Id +κ. Conversely, suppose that f is a homomorphism +from Hκd to Id +κ. Take distinct elements x, y of κd and a hyperedge of Hκd that contains x +and y. Since the f-image of this hyperedge is in Id +κ, we have f(x) ̸= f(y). +For (2), it is clear that any ⊥-preserving map ι is an order homomorphism for Id +κ. +Conversely, suppose that ι is an order homomorphism for Id +κ. To see that ι-preserves ⊥, +it is enough to show that ι(u⌢⟨α⟩) ⊥ ι(u⌢⟨β⟩) holds for all u ∈ κd and all α, β < κ with +α ̸= β. Seeking a contradiction, suppose that +ι(u⌢⟨α⟩) ⊆ ι(u⌢⟨β⟩). +Consider the sequence x = ⟨xγ : γ < κ⟩ obtained by choosing an arbitrary element xγ of +Nι(u⌢⟨γ⟩) for all γ ∈ κ − {α} and letting xα = xβ. Then x is not injective. However, since +xα ∈ Nι(u⌢⟨β⟩) ⊆ Nι(u⌢⟨α⟩), we have x ∈ � +γ ω, even if we assume that f is injective.117 To see this, +let f := [θ] for the map θ defined in Remark 5.4 (2). +6. Applications +In this section, we prove a number of applications of ODDκ +κ(X). We obtain analogues +of classical results in descriptive set theory such as the Hurewicz dichotomy [Hur28, +Section 6],118 its extension by Kechris, Louveau and Woodin [KLW87, Theorem 4] and +the Jayne-Rogers theorem [JR82, Theorem 5]. We thereby lift results of Carroy, Miller +and Soukup [CMS20] to the uncountable setting. We also derive the asymmetric Baire +property [Sch17, Section 3] from the open dihypergraph dichotomy. +This implication +is new even in the countable setting. +We further find that surprisingly, ODDκ +κ(κκ) is +already an interesting principle with several applications. For instance, it implies the open +dihypergraph dichotomy for κ-analytic sets and therefore has the consistency strength of +at least an inaccessible cardinal. +We further show that it implies the determinacy of +V¨a¨an¨anen’s perfect set game [V¨a¨a91, Section 2] for all subsets of κκ and thereby extend +a result of V¨a¨an¨anen. +Since the Kechris-Louveau-Woodin dichotomy for definable sets and the results on +V¨a¨an¨anen’s game follow from ODDκ +κ(Dκ), a Mahlo cardinal suffices for these applications. +The remaining applications for definable sets follow from ODDκ +κ(Dκ, Dκ) and thus an +117I.e., a continuous homomorphism from Hκκ to Htop +κ +need not be a closed map. +118Hurewicz’ result for Polish spaces was proved for analytic sets by Kechris and Saint Raymond +[SR75,Kec77]. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +73 +inaccessible cardinal suffices. We do not know the precise consistency strength of these +statements (except for the results in Subsection 6.3.1). +6.1. The Hurewicz dichotomy. We prove two different generalizations of the Hurewicz +dichotomy for subsets of the space κκ. The first one is given in the next definition. Recall +that a topological space is κ-compact (or κ-Lindel¨of ) if every open cover has a subcover +of size strictly less than κ. Note that every κ-compact subset of κκ is closed. Call a space +Kκ if it is a union of at most κ many κ-compact subsets. +Definition 6.1 ([LMS16]). The topological Hurewicz dichotomy119 for a subset X of κκ +is the following statement: +THDκ(X): either X is contained in a Kκ subset of κκ, or X contains a +closed subset of κκ which is homeomorphic to κκ. +The alternatives are mutually exclusive,120 since any κ-compact subset of κκ is bounded +by an element of κκ. +We first obtain THDκ(X) as a special case of ODDκ +κ(X, Dκ). We use the dihypergraph +Htop +κ += Iκ ∩ Nκ from Definition 5.37. A subset Y of κκ is Htop +κ -independent if and only if +every injective sequence of length κ in Y has a convergent subsequence. Moreover, note +that Htop +κ +∈ Dκ is box-open on κκ. +Lemma 6.2. A subset Y of κκ is Htop +κ -independent if and only if its closure is κ-compact. +Proof. Suppose Y is not κ-compact. Suppose that ⟨Uα : α < κ⟩ is a sequence of pairwise +disjoint121 basic open sets with Y ⊆ � +α<κ Uα and Y ∩Uα ̸= ∅ for all α < κ. To show that +Y is not Htop +κ -independent, take an (injective) sequence y = ⟨yα : α < κ⟩ with yα ∈ Y ∩Uα +for each α < κ. Suppose that y has a subsequence z converging to some x ∈ Y . Note that +z has length κ, since y is injective. Let β < κ with x ∈ Uβ. Since at most one element of +z is in Uβ, z cannot converge to x. +Suppose Y is not Htop +κ -independent. Let y = ⟨yα : α < κ⟩ ∈ Htop +κ +∩ κY . Since y has no +convergent subsequences, {yα : α < κ} is a closed discrete subset of Y . Hence Y cannot +be κ-compact. +□ +Theorem 6.3.122 Suppose X is a subset of κκ. +(1) Htop +κ ↾X has a κ-coloring if and only if X is contained in a Kκ subset of κκ. +(2) There exists a continuous homomorphism from Hκκ to Htop +κ ↾X if and only if X +contains a closed subset of κκ that is homeomorphic to κκ. +Thus, THDκ(X) is equivalent to ODDHtop +κ ↾X +κ +. +119This is called the Hurewicz dichotomy in [LMS16]. +120This also follows from Theorem 6.3 below. +121We may assume that the basic open sets Uα’s are disjoint by replacing each Uα with a family of ≤κ +many pairwise disjoint basic open sets whose union is Uα \ � +β<α Uβ and then removing those sets whose +intersection with Y is empty. +122This was proved for κ = ω in [CMS20, Proposition 2.1] with a different proof which uses compactness +and thus does not generalize to the uncountable setting. + +74 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Proof. For (1), suppose first that Htop +κ ↾X has a κ-coloring. Take Htop +κ -independent sets Xα +for α < κ with X = � +α<κ Xα. Since Xα is Htop +κ -independent, its closure Xα is κ-compact +by the previous lemma. Thus, X is a subset of the Kκ set � +α<κ Xα. Conversely, suppose +that X ⊆ � +α<κ Yα, where each Yα is κ-compact. Since Yα is Htop +κ -independent by the +previous lemma, we obtain a κ-coloring of X. +(2) follows from Lemma 5.39 (2) and Theorem 5.40 (2) for H := Htop +κ ↾X. +□ +We sketch an alternative proof of the equivalence THDκ(X) ⇔ ODDHtop +κ ↾X +κ +. +If κ is +weakly compact, then the proof of [CMS20, Proposition 2.1] works without major changes. +Weak compactness is needed to ensure that all bounded subsets of κκ are κ-compact. If κ +is not weakly compact, then κ2 is homeomorphic to κκ [HN73, Theorem 1]. In this case, +THDκ(X) already follows from the κ-perfect set property for X [LMS16, Proposition 2.7]. +We next consider a different version of the Hurewicz dichotomy for κκ. A node t in a +tree T is called <κ-splitting (resp. κ-splitting) if it has <κ many (resp. precisely κ many) +direct successors in T. +Definition 6.4. Suppose T is a subtree of <κκ. +(a) T is called <κ-splitting if every node in T is <κ-splitting. +(b) T is κ-superperfect if it is <κ-closed and every s ∈ T extends to a κ-splitting node +t ∈ T. +(c) A subset X of κκ is κ-superperfect if X = [T] for some κ-superperfect subtree T +of <κκ. +Definition 6.5 ([LMS18]). The Hurewicz dichotomy for a subset X of κκ is the statement: +HDκ(X): Either X contains a κ-superperfect subset, or there exists a +sequence ⟨Tα : α < κ⟩ of <κ-splitting subtrees of <κκ with X ⊆ � +α<κ[Tα]. +We will obtain HDκ(X) as a special case of ODDκ +κ(X, Dκ). +Definition 6.6. Let Hsuper +κ +denote the κ-dihypergraph on κκ which consists of all se- +quences ⟨xα : α < κ⟩ such that all the xα’s split at the same node.123 +Note that Hsuper +κ +∈ Dκ is box-open on κκ. +Lemma 6.7. A subset Y of κκ is Hsuper +κ +-independent if and only if T(Y ) is <κ-splitting. +Proof. If T(Y ) has a κ-splitting node, then Y contains a sequence in Hsuper +κ +. The converse +is similar. +□ +Theorem 6.8. Suppose X is a subset of κκ. +(1) Hsuper +κ +↾X admits a κ-coloring if and only if there exists a sequence ⟨Tα : α < κ⟩ of +<κ-splitting trees with X ⊆ � +α<κ[Tα]. +(2) There exists a continuous homomorphism from Hκκ to Hsuper +κ +↾X if and only if X +contains a κ-superperfect subset. +Thus, HDκ(X) is equivalent to ODDHsuper +κ +↾X +κ +. +123I.e., the xα’s extend pairwise different immediate successors of the some t ∈ <κκ. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +75 +Proof. For (1), suppose first that Hsuper +κ +↾X has a κ-coloring. Take Hsuper +κ +-independent +sets Xα for α < κ with X = � +α<κ Xα. Then X ⊆ � +α<κ Xα = � +α<κ[T(Xα)]. Since +Xα is Hsuper +κ +-independent, T(Xα) is <κ-splitting by the previous lemma. +Conversely, +suppose that X ⊆ � +α<κ[Tα], where each Tα is <κ-splitting. We may assume that each +Tα is pruned. Since each [Tα] is Hsuper +κ +-independent by the previous lemma, we obtain a +κ-coloring of X. +For (2), suppose first that there exists a continuous homomorphism from Hκκ to +Hsuper +κ +↾X. Then there is a ⊥-preserving continuous order homomorphism ι for (X, Hsuper +κ +) +by Lemmas 3.11 and 5.38 (2). Then T(ι) is a κ-superperfect tree, since it is <κ-closed +by Corollary 5.6 and each ι(t) is a κ-splitting node of T(ι). Since Hsuper +κ +⊆ Htop +κ , [ι] is a +closed map by Lemma 5.39 (3). In particular, ran([ι]) is closed. Thus, ran([ι]) = [T(ι)] is +a κ-superperfect subset of X. +Conversely, suppose that T is a κ-superperfect tree with [T] ⊆ X. Let ι : <κκ → T be +any order preserving map such that for all u ∈ <κκ, ⟨ι(u)⌢⟨α⟩ : α < κ⟩ is an injective +enumeration of the immediate successors of some κ-splitting node t ⊇ ι(u) of T. Then ι +is an order homomorphism for Hsuper +κ +, so [ι] is a continuous homomorphism from Hκκ to +Hsuper +κ +↾X by Lemma 3.11. +□ +The previous two theorems and the main result, Theorem 1.4, yield a new proof of the +following theorem from [LMS18]. Given a class C, let HDκ(C) state that HDκ(X) holds +for all subsets X ∈ C of κκ. THDκ(C) is defined analogously. +Corollary 6.9. Both HDκ(Dκ) and THDκ(Dκ) hold in all Col(κ, <λ)-generic extensions +if λ > κ is inaccessible in V . +Figure 6 summarizes the implications between the options in the two different variants +of the Hurewicz dichotomy, under varying assumptions on κ. +(TH2) X contains a closed subset of κκ which +is homeomorphic to κκ. +(HD2) X contains a κ-superperfect subset. +(TH1) X is contained in a Kκ subset of κκ. +(HD1) X is contained in a set of the form +� +α<κ[Tα], where each Tα is <κ-splitting. +A +B: A and B are equivalent for all X when κ is weakly compact or κ = ω. +A +B: A implies B for all X, and the reverse implication fails for X = κ2 when +κ > ω is not weakly compact. +Figure 6. The options in the two versions of the Hurewicz dichotomy. +It is easy to see that (TH1) ⇒ (HD1) and (HD2) ⇒ (TH2).124 If κ is weakly compact +or κ = ω, then the converse implications also hold by [LMS16, Lemma 2.6 & Proposi- +tion 2.11]. In particular, HDκ(X) ⇔ THDκ(X). However, if κ > ω is not weakly compact, +124This also follow from the previous two theorems, since Hsuper +κ +⊆ Htop +κ . + +76 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +then (HD1) holds for κ2 since <κ2 is <κ-splitting, while (TH2) holds for κ2 since κ2 is +homeomorphic to κκ. +6.2. The Kechris-Louveau-Woodin dichotomy. Define π : <κκ → <κ2 by letting +π(t) := +� +α κ is inaccessible in V . +(2) KLWκ(Dκ) if λ > κ is Mahlo in V . +Remark 6.21. Instead of the dihypergraph CY defined in the beginning of this sub- +section, we can instead work with the box-open dihypergraph CY ∩ Iκ consisting of all +injective sequences in CY . All proofs in this and the following subsections can be adapted. +(i) The notion of independence is the same for CY and CY ∩ Iκ, so the existence of +a κ-coloring is equivalent. To see this, note that the CY ∩ Iκ-independence of A +is also equivalent to Y ∩ A′ = ∅, since each x ∈ A′ is the limit of an injective +sequence in A \ {x}. +(ii) Lemma 6.13 and Theorem 6.19 (1) hold for CY ∩ Iκ by the previous observation. +(iii) Lemmas 6.14 and 6.15 need to be modified as follows: +(a) If f is a continuous injective reduction of (Rκ, Qκ) to (X, Y ), then f ◦ [π] +is a continuous homomorphism from Hκκ to (CY ↾X) ∩ Iκ +κ. This follows from +Lemma 6.14, since any injective map is a homomorphism from Hκκ to Iκ +κ by +Lemma 5.38 (1). +130If X and Y are disjoint, then it is equivalent that there exists a Σ0 +2(κ) set A separating X from Y . + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +81 +(b) If t = ⟨tα : α < κ⟩ is a sequence of elements of T(X), then � +α<κ Ntα ∩ X ⊆ +CY ∩ Iκ is equivalent to the existence of a sequence t of pairwise incompat- +ible elements converging to some y ∈ Y . +This follows from Lemma 6.15 +by observing that � +α<κ Ntα ∩ X ⊆ Iκ if and only if the tα’s are pairwise +incompatible. +(iv) To see that Theorem 6.19 (2) holds for (CY ∩ Iκ)↾X, note that an order homo- +morphism ι for (X, CY ∩ Iκ) is also an order homomorphism for (X, CY ). One +therefore obtains a nice reduction for (X, Y ) by Lemma 6.18. This shows (d) ⇒ +(a). The other implications follow from (iii) (a) and Theorem 6.19 (2). +6.3. Applications of ODDκ +κ(κκ). Here we study only dihypergraphs with domain κκ. +There is no immediate reason why ODDκ +κ(κκ) should have interesting applications. It +implies ODDκ +κ(X) for subsets X of κκ that are continuous images of κκ by Lemma 3.5, +but for uncountable κ not all closed subsets of κκ are of this form [LS15, Theorem 1.5]. +However, we will show that in fact ODDκ +κ(κκ) ⇒ ODDκ +κ(Σ1 +1(κ)) and a similar implication +holds for the restrictions to dihypergraphs in Dκ. +6.3.1. The perfect set property for closed sets. We begin with a short proof of a +special case that shows ODDκ +κ(κκ, Dκ) already implies the perfect set property PSPκ(X) +for closed subsets X of κκ.131 It follows that ODDκ +κ(κκ, Dκ) has the same consistency +strength as an inaccessible cardinal. We will extend this case in a different direction in +Subsection 6.4 to study the determinacy of V¨a¨an¨anen’s perfect set game. +We use the dihypergraph CX from Section 6.2. It consists of all convergent sequences +x in κκ with limα<κ(x) ∈ X \ ran(x). Note that CX ∈ Dκ for whenever X ∈ Dκ, and +in particular, whenever X is a closed subset of κκ. +Recall that a subset A of κκ is +CX-independent if and only if A′ ∩ X = ∅, where A′ denotes the set of limit points of A. +Lemma 6.22. Suppose X is a subset of κκ. +(1) If CX has a κ-coloring, then |X| ≤ κ. +(2) If |X| ≤ κ, then CX has a κ-coloring. +Proof. For (1), take CX-independent sets Aα for α < κ with κκ = � +α<κ Aα. For each +α < κ, let Bα := X ∩ Aα. Since Bα is a subset of X with no limit points in X, it is +discrete, so |Bα| ≤ κ. Since X ⊆ � +α<κ Bα, we have |X| ≤ κ. +For (2), it suffices to show that κκ \ X is the union of κ many CX-independent sets. +Write κκ \ X as the union of basic open sets Nsi for i < α, where α ≤ κ. Each Nsi is +CX-independent, since it is closed and thus sequences in Nsi cannot converge to elements +of X. +□ +Remark 6.23. The converse of (1) fails for any subset X of κκ of size κ that is dense in +some Nt.132 To see that CX does not have a κ-coloring, let ⟨Xα : α < κ⟩ be a sequence of +131It is easy to see that the perfect set property PSPκ(X) follows from ODDκ +κ(X, Dκ) for any subset +X of κκ. To see this, note that the range of any continuous homomorphism from Hκκ to the complete +κ-hypergraph Kκ +X on X contains a κ-perfect subset by Lemma 5.10 (1). +132In fact, there exists a continuous homomorphism from Hκκ to CX for any subset X that is somewhere +dense by Lemma 6.33 below and Theorem 6.19 (2). + +82 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +CX-independent sets covering κκ. We can assume that Xα is closed, since the closure of +any CX-independent set is also CX-independent.133 Then some Xα ∩ Nt has nonempty +interior in Nt, so it contains a nonempty basic open set Nu. Since X ∩ Nu ̸= ∅, Xα is not +CX-independent. The previous argument shows that the existence of a κ-coloring for CX +implies that X is nowhere dense. +The converse of (2) fails for the set X of all t⌢⟨2⟩⌢⟨0⟩κ with t ∈ <κ2. Note that +X = X ∪ κ2. Now CX admits a κ-coloring, since κ2 is CX-independent and κκ \ X is +covered by κ many CX-independent sets as in the proof of (2). +Theorem 6.24. Suppose X is a closed subset of κκ. +(1) CX has a κ-coloring if and only if |X| ≤ κ. +(2) There exists a continuous homomorphism from Hκκ to CX if and only if X has a +κ-perfect subset. +In particular, PSPκ(X) is equivalent to each of ODDCX +κ +and KLWκ(κκ, X).134 +Proof. (1) was proved in the previous lemma. For (2), recall from Theorem 6.19 (2) that +the existence of a continuous homomorphism from Hκκ to CX is equivalent to the existence +of a nice reduction for (κκ, X), i.e., a ⊥- and strict order preserving map Φ : <κ2 → <κκ +with [Φ](Qκ) ⊆ X, where the last inclusion is equivalent to ran([Φ]) ⊆ X since X is closed +and [Φ](Qκ) is a dense subset of ran([Φ]). By Lemma 5.8, the existence of such a nice +reduction is thus equivalent to the existence of a κ-perfect subset of X. +□ +The previous theorem also holds for CX ∩ Iκ instead of CX by Remark 6.21. +6.3.2. ODDκ +κ(X) for κ-analytic sets. The following proofs use ideas from the Kechris- +Louveau-Woodin dichotomy in Subsection 6.2. We first fix some notation. Suppose that +X = ⟨Xα : α < κ⟩ is a sequence of subsets of κκ. We say that X converges to x ∈ κκ if +for every γ < κ, there exists some α < κ with � +α<β<κ Xβ ⊆ Ny↾γ.135 Moreover, if H is a +κ-dihypergraph on κκ, then we write Hy for the set {x ∈ κ(κκ) : ⟨y⟩⌢x ∈ H}. +Towards proving ODDκ +κ(X) for a subset X of κκ, suppose that H is a κ-dihypergraph +on κκ. Let CX,H denote the (κ × κ)-dihypergraph on κκ which consists of all sequences +⟨x(α,i) : (α, i) ∈ κ × κ⟩ with x(α,i) ∈ κκ such that the sets Xα := {x(α,i) : i < κ} converge +to an element y of X such that xα := ⟨x(α,i) : i < κ⟩ ∈ Hy for all α < κ (see Figure 9). +Moreover, let C +X,H denote the κ-dihypergraph on κκ obtained in the natural way from +CX,H via a bijection κ × κ → κ. If H is a box-open dihypergraph on κκ, then CX,H and +C +X,H are box-open on κκ. Note that both are in Dκ whenever X, H ∈ Dκ. We will show +that if X is closed and H is a box-open dihypergraph on κκ,136 then ODDH↾X +κ +is equivalent +to ODDC +X,H +κ +(equivalently, ODDCX,H +κ +). Since every Σ1 +1(κ) subset of κκ is a continuous +image of a closed subset, the results described at the beginning of the subsection will +follow from this result by Lemma 3.5. +133This is is clear for CX but is also true for box-open dihypergraphs in general by Corollary 3.14. +134Recall that ODDCX +κ +and KLWκ(κκ, X) are equivalent by Theorem 6.19. +135Note that if Xα = Ntα for each α < κ, then X converges to x if and only if ⟨tα : α < κ⟩ converges +to x (see page 77). +136The following Lemma 6.25 only uses that H↾X is box-open on X. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +83 +xβ ∈ Hy +x(β,i) +xα ∈ Hy +x(α,i) +y +Figure 9. CX,H +In the next two lemmas, X is a subset of κκ and H is a κ-dihypergraph on κκ. +Lemma 6.25. +(1) If CX,H has a κ-coloring, then H↾X has a κ-coloring. +(2) Suppose X is closed and H↾X is box-open on X. If H↾X has a κ-coloring, then +CX,H has a κ-coloring. +Proof. For (1), suppose that Y ⊆ X is CX,H-independent. We define a κ-coloring for +H↾Y as follows. For each y ∈ Y , there exists some c(y) ⊊ y such that Hy↾(Y ∩ Nc(y)) = ∅ +because otherwise, it would be possible to construct a hyperedge of CX,H↾Y by choosing +an arbitrary ⟨x(α,i) : i < κ⟩ ∈ Hy↾(Y ∩ Ny↾α) for each α < κ. We claim that for each +t ∈ <κκ, the set Yt := {y ∈ Y : c(y) = t} is H-independent. Otherwise, pick a sequence +⟨yi : i < κ⟩ ∈ H↾Yt. Then ⟨yi : 1 ≤ i < κ⟩ ∈ Hy0↾Yt. Since Yt ⊆ Y ∩ Nt and c(y0) = t, this +contradicts the definition of c. +For (2), first observe that κκ \ X is the union of κ many CX,H-independent sets. To +see this, write κκ \ X as the union of basic open sets Nsi for i < α, where α ≤ κ. Each +Nsi is CX,H-independent, since it is closed and thus sequences in Nsi cannot converge to +elements of X. +It thus suffices to show that every H-independent subset Y of X is CX,H-independent. +Let Y be such a set. +Then Y ⊆ X since X is closed. +Y is also H-independent by +Corollary 3.14, since H↾X is box-open on X. We claim that Y is CX,H-independent. +Otherwise pick some hyperedge ⟨x(α,i) : (α, i) ∈ κ × κ⟩ of CX,H↾Y . The limit y of the sets +Xα := {x(α,i) : i < κ} is in Y . But then ⟨y⟩⌢⟨x(0,i) : i < κ⟩ ∈ H↾Y . +□ +Lemma 6.26. Let ι : <κ(κ × κ) → <κκ be a strict order preserving map and let Xt +α := +� +i<κ Nι(t⌢⟨(α,i)⟩) for all t ∈ <κ(κ×κ) and α < κ. The following statements are equivalent: +(1) ι is an order homomorphism for (κκ, CX,H). +(2) For each t ∈ <κ(κ × κ), the sequence ⟨Xt +α : α < κ⟩ converges to some xt ∈ X with +� +i<κ Nι(t⌢⟨(α,i)⟩) ⊆ Hxt for all α < κ. +Proof. This is similar to the proof of Lemma 6.15. For (2) ⇒ (1), let t ∈ <κ(κ × κ), and +suppose that ⟨Xt +α : α < κ⟩ converges to xt ∈ X. Take an arbitrary sequence a = ⟨a(α,i) : + +84 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +α, i < κ⟩ with a(α,i) ∈ Nι(t⌢⟨(α,i)⟩) for all (α, i) ∈ κ × κ. Then Aα := {a(a,i) : i < κ} is a +subset of Xt +α for all α < κ, so the sets Aα converge to xt. If � +i<κ Nι(t⌢⟨(α,i)⟩) ⊆ Hxt for +each α < κ, then ⟨a(α,i) : i < κ⟩ ∈ Hxt for each α < κ, so a ∈ CX,H. This shows that ι is +an order homomorphism for (κκ, CX,H). +For (1) ⇒ (2), fix t ∈ <κ(κ × κ) and a = ⟨a(α,i) : α, i < κ⟩ with a(α,i) ∈ Nι(t⌢⟨(α,i)⟩) for +all (α, i) ∈ κ×κ. Since a ∈ CX,H, the sets Aα := {a(a,i) : i < κ} converge to some xt ∈ X. +We first show that ⟨Xt +α : α < κ⟩ converges to xt. Otherwise, there exists some γ < κ and +a sequence ⟨xα : α < κ⟩ with xα ∈ Xt +α and xα /∈ Nxt↾γ for unboundedly many α < κ. +Take any sequence b = ⟨b(α,i) : α, i < κ⟩ with b(α,i) ∈ Nι(t⌢⟨(α,i)⟩) and xα ∈ {b(α,i) : i < κ}. +Since b ∈ CX,H, the sets Bα := {b(α,i) : i < κ} converge to some yt ∈ X. Now xt ̸= yt +since Bα ̸⊆ Nxt↾γ for unboundedly many α < κ. For all (α, i) ∈ κ × κ, let +c(α,i) := +� +� +� +a(α,i) +if α is even, +b(α,i) +if α is odd. +Since ι is an order homomorphism for (κκ, CX,H), the sets Cα := {c(α,i) : i < κ} converge. +But they cannot converge to both xt and yt at the same time. It now suffices to show for +any α < κ that any a = ⟨a(α,i) : i < κ⟩ in � +i<κ Nι(t⌢⟨(α,i)⟩) is also in Hxt. This follows +from the fact that we may extend a to a hyperedge of CX,H by choosing a(β,i) ∈ Nι(t⌢⟨(β,i)⟩) +arbitrarily for all β ̸= α and i < κ. This fact holds since ι is an order homomorphism for +(κκ, CX,H). +□ +Remark 6.27. For any order homomorphism ι : <κ(κ×κ) → <κκ for (κκ, CX,H), we have +ran(ι) ⊆ T(X) and ran([ι]) ⊆ X. To see this, note that Xt +α ⊆ Nι(t) for each t ∈ <κ(κ×κ). +Since the sets Xt +α converge to some xt ∈ X by the previous lemma, we have xt ∈ Nι(t) +and thus ι(t) ∈ T(X). +Theorem 6.28. Let X be a closed subset of κκ, and let H be a box-open κ-dihypergraph +on κκ. +(1) H↾X has a κ-coloring if and only if CX,H has a κ-coloring. +(2) There is a continuous homomorphism from Hκκ to H↾X if and only if there is a +continuous homomorphism from Hκ(κ×κ) to CX,H. +In particular, ODDH↾X +κ +is equivalent to ODDCX,H +κ +. +Proof. (1) was proved in Lemma 6.25. For (2), define the map πκ×κ : <κ(κ × κ) → <κκ +by letting +πκ×κ(t) = +� +α ω. For instance, there exists a closed subset X of ω1ω1 such +that Vω+1(X) is not determined, and it is consistent that there exists such a set X of +size ω2 [V¨a¨a91, Section 3].141 The next property can be understood as the determinacy +of V¨a¨an¨anen’s game of length κ with an extra condition, or as a variant of the Cantor- +Bendixson analysis by the previous discussion. Let X be a subset of κκ. +CB1 +κ(X): Either |X| ≤ κ and I wins Vκ(X), or II wins Vκ(X). +Equivalently, CB1 +κ(X) states that either X is κ-scattered and of size ≤κ, or X contains a +κ-crowded subset.142 This principle may fail even for closed sets; in fact, it is consistent +that there exists an ω1-scattered closed subset of ω1ω1 of size ω2 [V¨a¨a91, Theorem 3].143 +For the next theorem, we use the dihypergraph CX from Section 6.2. It consists of +all convergent sequences x in κκ with limα<κ(x) ∈ X \ ran(x). The next theorem shows +ODDκ +κ(κκ, Dκ) ⇒ CB1 +κ(Dκ) and ODDκ +κ(κκ) ⇒ CB1 +κ(P(κκ)). +Theorem 6.31. +Suppose X is a subset of κκ. +(1) If CX has a κ-coloring, then |X| ≤ κ and I wins Vκ(X). +(2) There exists a continuous homomorphism from Hκκ to CX if and only if II wins +Vκ(X). +In particular, each of ODDCX +κ +and KLWκ(κκ, X)144 implies CB1 +κ(X). +It is open whether the converse of (1) holds for all subsets X of κκ. It is equivalent +that CB1 +κ(X) ⇒ ODDCX +κ +for all X. Note that this holds for all closed subsets X of κκ by +Lemma 6.22. +Given the characterisation in Theorem 6.19, the previous theorem follows from the next +two lemmas. It also holds for CX ∩ Iκ instead of CX by Remark 6.21. +Lemma 6.32. Let X ⊆ κκ. If CX has a κ-coloring, then |X| ≤ κ and I wins Vκ(X). +Proof. We have already shown |X| ≤ κ in Lemma 6.22 (1). For the second claim, recall +that a subset A of κκ is CX-independent if and only if A′ ∩ X = ∅, where A′ denotes the +set of limit points of A. Take CX-independent sets Aα with κκ = � +α<κ Aα. We define +the following strategy for player I. In rounds of the form α + 1, I chooses γα+1 so that +140I.e., α + β < ξ for all α, β < ξ. +141See [Gal16, Secion 1.5] for variants of these results. +142If II wins Vκ(X), then Kerκ(X) ̸= ∅ is a κ-crowded subset of X. +143See [Gal16, Section 1.5] for variants of this result. +144Recall that ODDCX +κ +and KLWκ(κκ, X) are equivalent by Theorem 6.19. + +88 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +(i) xα↾γα+1 ̸= xβ↾γα+1 for all β < α, and +(ii) Aα ∩ Nxα↾γα+1 ⊆ {xα}. +I can achieve the first property because II plays an injective sequence and the second +property because xα ∈ X cannot be a limit point of Aα. Suppose II wins a run of the +game where I has used this strategy. Let y := � +α<κ xα↾γα+1. Then y ∈ Aβ for some +β < κ. Since y ∈ Nxβ↾γβ+1, we must have y = xβ by (ii). This contradicts (i) for any +α > β. +□ +Lemma 6.33. 145 The following statements are equivalent for any X ⊆ κκ and x ∈ X. +(1) x ∈ Kerκ(X). +(2) There exists Y ⊆ X such that x ∈ Y and Y is κ-perfect. +(3) There is a nice reduction146 Φ for (κκ, X) with x ∈ [Φ] +� +Qκ). +Proof. For (3)⇒(2), suppose Φ is a nice reduction for (κκ, X). Let Y := [Φ] +� +Qκ). Then +x ∈ Y ⊆ X. Moreover, Y = ran([Φ]) is κ-perfect, since it is a closed homeomorphic copy +of κ2 by Corollary 5.2. +For (2)⇒(1), let Y be a subset of X with x ∈ Y such that Y is κ-perfect. Then T(Y ) is +a κ-perfect tree. It is straightforward to construct a winning strategy τ for II in Vκ(Y, x) +using the fact that T(Y ) is cofinally splitting in successor rounds of the game and the +fact that T(Y ) is <κ-closed in limit rounds. Since Y ⊆ X, τ is also a winning strategy +for II in Vκ(X, x). +For (1)⇒(3), suppose that τ is a winning strategy for II in Vκ(X, x). We construct a +nice reduction for (κκ, X) by having II use τ repeatedly in response to different partial +plays of I. In detail, we construct a ⊥- and strict order preserving function Φ: <κ2 → +T(X), a continuous strict order preserving function ⟨γs : s ∈ Sκ⟩147 of ordinals below κ +and a function ⟨xs : s ∈ <κκ⟩ of elements of X such that the following hold for all s ∈ <κ2: +(i) γ∅ = 0 and x∅ = x. +(ii) Φ(s) = xs↾γs⌢⟨1⟩. +(iii) xs = τ +� +⟨γt : t ⊆ s, t ∈ Sκ⟩ +� +. +For all s ∈ <κκ, we define ps := ⟨γt, xt : t ⊆ s, t ∈ Sκ⟩. (iii) states that ps is a partial +run of Vκ(X) where II uses the strategy τ. Note that xs is constant in intervals disjoint +from Sκ, but takes a new value at each s ∈ Sκ. If s ∈ Sκ, then �{pu : u ⊊ s, u ∈ Sκ} is +extended to ps by I playing γs as defined below and II choosing xs using τ. Otherwise, +s = t⌢⟨0⟩α for some α < κ. Then ps = pt, so xs = xt. +We shall construct xs, γs and Φ(s) by recursion on lh(s). In fact we only need to define +γs, since Φ(s) and xs are then determined by (i)-(iii). It suffices to choose I’s moves γs⌢⟨1⟩ +so that Φ is ⊥- and strict order preserving. +145Our proof is virtually the same as in [Szir18, Proposition 2.69]. The notation in [Szir18] is different +from ours. There, a subset Y of κκ is called κ-strongly dense in itself if Y is κ-perfect and Kerξ(X) +denotes V¨a¨an¨anen’s definition of the kernel. +146See Definition 6.11. +147Recall the notations Qκ and Sκ from Subsection 6.2. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +89 +It remains to construct xs, γs⌢⟨1⟩ and Φ(s) simultaneously. Suppose xt, γt⌢⟨1⟩ and +Φ(t) have been constructed for all t ⊊ s. Let xs be as in (iii). The next claim shows that +Φ(s) := xs↾γs⌢⟨1⟩ extends Φ(t) for all t ⊊ s if γs⌢⟨1⟩ is chosen sufficiently large. +Claim. Φ(t) ⊊ xs for all t ⊊ s. +Proof. If s = t⌢⟨0⟩α, then Φ(t) ⊆ xt = xs. Otherwise, take the least α ≥ lh(t) with +s(α) = 1. Then s ⊇ t⌢⟨0⟩α⌢⟨1⟩. By (ii) applied to t⌢⟨0⟩α, we have +Φ(t) ⊆ Φ(t⌢⟨0⟩α⌢⟨1⟩) = (xt⌢⟨0⟩α↾γt⌢⟨0⟩α⌢⟨1⟩) ⊆ xs. +The last inclusion holds since xs is chosen according to the rules of Vκ(X). +□ +Suppose s = t⌢⟨i⟩. Since II always has to play an injective sequence in Vκ(X, x), +we have xt⌢⟨0⟩ = xt ̸= xt⌢⟨1⟩. +Thus, we may choose γt⌢⟨0,1⟩ and γt⌢⟨1,1⟩ so that +Φ(t⌢⟨0⟩) = xt⌢⟨0⟩↾γt⌢⟨0,1⟩ and Φ(t⌢⟨1⟩) = xt⌢⟨1⟩↾γt⌢⟨1,1⟩ are incompatible. This com- +pletes the construction. +Claim. [Φ](π(u)⌢⟨0⟩κ) = xπ(u) for all u ∈ <κκ. +Proof. We have [Φ](π(u)⌢⟨0⟩κ) = � +α<κ Φ(π(u)⌢⟨0⟩α) = xπ(u). The first equality holds +by the definition of [Φ]. The last one holds since ⟨Φ(π(u)⌢⟨0⟩α) : α < κ⟩ is a strictly +increasing chain of initial segments of xπ(u) by (ii) and since xs = xt if s = t⌢⟨0⟩α. +□ +By the previous claim, [Φ](π(∅)⌢⟨0⟩κ) = x∅ = x and [Φ](Qκ) ⊆ {xs : s ∈ <κκ} ⊆ X. +So Φ is as required in (3). +□ +Remark 6.34. Lemmas 6.32 and 6.33 generalize the following observations from closed +subsets to arbitrary subsets of κκ: +(i) If X is closed and |X| ≤ κ, then I wins Vκ(X) [V¨a¨a91, Proposition 3]. +(ii) If X is closed, then Kerκ(X) is the union of all κ-perfect subsets of X [V¨a¨a91, +Lemma 1 & Proposition 1]. +These observations show that CB1 +κ(X) is equivalent to the κ-perfect set property PSPκ(X) +for closed sets X. Together with Theorem 6.31, this gives a alternative proof of the fact +that ODDκ +κ(κκ, Dκ) implies PSPκ(X) for all closed sets X. +The next property is an analogue of Cantor-Bendixson analysis: it ensures that a +subset X of κκ can be turned into a κ-crowded set by first removing a κ-scattered subset +consisting of up to κ many points. +CB2 +κ(X) : X = Kerκ(X) ∪ Scκ(X) and |Scκ(X)| ≤ κ. +Note that CB2 +κ(X) implies that Vκ(X, x) is determined for every x ∈ X, hence Vκ(X) is +determined. +For i = 1, 2, let CBi +κ(C) state that CBi +κ(X) holds for all subsets X ∈ C of κκ. +Theorem 6.35. Suppose that C is a class. +(1) CB2 +κ(C) implies CB1 +κ(C). +(2) If X ∩ Nt ∈ C for all X ∈ C with X ⊆ κκ and t ∈ κκ, then CB1 +κ(C) ⇔ CB2 +κ(C). +In particular ODDκ +κ(κκ, Dκ) ⇒ CB2 +κ(Dκ) and ODDκ +κ(κκ) ⇒ CB2 +κ(P(κκ)). + +90 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Proof. The last assertion in the proposition follows immediately from (1), (2) and Theo- +rem 6.31. For (1), let X ⊆ κκ and assume CB2 +κ(X). To see CB1 +κ(X), suppose that II does +not win Vκ(X). Then Kerκ(X) = ∅. By CB2 +κ(X), X = Scκ(X) is of size κ, as required. +Now, assume that C is as in (2) and that CB1 +κ(C) holds. Let X ∈ C. To show CB2 +κ(X), +let S denote the set of those s ∈ <κκ such that X ∩ Ns is κ-scattered set of size ≤ κ. We +will prove that +X \ Kerκ(X) ⊆ � +s∈S +(X ∩ Ns) ⊆ Scκ(X). +This is sufficient for CB2 +κ(X) since Scκ(X)∩Kerκ(X) = ∅ and � +s∈S(X ∩Ns) has size ≤ κ. +To see the first inclusion, let x ∈ X \ Kerκ(X). Then II does not win Vκ(X, y), so +there is some γ such that II does not win Vκ(X ∩ Ny↾γ). Since X ∩ Ny↾γ ∈ C, we have +CB1 +κ(X ∩ Ny↾γ). Therefore |X ∩ Ny↾γ| ≤ κ and I wins X ∩ Ny↾γ, i.e., y↾γ ∈ S. For the +second inclusion, let y ∈ � +s∈S(X ∩Ns). Then y↾γ ∈ S for some γ < κ. I has the following +winning strategy in Vκ(X, y): I first plays γ0 = 0 and γ1 = γ. In subsequent rounds, I +uses their winning strategy in Vκ(X ∩ Ny↾γ) to determine their moves in Vκ(X, y). Thus, +y ∈ Scκ(X). +□ +Using the main result Theorem 1.4, the results in this subsection yield: +Corollary 6.36. The following hold in all Col(κ, <λ)-generic extensions of V : +(1) CB2 +κ(Dκ) if λ > κ is inaccessible in V . +(2) CB2 +κ(P(κκ)) if λ > κ is Mahlo in V . +We can equivalently replace CB2 +κ(C) by CB1 +κ(C), where C denotes Dκ and P(κκ), respec- +tively. Therefore in the situation of (2), V¨a¨an¨anen’s game is determined for all subsets of +κκ. This strengthens two results of V¨a¨an¨anen [V¨a¨a91, Theorem 1 & Theorem 4] showing +that the principle I(ω) that is equiconsistent with a measurable cardinal148 implies (a) +II wins Vω1(X) for all subsets X of ω1ω1 of size >ω1149 and (b) CB2 +ω1(X) for all closed +subsets X of ω1ω1.150 +Using the previous corollary and [Szir18, Corollary 4.35], we can lower the consistency +strength of the dichotomy on the size of complete subhypergraphs (cliques) of finite di- +mensional Π0 +2(κ) dihypergraphs on Σ1 +1(κ) sets studied in [SzV17] and its generalization +to families of hypergraphs. Given a family H of dihypergraphs on a set X, we say that +Y ⊆ X is an H-clique if KdH +Y +⊆ H for each H ∈ H, where dH denotes the arity of H. +DKκ(X): If H is a set of κ many finite dimensional Π0 +2(κ) dihypergraphs on +X and there is an H-clique of size κ+, then there is a κ-perfect H-clique.151 +148I(ω) holds after L´evy-collapsing a measurable cardinal to ω2 [GJM78] and it implies the existence +of a precipitous ideal on ω2. +149The latter statement follows from CB1 +ω1(X) for all X ⊆ ω1ω1. +150CB1 +ω1(X) and CB2 +ω1(X) are equivalent in this situation by Theorem 6.35. +151Π0 +2(κ) can equivalently be replaced by open. DKκ(X) is a statement about complete subhypergraphs +of certain product-open ω-dihypergraphs, as Y is an H-clique if and only if Kω +Y ⊆ � +H∈H p−1 +ω,dH (H)↾X. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +91 +DKκ(X) extends a dichotomy of Doleˇzal and Kubi´s [DK16] to uncountable cardinals.152 +By [Szir18, Corollary 4.35],153 DKκ(Σ1 +1(κ)) follows from ♦i +κ and +CB− +κ : every subset of κκ of size κ+ has a κ-crowded subset. +Since CB− +κ follows from CB1 +κ(P(κκ)), Corollary 6.36 yields: +Corollary 6.37. DKκ(Σ1 +1(κ)) holds in all Col(κ, <λ)-generic extensions of V if λ > κ is +a Mahlo cardinal in V . +This strengthens a joint result of V¨a¨an¨anen and the second-listed author [SzV17] show- +ing that DKκ(Σ1 +1(κ)) holds in Col(κ, <λ)-generic extensions for any measurable cardinal +λ > κ.154 It is open whether one can obtain DKκ(Σ1 +1) directly from ODDκ +κ(κκ) by con- +structing a suitable dihypergraph. Moreover, we have not considered the case of <κ- +dimensional dihypergraphs. +Remark 6.38. Galgon showed that CB2 +κ(X) for all closed subsets X of κκ can be obtained +by L´evy-collapsing an inaccessible cardinal to κ+ [Gal16, Theorem 1.4.5]. This statement +is in fact equivalent to PSPκ(X) for all closed subsets X of κκ by a result of the second- +listed author [Szir18, Proposition 2.16]. Our proof of Theorem 6.35 generalizes the proof +of the latter. +Remark 6.39. Suppose X and Y are subsets of κκ. We study a more general game +VY +κ (X) that is defined by modifying the winning condition in Vκ(X). II wins a run if and +only if � +α<κ xα↾γα+1 ∈ Y . We claim that ODDCX +κ +155 implies that this game is determined. +If CX↾Y has a κ-coloring then |Y ∩ X| ≤ κ and I wins VY +κ (X) as in Lemmas 6.22 +and 6.32.156 In more detail, we use that a subset A of κκ is CX↾Y -independent if and +only if (A ∩ Y )′ ∩ X = ∅ and replace condition (ii) in the proof of Lemma 6.32 by +Aα ∩ Y ∩ Nxα↾γα+1 ⊆ {xα}. +Now suppose that there is a continuous homomorphism from Hκκ to CX↾Y . Let Φ +be a nice reduction for (Y, X) by Theorem 6.19.157 We define the following strategy for +II in VY +κ (X). Suppose I has played ⟨γβ : β ≤ α⟩ and II has played ⟨xβ : β < α⟩. We +assume that each xβ is in [Φ](Qκ) and that II has also constructed a strictly increasing +sequence ⟨sβ : β < α⟩ in Sκ with xη↾γη+1 ⊆ Φ(sβ) ⊆ xβ for all η < β < α. Note that +tα := � +β<α xβ↾γβ+1 is in T(Φ). If α = β + 1, this holds since xβ ∈ ran([Φ]). If α ∈ Lim, +this holds since tα = � +β<α Φ(sβ) ⊆ Φ(� +β<α sβ). Now II picks some sα ∈ Sκ extending +152They proved DKω(Σ1 +1), extending the special cases for a single Π0 +2 graph [She99] and a single finite +dimensional Π0 +2 hypergraph [Kub03]. +153In [Szir18], DKκ(X), ♦i +κ and CB− +κ are denoted by PIFκ(X), DIκ and DISPκ, respectively. Moreover, +DKκ(X) is formulated as a dichotomy about the size of independent sets with respect to κ many finite +dimensional Σ0 +2(κ) dihypergraphs. +154They showed that the combination of I−(κ) and ♦κ implies DKκ(Σ1 +1(κ)). Note that I−(ω1) is the +same as I(ω). The consistency of I−(κ) can be proved similarly to that of I(ω) in [GJM78]. +155Equivalently, KLWκ(Y, X). +156Conversely, if |Y ∩ X| ≤ κ, then CX↾Y +has a κ-coloring. +This follows from the proof of +Lemma 6.22 (2). +157It suffices to assume that Φ : <κ2 → <κκ is a strict order preserving map with [Φ](Rκ) ⊆ Y , +[Φ](Qκ) ⊆ X and |[Φ](Qκ) ∩ NΦ(s)| ≥ κ for all s ∈ <κκ. + +92 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +� +β<α sβ with tα ⊆ Φ(sα). II plays some xα ∈ [Φ] +� +Qκ) ⊆ X that extends Φ(sα) and is +distinct from xβ for each β < α. Suppose ⟨γα, xα : α < κ⟩ is a run according to this +strategy and II has constructed ⟨sα : α < κ⟩. Let y := � +α<κ sα. Then y ∈ Rκ and +[Φ](y) = � +α<κ +Φ(sα) = � +α<κ +xα↾γα+1. +Since this is in [Φ](Rκ) ⊆ Y , II wins. +Note that the converse of the previous implication holds as well: If II wins VY +κ (X), +then there is a nice reduction for (Y, X). To see this, suppose that τ is a winning strategy +for II in VY +κ (X, x). Construct functions Φ: <κ2 → T(X), ⟨γs : s ∈ Sκ⟩ and ⟨xs : s ∈ <κκ⟩ +just as in the proof of Lemma 6.33. We claim that Φ is a nice reduction for (Y, X). We +have already seen in the proof of Lemma 6.33 that Φ is a nice reduction to (κκ, X), so it +suffices to show that [Φ](Rκ) ⊆ Y . Let z ∈ Rκ. It suffices to show that [Φ](z) is produced +during a run of VY +κ (X) where II uses τ. Since s ∈ Sκ for cofinally many initial segments +s of z and Φ(s) = xs↾γs⌢⟨1⟩ for all s ∈ <κκ by the definition of Φ, +[Φ](z) = �{Φ(s) : s ⊆ z, s ∈ Sκ} = �{xs↾γs⌢⟨1⟩ : s ⊆ z, s ∈ Sκ}. +Thus, [Φ](z) is produced during the run ⟨γs, xs : s ⊆ z, s ∈ Sκ⟩ of VY +κ (X) where II uses τ. +6.5. The asymmetric Baire property. A subset X of κκ is called κ-meager if it is +the union of κ many nowhere dense subsets of κκ and κ-comeager if its complement is +κ-meager. Moreover, if Y is a subset of κκ then X is called κ-meager in Y if X ∩ Y is +κ-meager158 and κ-comeager in Y if Y \ X is κ-meager. +A subset A of κκ has the κ-Baire property if there exists an open subset U of κκ such +that X △ U is κ-meager. Halko and Shelah [HS01] observed that the club filter +Clubκ = {x ∈ κ2 : {α < κ : x(α) = 1} contains a club} +does not have the κ-Baire property. +Note that Clubκ is κ-analytic. +To see that the +κ-Baire property fails for Clubκ, suppose first that Clubκ is κ-comeager in some basic +open set Nt. Then there is a sequence ⟨Uα : α < κ⟩ of open dense subsets of Nt such +that � +α<κ Uα ⊆ Clubκ. We construct a strictly increasing sequence ⟨tα : α < κ⟩ in <κκ +with t0 = t as follows. If α = β + 1, choose tα ⊋ tβ with Ntα ⊆ Uβ. If α is a limit, +let tα := � +β<α tβ ∪ {⟨α, 0⟩}. Let x := � +α<κ tα. Then x ∈ � +α<κ Uα and x /∈ Clubκ, a +contradiction. One can similarly see that Clubκ is not κ-meager. +We consider a variant of the Baire property from [Sch17]. Its definition uses the fol- +lowing types of strict order preserving maps. +Definition 6.40 ([Sch17, Definition 3.1]). Let S ⊆ <κκ, and let t ∈ <κκ. +(a) S is dense above t if for any u ⊇ t there exists s ∈ S with s ⊇ u. S is nowhere +dense if it is not dense above any t ∈ <κκ. +(b) A strict order preserving map ι : <κκ → <κκ is dense if for all t ∈ <κκ, the set +{ι(t⌢⟨α⟩) : α < κ} is dense above ι(t). +158If Y is open, then this is equivalent to the fact that X ∩ Y is κ-meager as a subset of the space Y , +but this is not true for arbitrary subspaces. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +93 +Whether a subset of κκ is comeager can be characterized via dense strict order pre- +serving maps: X is comeager in a basic open set Nt if and only if there exists a dense +continuous strict order preserving map ι : <κκ → <κκ with ι(∅) = t and ran([ι]) ⊆ X +[Sch17, Lemma 3.2]. +Definition 6.41 ([Sch17, Definition 3.3]). A subset X of κκ has the asymmetric Baire +property ABPκ(X)159 if either X is κ-meager or there exists a dense strict order preserving +map ι : <κκ → <κκ with ran([ι]) ⊆ X. +This property was characterised via a variant of the Banach-Mazur game in [Sch17, +Section 3.1]. It is weaker than the κ-Baire property. In fact, an alternative definition +of the κ-Baire property is obtained by asking that the dense strict order preserving map +is continuous, assuming the class of sets under consideration is closed under continuous +preimages of shift maps for each t ∈ <κκ that send x ∈ κκ to t⌢x. ABPω(X) is equivalent +to the Baire property for X, since continuity is trivial for ω. +We obtain the asymmetric Baire property ABPκ(X) as a special case of ODDκ +κ(X, Dκ). +We use the dihypergraph Dκ from Definition 5.29. Note that Dκ ∈ Dκ is a box-open +dihypergraph on κκ. +Lemma 6.42. A subset Y of κκ is Dκ-independent if and only if it is nowhere dense. +Proof. Suppose Y is somewhere dense, i.e., Y is dense in Nt for some t ∈ <κκ. +Fix +yu ∈ Nu ∩ Y for all u ⊇ t. Then any injective enumeration of the yu’s is a hyperedge of +Dκ↾Y . Conversely, if Dκ↾Y is nonempty, then Y is dense is some basic open set Nt. +□ +The next lemma shows that the property of being an order homomorphism for (X, Dκ) +in inherited to supersets of X. +Lemma 6.43. Let X ⊆ κκ. +(1) Let ⟨tα : α < κ⟩ be a sequence of elements of T(X). Then � +α<κ Ntα ∩ X ⊆ Dκ160 +if and only if {tα : α < κ} is dense above some s ∈ <κκ. +(2) Any order homomorphism ι for (X, Dκ) is an order homomorphism for Dκ. +Proof. We first show (1). +If {tα : α < κ} is dense above s, then every sequence in +� +α<κ Ntα ∩ X is dense in Ns. Conversely, suppose that {tα : α < κ} is nowhere dense. It +suffices to construct a sequence x ∈ � +α<κ Ntα ∩ X which is nowhere dense in κκ. Let I +denote the set of α < κ such that T(X) is dense above some t ⊇ tα, i.e., succ(t) ⊆ T(X) +as T(X) is downwards closed.161 Then κ \ I is precisely the set of those α < κ such that +(<κκ) \ T(X) is dense above tα. +We first claim that any x ∈ � +α∈(κ\I) Ntα ∩ X is nowhere dense in κκ. Otherwise x is +dense in Nt for some t ∈ <κκ. Pick some α ∈ (κ \ I) with xα ∈ Nt. Since x ∈ κX, T(X) +is dense above t ∪ tα. But then α ∈ I. +159ABPκ(X) was called the almost Baire property in [Sch17]. Note that X is κ-meager if and only if +there exists a continuous dense strict order preserving map ι : <κκ → <κκ with ι(∅) = ∅ and ran([ι]) ⊆ +κκ \ X, by [Sch17, Lemma 3.2]. Therefore our definition of ABPκ(X) is equivalent to the one given in +[Sch17, Definition 3.3]. +160Note that � +α<κ Ntα ∩ X ̸= ∅, as tα ∈ T(X) for all α < κ. +161Recall that succ(t) = {u ∈ <κκ : t ⊆ u}. + +94 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +It therefore suffices to find a sequence y ∈ � +α∈I Ntα ∩ X that is nowhere dense in κκ. +To this end, we construct a sequence ⟨uα : α ∈ I⟩ such that the following hold for all +α ∈ I: +(i) uα ∈ T(X) and uα ⊋ tα. +(ii) uα ̸⊆ tγ for all γ < κ. +(iii) For all β ∈ I ∩ α, either uα ⊥ uβ or uα = uβ. +Suppose that α ∈ I and uβ has been constructed for all β ∈ I ∩α. If there exists β ∈ I ∩α +with uβ ⊋ tα, then let uα := uβ. Otherwise, we have tα ⊥ uβ for all β < α by (ii). As +α ∈ I, there exists s ⊇ tα with succ(s) ⊆ T(X). Since {tγ : γ < κ} is nowhere dense, +there exists uα ⊋ s in T(X) such that uα ̸⊆ tγ for all γ < κ. Then uα ⊋ tα, so uα ⊥ uβ +for all β ∈ I ∩ α. +Since uα ∈ T(X) for all α < κ, we may choose a sequence y ∈ � +α∈I Nuα ∩ X ⊆ +� +α∈I Ntα ∩ X such that yα = yβ whenever uα = uβ. Since the uα’s are pairwise incom- +parable, y is nowhere dense in κκ. +(1) shows that the property � +α<κ Ntα ∩ X ⊆ Dκ is inherited to supersets of X. (2) +follows. +□ +Lemma 6.44. +(1) Any dense strict order preserving map ι : <κκ → <κκ is an order homomorphism +for Dκ. +(2) Conversely, if there exists an order homomorphism ι for Dκ, then there exists a +dense strict order preserving map ι′ : <κκ → <κκ with ran([ι′]) ⊆ ran([ι]). +Proof. (1) is clear.162 For (2), suppose there is an order homomorphism ι for Dκ. By +Lemma 6.43 (1) there exists a function s: <κκ → <κκ such that for all t ∈ <κκ, ⟨ι(t⌢⟨α⟩) : +α < κ⟩ is dense above s(t). Since ι is strict order preserving, s(t) ⊇ ι(t). We will define +a continuous strict order preserving map e : <κκ → <κκ such that ι′ := s ◦ e is dense and +strict order preserving. +We define e(t) by recursion on lh(t). Let e(∅) := ∅. If lh(t) ∈ Lim, then let e(t) = +� +u⊊t e(u). Suppose e(t) has been defined. Let ι′(t) := s(e(t)). Let ⟨e(t⌢⟨β⟩) : β < κ⟩ be +an injective enumeration of the set of those e(t)⌢⟨α⟩ such that ι(e(t)⌢⟨α⟩) ⊋ ι′(t). It is +clear from the construction that ι′ is dense and strict order preserving. Since +ι(e(t)) ⊆ s(e(t)) = ι′(t) ⊆ ι +� +e(t⌢⟨α⟩) +� +for all t ∈ <κκ and all α < κ, we have [ι′] = [ι] ◦ [e]. Therefore ran([ι′]) ⊆ ran([ι]). +□ +Theorem 6.45. Suppose X is a subset of κκ. +(1) Dκ↾X admits a κ-coloring if and only if X is κ-meager. +(2) There exists a continuous homomorphism from Hκκ to Dκ↾X if and only if there +exists a dense strict order preserving map ι : <κκ → <κκ with ran([ι]) ⊆ X. +Thus, ABPκ(X) is equivalent to ODDDκ↾X +κ +. +162This is the easy direction of Lemma 6.43. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +95 +Proof. (1) holds since a subset of κκ is Dκ-independent if and only if it is nowhere dense +by Lemma 6.42. +For (2), suppose there exists a continuous homomorphism from Hκκ to Dκ↾X. We obtain +an order homomorphism ι for (X, Dκ) by Lemma 3.11. ι is also an order homomorphism +for Dκ by Lemma 6.43. We then obtain the required map from Lemma 6.44 (2). The +converse holds by Lemma 6.44 (1). +□ +Remark 6.46. Similarly to Theorem 6.45, we obtain that ABPκ(X) is equivalent to +ODDD− +κ ↾X +κ +, where D− +κ is the dihypergraph studied in Remark 5.34. This is because D− +κ ⊆ +Dκ and Lemma 6.42 and Corollary 6.44 (1) hold for D− +κ instead of Dκ. +Theorem 6.45 and the main theorem, Theorem 1.4, yield a new proof of the following +result from [Sch17]. Given a class C, let ABPκ(C) state that ABPκ(X) holds for all subsets +X ∈ C of κκ. +Corollary 6.47. ABPκ(Dκ) holds in all Col(κ, <λ)-generic extensions if λ > κ is inac- +cessible in V . +6.6. The Jayne-Rogers theorem. We will provide an analogue of Jayne and Rogers’ +characterization of ∆0 +2-measurable functions [JR82, Theorem 5]. They proved the follow- +ing principle for functions with analytic domain in the countable setting. +Suppose throughout this subsection that X is a subset of κκ and f : X → κκ is a +function. We say that f is κ-continuous with closed pieces if X is the union of κ many of +its closed subsets C such that f↾C is continuous. +Definition 6.48. Let JRf +κ denote the statement that the following conditions are equiv- +alent: +(1) f is ∆0 +2(κ)-measurable.163 +(2) f is κ-continuous with closed pieces. +JRκ(X) denotes the statement that JRg +κ holds for all g : X → κκ. +We will obtain JRf +κ as a special case of ODDκ +κ(G(f)), where G(f) denotes the graph +of f.164 If X = dom(f) is in Dκ, then ODDκ +κ(G(f), Dκ) suffices. Our proof is similar to +that of [CM15, Theorem 13 & Proposition 18]. Regarding the fact that G(f) is a subset +of κκ × κκ, note that in the definition of ODDκ +κ(C, D),165 the space κκ could be replaced +by an arbitrary topological space and in particular by κκ × κκ. Since there is a definable +homeomorphism k from κκ × κκ to κκ, the definition of ODDκ +κ(G(f), Dκ) is equivalent to +the definition of ODDκ +κ(k(G(f)), Dκ). +Recall from Section 6.2 that CX denotes the set of all convergent sequences x in κκ with +limα<κ(x) ∈ X \ ran(x). We will work with the κ-dihypergraph Jf on G(f) consisting of +those sequences that project to CX and witness discontinuity of f in the following sense: +Let Jf consist of all sequences ⟨(xα, f(xα)) : α < κ⟩ such that x = ⟨xα : α < κ⟩ ∈ CX +and f(limα<κ(x)) is not in the closure of {f(xα) : α < κ}. Note that Jf is the restriction +163Equivalently, Π0 +2(κ)-measurable. +164I.e., G(f) consists of those pairs (x, y) in X × κκ such that f(x) = y. +165See Definition 1.3. + +96 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +of the dihypergraph JX on κκ × κκ which consists of those sequences ⟨(xα, yα) : α < κ⟩ +such that x = ⟨xα : α < κ⟩ ∈ CX and f(limα<κ(x)) is not in the closure of {yα : α < κ}. +Clearly JX is a box-open dihypergraph on κκ × κκ and JX ∈ Dκ when X ∈ Dκ. +Let pk denote the projection onto the kth coordinate for k ∈ {0, 1}. +Lemma 6.49. Suppose A ⊆ G(f) and C is the closure of p0(A) in X. +Then A is +Jf-independent if and only if f↾C is continuous. +Proof. First, suppose that A is Jf-independent. +Assuming f↾C is not continuous at +y ∈ C, take an injective sequence x = ⟨xα : α < κ⟩ in p0(A) converging to y such that +⟨f(xα) : α < κ⟩ does not converge to f(y). Then there exists a subsequence ⟨xαi : i < κ⟩ +of x and some γ < κ with f(xαi) /∈ Nf(y)↾γ for all i < κ. Since f(y) /∈ {f(xαi) : i < κ}, +the sequence ⟨(xαi, f(xαi)) : i < κ⟩ contradicts the fact that A is Jf-independent. +Conversely, suppose that f↾C is continuous. It suffices to show that for any sequence +x = ⟨xα : α < κ⟩ in C with x ∈ CX, the sequence ⟨(xα, f(xα)) : α < κ⟩ is not in Jf. But +this follows from the fact that f(limα<κ x) = limα<κ(⟨f(xα) : α < κ⟩) by the continuity +of f↾C. +□ +Recall from Subsection 5.1 that for a strict order preserving map ι: <κκ → <κκ, Limι +t +denotes the union of all limit points of sets of the form {[ι](xα) : α < κ} where xα ∈ +Nt⌢⟨α⟩ ∩ X for all α < κ. We will “thin out” ι to a map ι ◦ e such that Limι◦e +t +contains +at most one element for any t ∈ <κκ. Recall the definition of ∧-homomorphism from the +paragraph before Lemma 6.18. A map e: <κκ → <κκ is a ∧-homomorphism if and only +if e(t⌢⟨α⟩ ∧ e(t⌢⟨α⟩ = e(t) for all t ∈ <κκ and all α < β < κ.166 This is a strong form of +⊥-preservation. +Lemma 6.50. If ι : <κκ → <κκ is strict order preserving, then there is a strict order +preserving ∧-homomorphism e : <κκ → <κκ with |Limι◦e +t | ≤ 1 for all t ∈ <κκ. +Proof. We construct e(t) by recursion on t. If t is a limit, let e(t) := � +s⊊t e(s). Now +suppose that e(t) has been constructed. If there exists no sequence x = ⟨xα : α < κ⟩ +in � +α<κ Ne(t)⌢⟨α⟩ such that ⟨[ι](xα): α < κ⟩ has any convergent subsequence, then let +e(t⌢⟨α⟩) := e(t)⌢⟨α⟩ for all α < κ. Otherwise fix a strictly increasing sequence ⟨αi : i < κ⟩ +and a sequence ⟨xi : i < κ⟩ in � +i<κ Ne(t)⌢⟨αi⟩ such that ⟨[ι](xi): i < κ⟩ converges to some +y. For each i < κ, find some e(t⌢⟨i⟩) with e(t) ⊊ e(t)⌢⟨αi⟩ ⊊ e(t⌢⟨i⟩) ⊊ xi such that +⟨ι(e(t⌢⟨i⟩)): i < κ⟩ converges y. +□ +We need to extend the notion of order homomorphisms to dihypergraphs on κκ × κκ. +Let <κκ ⊗ <κκ := � +α<κ(ακ × ακ). +Given a strict order preserving map θ : <κκ → +(<κκ ⊗ <κκ), let θk := pk ◦θ for k ∈ {0, 1} and define [θ]: κκ → κκ × κκ by letting +[θ](x) := ([θ0](x), [θ1](x)). In the following definition and lemma, we assume Y is a subset +of κκ × κκ and H is a κ-dihypergraph on κκ × κκ. +166Note that if e: <κκ → <κκ is strict order preserving and continuous, then it is a ∧-homomorphism +if and only if in the standard κ-ultrametric, the distance of no two elements can shrink under the map. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +97 +Definition 6.51. We say that a strict order preserving map θ : <κκ → (<κκ ⊗ <κκ) is an +order homomorphism for (Y, H) if ran([θ]) ⊆ Y 167 and for all t ∈ <κκ, +� +α<κ +(Nθ0(t⌢⟨α⟩) × Nθ1(t⌢⟨α⟩)) ∩ Y ⊆ H. +Lemma 6.52. If H↾Y is box-open on Y , then the existence of the following objects is +equivalent: +(1) a continuous homomorphism from Hκκ to H↾Y , +(2) an order homomorphism for (Y, H). +Proof. Let k denote the natural coding of elements (u, v) of ≤κκ ⊗ ≤κκ168 as elements +k(u, v) of ≤κκ such that lh(k(u, v)) = 2 · lh(u) = 2 · lh(v).169 k induces the dihypergraph +H := kκ(H) on κκ. Moreover, θ: <κκ → <κκ × <κκ is an order homomorphism for (Y, H) +if and only if k ◦ θ is an order homomorphism for (k(Y ), H). +First, if θ is an order homomorphism for (Y, H), then [k ◦ θ] = k ◦ [θ] is a continuous +homomorphism from Hκκ to H↾k(Y ) by Lemma 3.11. Then [θ] is a continuous homomor- +phism from Hκκ to H↾Y . Now, suppose that h is a continuous homomorphism from Hκκ to +H↾Y . Then k ◦h is a continuous homomorphism from Hκκ to H↾k(Y ). Since H↾k(Y ) is a +box-open dihypergraph on k(Y ), there exists an order homomorphism ι for (k(Y ), H) by +Lemma 3.11, and by composing ι with a ∧-homomorphism, we may assume that lh(ι(t)) +is even for all t ∈ <κκ. Then ψ := ι ◦ k−1 is an order homomorphism for (Y, H). +□ +Theorem 6.53. +(1) Jf admits a κ-coloring if and only if f is κ-continuous with closed pieces. +(2) There exists a continuous homomorphism from Hκκ to Jf if and only if there exists +a continuous map g : κ2 → X with the following properties: +(a) f ◦ g is a reduction of Rκ to a closed subset of κκ. +(b) (f ◦ g)↾Rκ is continuous. +In (2), we can equivalently take g to be injective, or even a homeomorphism onto a closed +subset of κκ. +Proof. For (1), suppose first that G(f) = � +α<κ Aα, where each Aα is an Jf-independent +subset of G(f). For each α < κ, let Cα be the closure of p0(Aα) in X. Then f↾Cα is +continuous by Lemma 6.49, and X = � +α<κ Cα, so f is κ-continuous with closed pieces. +Conversely, suppose that the sequence ⟨Cα : α < κ⟩ witnesses that X is κ-continuous +with closed pieces. Since each f↾Cα is continuous, the sets Aα := {(x, f(x)) : x ∈ Cα} +are Jf-independent by Lemma 6.49. Since X = � +α<κ Cα, we have G(f) = � +α<κ Aα. +For (2), suppose that g : κ2 → X is a continuous map with (a) and (b). Recall the +definition of π in Subsection 6.2. Let h := (g ◦ [π], f ◦ g ◦ [π]) : κκ → X × κκ. Since +([π]↾Rκ) : Rκ → κκ is a homeomorphism, h is continuous. +It remains to show that h is a homomorphism from Hκκ to Jf. To see this, suppose +that ⟨yα : α < κ⟩ is a hyperedge in Hκκ with t⌢⟨α⟩ ⊆ yα for all α < κ. Then t⌢⟨0⟩α⌢⟨1⟩ ⊆ +167For Y = G(f) and H = Jf, this is equivalent to f ◦ [θ0] = [θ1]. +168We write ≤κκ ⊗ ≤κκ := (<κκ ⊗ <κκ) ∪ (κκ × κκ). +169That is, for all α < lh(u), let k(u, v)(2 · α) = u(α) and let k(u, v)(2 · α + 1) = v(α). + +98 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +[π](yα) for all α < κ, so limα<κ[π](yα) = π(t)⌢⟨0⟩κ. Since g is continuous, limα<κ(g ◦ +[π])(yα) = xt := g(π(t)⌢⟨0⟩κ). Since g ◦ f is a reduction of Rκ, g(Rκ) and g(Qκ) do not +overlap. Hence (g ◦ [π])(yα) ̸= xt for all α < κ and ⟨(g ◦ [π])(yα) : α < κ⟩ ∈ CX. Suppose +that f ◦ g is a reduction of Rκ to the closed set C. Then f(xt) /∈ {zα : α < κ} ⊆ C, where +zα := (f ◦ g ◦ [π])(yα). Thus ⟨h(yα) : α < κ⟩ ∈ Jf as required. +For the converse, suppose that there is a continuous homomorphism from Hκκ to Jf. +Since Jf is box-open on G(f), there is an order homomorphism θ = (θ0, θ1) for (G(f), Jf) +by Lemma 6.52. +Then θ0 is an order homomorphism for (ran([θ0]), CX). +Therefore +⟨θ0(t⌢⟨α⟩) : α < κ⟩ converges to some xθ0 +t +∈ X by Lemma 6.15. We now show that +θ can be assumed to have some additional properties. +Claim. There exists an order homomorphism ψ for (G(f), Jf) such that for all t ∈ <κκ, +f(xψ0 +t ) is not in the closure of ran([ψ1]). +Proof. Let θ be any order homomorphism for (G(f), Jf). We will construct a strict order +preserving ∧-homomorphism e : <κκ → <κκ such that for ψ := θ ◦ e we have +∀t ∈ <κκ f(xψ0 +t ) /∈ ran([ψ1]). +This suffices, since ψ will also be an order homomorphism for (G(f), Jf) because e is a +∧-homomorphism and Jf is closed under subsequences. +Case 1. [θ1]↾Nt is constant for some t ∈ <κκ. +In this case, let y denote the unique element of [θ1](Nt). Let e(u) := t⌢u for all u ∈ <κκ. +For ψ := θ ◦ e, ran([ψ1]) has the unique element y. It thus suffices to show f(xψ0 +u ) ̸= y for +all u ∈ <κκ. To see this, pick yα with t⌢⟨α⟩ ⊆ yα for α < κ. Then f(xψ0 +t ) is not in the +closure C of {[ψ1](yα): α < κ}, since θ is an order homomorphism for (G(f), Jf). But C +has the unique element y. +Case 2. |{f(xθ0 +t ): s ⊆ t ∈ <κκ}| < κ for some s ∈ <κκ and Case 1 does not occur. +We first claim that there exists some u ∈ <κκ such that Au := {t : f(xθ0 +u ) = f(xθ0 +t )} is +dense above some v ∈ <κκ.170 Let u = ⟨uα : α < γ⟩ be a sequence in <κκ such that γ < κ +and for any t ∈ <κκ, there exists some α < γ with f(xθ0 +t ) = f(xθ0 +uα). Supposing each Auα +is nowhere dense, we can construct a strictly increasing sequence ⟨sα : α < γ⟩ in <κκ with +s ⊆ s0 such that for each α < γ, there exists no t ⊇ sα with f(xθ0 +t ) = f(xθ0 +uα). Then for +t = � +α<γ sα, we have f(xθ0 +t ) ̸= f(xθ0 +uα) for all α < γ. But this contradicts the choice of u. +Since [θ1]↾Nv is not constant, there exists some y ∈ Nv with [θ1](y) ̸= f(xθ0 +u ). Take +w ⊊ y with v ⊆ w and θ1(w) ⊥ f(xθ +u). Since Au is dense above w, we can construct +a strict order preserving ∧-homomorphism e : <κκ → <κκ with e(∅) ⊇ w such that the +value f(xθ0 +e(t)) = f(xθ0 +u ) for all t ∈ <κκ. Let ψ := θ ◦ e. Then ran([ψ1]) is a subset of Nw +and therefore does not contain f(xψ0 +u ). Since e is a ∧-homomorphism, xψ0 +t += xθ0 +e(t) = xθ0 +u +for any t ∈ <κκ, so f(xψ0 +t ) /∈ ran([ψ1]). +Case 3. Neither Case 1 nor Case 2 occurs. +170See Definition 6.40. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +99 +By Lemma 6.50 for θ1, there exists a strict order preserving ∧-homomorphism ϵ : <κκ → +<κκ such that |Limθ1◦ϵ +t +| ≤ 1 for all t ∈ <κκ. Since Jf is closed under subsequences, θ ◦ ϵ +is an order homomorphism for (G(f), Jf). Thus, we may assume that |Limθ1 +t | ≤ 1 for all +t ∈ <κκ, by replacing θ with θ ◦ ϵ if necessary. Fix an enumeration ⟨tα : α < κ⟩ of <κκ +such that tα ⊊ tβ implies α < β. +Subclaim. There exists an order preserving ∧-homomorphism e: <κκ → <κκ such that +ψ1 := θ1 ◦ e has the following properties for all α, β < κ: +(i) f(xψ0 +tβ ) /∈ Limψ1 +tα if α ≤ β. +(ii) f(xψ0 +tα ) /∈ Nψ1(tβ) if α < β. +Proof. Note that (i) already holds for θ1 and α = β. To see this, observe that f(xθ0 +t ) /∈ +Limθ1 +t +for all t ∈ <κκ, since ⟨θ0(t⌢⟨α⟩) : α < κ⟩ converges to xθ0 +t +and θ is an order +homomorphism for (G(f), Jf). Since this property remains true for ψ1 = θ1 ◦ e for any +choice of e, we only need to ensure (i) in the case α < β. +We first show how to arrange (i). +We define e(tβ) by recursion on β. +Let Bβ := +� +α<β Limθ1 +tα. Since |Limθ1 +tα| ≤ 1 for all α < κ, we have |Bβ| < κ. Let u := e(tγ)⌢⟨η⟩ +if tβ = tγ⌢⟨η⟩ for some γ < β and η < κ, and if tβ has limit length, then let u := +� +s⊊tβ e(s).171 Since |{f(xθ0 +t ): u ⊆ t ∈ <κκ}| ≥ κ by the case assumption, we can find +some t extending e(tγ)⌢⟨η⟩ with f(xψ0 +t ) /∈ Bβ. Then e(tβ) := t satisfies (i). Note that e +is a ∧-homomorphism. +Assume θ satisfies (i). We next show how to arrange (ii). Again, we define e(tβ) by +recursion on β. We let e be continuous at nodes tβ of limit length. Suppose that tβ is a +direct successor of tγ with tβ = tγ⌢⟨η⟩. Since [θ1](Ne(tγ⌢⟨η⟩)) has at least 2 elements by +the case assumption, we can pick some proper extension u0 of e(tγ)⌢⟨η⟩ such that θ1(u0) +is incompatible with f(xθ0 +t0 ). +Continuing in the same fashion, we construct a strictly +increasing sequence ⟨uα : α ≤ γ⟩ such that θ1(uα) is incompatible with f(xθ0 +tα) for all +α ≤ γ. Let e(tβ) := uγ. Then (ii) holds for tβ. Again, e is a ∧-homomorphism. +□ +Let ψ := θ ◦ e, where e is as in the previous subclaim. +We show that f(xψ0 +t ) /∈ +ran([ψ1]) for all t ∈ <κκ. +Suppose that this fails for some t ∈ <κκ. +Suppose that +t = tα. We first show that f(xψ0 +t ) /∈ ran([ψ1]). Otherwise, suppose that y ∈ κκ with +f(xψ0 +t ) = [ψ1](y) = � +u⊊y ψ1(u). +Take any β with α < β < κ and tβ ⊊ y. +Then +f(xψ0 +t ) ∈ Nψ1(tβ), contradicting (ii). We next show f(xψ0 +t ) /∈ ran([ψ1]). Suppose otherwise. +Since ran([ψ1]) = ran([ψ1]) ∪ � +β<κ Limψ1 +tβ by Lemma 5.1 (2), we have f(xψ0 +t ) ∈ Limψ1 +tβ for +some β < κ. Then α < β by (i). Since Limψ1 +tβ ⊆ � +η<κ Nψ1(tβ⌢⟨η⟩) ⊆ Nψ1(tβ), we have +f(xψ0 +t ) ∈ Nψ1(tβ). But this contradicts (ii). +□ +Let ψ be an order homomorphism for (G(f), Jf) as in the previous claim. Since ψ0 is +an order homomorphism for (ran([ψ0], CX), there exists a nice reduction Φ : <κ2 → <κκ +for (ran([ψ0]), X) and a strict order preserving ∧-homomorphism ϵ : Sκ → <κκ such that +ψ0 ◦ ϵ = Φ↾Sκ by Lemma 6.18. We will show g := [Φ] : κ2 → κκ is as required in (2). +171This is defined since each s ⊊ tβ is equal to tα for some α < β. + +100 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +We first show that (f ◦ g)↾Rκ is continuous. It is clear from the definitions that g is +a homeomorphism onto ran(g) with g(Qκ) ⊆ X and g(Rκ) ⊆ ran([ψ0]) ⊆ X. Therefore +ran(g) ⊆ X, and ran(g) is a closed subset of κκ by Lemma 5.1 (1). We have f ◦[ψ0] = [ψ1], +since ran([ψ]) ⊆ G(f). Thus (f ◦ g)↾Rκ = [ψ1 ◦ ϵ] and hence it is continuous. +We now show that f ◦g is a reduction of Rκ to a closed subset of κκ. Since (f ◦g)(Rκ) ⊆ +ran([ψ1]), it suffices to show that (f ◦ g)(Qκ) and ran([ψ1]) are disjoint. Suppose that +z ∈ Qκ. Then z = s⌢⟨0⟩κ for some s ∈ Sκ. For each α < κ, let tα := s⌢⟨0⟩α⌢⟨1⟩. +We claim that g(z) = xψ +ϵ(s). +This suffices, since f(xψ +ϵ(s)) /∈ ran([ψ1]) by the choice of +ψ. Since ϵ is a ∧-homomorphism from Sκ to <κκ,172 there exists an injective sequence +⟨ηα : α < κ⟩ of ordinals below κ such that ψ0(ϵ(s)⌢⟨ηα⟩) ⊆ Φ(tα). On the one hand, +⟨Φ(tα) : α < κ⟩ converges to g(z) = [Φ](s⌢⟨0⟩κ). +On the other hand, ψ0(ϵ(s)⌢⟨α⟩) +converges to xψ +ϵ(s) and therefore so does ψ0(ϵ(s)⌢⟨ηα⟩). Since ψ0(ϵ(s)⌢⟨ηα⟩) ⊆ Φ(tα), we +must have g(z) = xψ +ϵ(s). +□ +Remark 6.54. Let J′ +f denote the variant of Jf where CX is replaced with CX ∩Iκ. Then +Theorem 6.53 holds for J′ +f as well. To see this, note that (1) holds for J′ +f since the proof of +Lemma 6.49 works for J′ +f. The backward direction of (2) holds for J′ +f since J′ +f is a subset +of Jf. For the forward direction of (2), take an injective continuous map g: κ2 → X with +(a) and (b). In the beginning of the proof, we now use the fact that g is injective to ensure +that ⟨(g ◦ [π])(yα) : α < κ⟩ ∈ CX ∩ Iκ. The additional statement in Theorem 6.53 for J′ +f +is immediate from the original Theorem 6.53. +We next obtain an analogue of Jayne and Rogers’ characterization of ∆0 +2-measurable +functions [JR82, Theorem 5]. +Corollary 6.55. +ODDJf +κ implies JRf +κ. +Proof. Note that any κ-continuous function with closed pieces is ∆0 +2(κ)-measurable. There- +fore it suffices to show that the condition in Theorem 6.53 (2) fails for any ∆0 +2(κ)- +measurable function f : X → κκ. To see this, suppose that there exists a continuous +map g : κ2 → X such that f ◦ g is a reduction of Rκ to a closed subset of κκ. Then Rκ +would be a Σ0 +2(κ) subset of κκ. +□ +The previous corollary shows that ODDκ +κ(G(f)) ⇒ JRf +κ. +If X = dom(f) is in Dκ, +then JX is in Dκ as well. +Since JX is box-open on κκ and Jf = JX↾ G(f), we have +ODDκ +κ(G(f), Dκ) ⇒ ODDJf +κ ⇒ JRf +κ in this situation. +Given a class C, let JRκ(C) state that JRκ(X) holds for all subsets X ∈ C of κκ. +Corollary 6.56. JRκ(Dκ) holds in all Col(κ, <λ)-generic extensions if λ > κ is inacces- +sible in V . +Proof. Work in such as extension and recall that ODDκ +κ(Dκ, Dκ) holds by Theorem 1.4. +Suppose X ∈ Dκ and f : X → κκ. If f ∈ Dκ, then G(f) ∈ Dκ. We thus have ODDJf +κ and +JRf +κ. If f /∈ Dκ, then JRf +κ(f) trivially holds, since all ∆0 +2(κ)-measurable functions with +domain X are in Dκ. +□ +172In particular, e(tα) ∧ e(tβ) = e(s) for all α < β < κ; see the paragraph before Lemma 6.18. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +101 +7. Open problems +As above, κ always denotes an infinite cardinal with κ<κ = κ. Concerning our main +results, we ask for a global version of Theorem 1.4 (2) for all regular cardinals and +arbitrary dihypergraphs. A difficulty is that in an iterated forcing extension as in the +proof of Theorem 4.57, new dihypergraphs are added by the tail forcings. +Problem 7.1. Assume there is a proper class of Mahlo cardinals. Is there a tame class +forcing extension of V where ODDκ +κ(Dκ) holds for all κ? +The consistency strength of ODDκ +κ(Dκ) is open; in particular, we do not know if +it implies that there is an inner model with a Mahlo cardinal. We further ask what the +consistency strength of each of its variants and applications is. +Problem 7.2. Do the following need a Mahlo cardinal for some uncountable κ? +(1) (a) ODDκ +κ(Dκ). +(b) ODDκ +κ(κκ). +(2) KLWκ(Dκ). +(3) (a) CB2 +κ(P(κκ)). +(b) CB− +κ . +(c) The determinacy of Vκ(X) for all subsets X of κκ. +(4) DKκ(Σ1 +1(κ)). +In (a), one can equivalently consider CB1 +κ(X). Note that we have (1)(a)⇒(2)⇒(3)(a)⇒ +(3)(b)+(c), while (1)(b)⇒(3)(a) and (3)(b)+♦i +κ ⇒(4). +The statements ODDκ +κ(κκ, Dκ), KLWκ(Dκ, Dκ) and CB2 +κ(Dκ) are each equiconsistent +with the existence of an inaccessible cardinal, since they each imply PSPκ(X) for all +closed sets X. +Problem 7.3. Do the following need an inaccessible cardinal for some uncountable κ? +(1) (a) HDκ(Dκ). +(b) THDκ(Dκ). +(2) ABPκ(Dκ). +(3) JRκ(Dκ). +(4) OGDκ(κκ). +Note that THDκ(Σ1 +1(κ)) is consistent relative to ZFC [LMS16], but we have not studied +the question whether the same holds for HDκ(Σ1 +1(κ)). In the countable setting, ABPω(X) +is equivalent to the Baire property for X, for all sets X of reals. +Thus, ABPω(Dω) +is consistent relative to ZFC by a well-known result of Shelah [She84, Theorem 7.16]. +Moreover, OGDκ(κκ) is not provable in ZFC for κ = ω2. To see this, note that in L, +there exists a κ-Kurepa tree T that is a continuous image of κκ for any successor cardinal +κ = µ+, where µ has uncountable cofinality [LS, Theorem 1.7]. Then PSPκ([T]) fails in +L, and therefore OGDκ(κκ) also fails. +Besides the applications studied in this paper, we have shown in current work that +ODDω +ω(X) is equivalent to the closed-sets covering property CCP(X) introduced by Di +Prisco and Todorˇcevi´c [DT98, Section 3] and have obtained a version of this result in the + +102 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +uncountable case. We expect that some further applications in [DT98, Section 3] lift to +the uncountable setting. It is likely that the higher dimensional version of the Kechris- +Louveau-Woodin dichotomy and the Lecomte-Zeleny dichotomy in [CMS20, Section 4] +can be extended to the uncountable setting. Carroy and Miller showed in unpublished +work that their basis results for the classes of non-Baire class 1 functions and functions +that are not σ-continuous with closed pieces [CM20] follow from the Kechris-Louveau- +Woodin dichotomy. Does this hold in the uncountable setting? Our results also suggest +to investigate the Menger dichotomy studied by Tall, Todorˇcevi´c and Tokg¨oz [TTT21] in +the uncountable setting. In the countable setting, this follows from the Kechris-Louveau- +Woodin dichotomy using compactifications. +Since the Baire property of subsets of ωω follows from instances of ODDω +ω(X), we ask +whether a similar result holds for Lebesgue measurability. Note that there is no analogous +proof, since the ideal of Lebesgue null sets is not generated by closed sets and it thus +cannot be generated by the independent sets with respect to a box-open dihypergraph on +the Cantor space. +We aim to understand implications and separations between applications of the +open dihypergraph dichotomy. In the countable setting, the perfect set property does +not imply the Kechris-Louveau-Woodin dichotomy and the Baire property in L(R)[U], +where U is a selective ultrafilter on ω that is generic over L(R). This follows from results +of Di Prisco and Todorˇcevi´c [DT98, Section 6] and the fact that a free ultrafilter on ω +induces a subset of the Cantor space without the Baire property. Can this be done in the +uncountable setting as well? +Problem 7.4. Does PSPκ(C) ⇒ ABPκ(C) hold if κ is uncountable and C is closed under +continuous preimages? +It is consistent that the converse fails in the countable case by [She84, Theorem 7.16]. +The perfect set property implies the topological Hurewicz dichotomy for any subset of +κκ if κ is not weakly compact. Conversely, the topological Hurewicz dichotomy does not +imply the perfect set property for Σ1 +1(κ) [LMS16],173 but this implication is open for Dκ. +Some positive results are known for the class of closed subsets of κκ:174 +PSPκ(Π0 +1(κ)) ⇔ CB2 +κ(Π0 +1(κ)) ⇔ KLWκ(κκ, Π0 +1(κ)). +In particular, PSPκ(Π0 +1(κ)) implies the determinacy of Vκ(X) for all closed sets X. Con- +cerning larger classes, note that CB2 +κ(C) ⇒ KLWκ(κκ, C) holds if and only if CX admits +a κ-coloring for every κ-scattered set X ∈ C of size κ by Theorems 6.31 and 6.35. More- +over, PSPκ(Dκ) ⇒ CB2 +κ(Dκ) holds if and only if every subset of κκ of size κ can be +decomposed as a union of a κ-scattered and a κ-crowded set. Since we originally ob- +tained OGDκ(Dκ)175 by an argument that closely resembles the proof of PSPκ(Dκ), we +ask whether these statements are equivalent. +We further ask which of the open dihypergraph dichotomies for different dimensions +2 ≤ d ≤ κ can be separated, and in particular: +173This can be shown by forcing for any regular uncountable cardinal κ with κ<κ = κ. +174See Theorems 6.31 and 6.35. +175In fact ODDd +κ(C) for any 2 ≤ d < κ. + +THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ +103 +Problem 7.5. Do the following implications hold if 3 ≤ d < κ and C is closed under +continuous preimages? +(1) ODD2 +κ(C) ⇒ ODDd +κ(C) +(2) ODDd +κ(C) ⇒ ODDκ +κ(C) +The restriction of (2) to definable dihypergraphs is also open. Note that ODD2 +ω(C) ⇒ +ODDω +ω(C, Dκ) fails for the class C = L(R)[U], where U is a selective ultrafilter on ω that +is generic over L(R). ODD2 +ω(C) holds by [DT98, Theorem 5.1]. However, ODDω +ω(C, Dκ) +does not hold since it implies the Baire property for all sets of reals in C,176 while the set +of reals induced by U does not have the Baire property. We ask if this implication fails +for uncountable cardinals κ as well. +The restriction ODDκ +κ(κκ) to dihypergraphs on κκ implies ODDκ +κ(Σ1 +1(κ)),177 the gen- +eralized Cantor-Bendixson property CB2 +κ(P(κκ)) and in particular, the determinacy of +V¨a¨an¨anen’s perfect set game for all subsets of κκ.178 This motivates the next problem +and its restriction to definable dihypergraphs. +Problem 7.6. Does ODDκ +κ(κκ) imply ODDκ +κ(Dκ)? +Summarizing and expanding some of the above questions, we ask which implications +in Figure 11 can be reversed.179 Note that OGDκ(Dκ) appears in the diagram in the form +of ODD2 +κ(Dκ) and the same holds for Σ1 +1(κ). +ODDκ +κ(Dκ) +ODDκ +κ(Dκ, Dκ) +ODDd +κ(Dκ) +PSPκ(Dκ) +ODDκ +κ(Σ1 +1(κ)) +ODDκ +κ(Σ1 +1(κ), Dκ) +ODDd +κ(Σ1 +1(κ)) +PSPκ(Σ1 +1(κ)) +PSPκ(Π0 +1(κ)) +X +κ = ω, d = 2 +Figure 11. Implications between ODD for dimensions 2 ≤ d ≤ κ and PSP. +A difficulty in separating the various dichotomies and applications discussed above is +that all models in which they are known to hold are obtained by L´evy collapses. One +might try to force over a model of ODDκ +κ(Dκ) to preserve some applications but break +others. +In the countable setting, Di Prisco and Todorˇcevi´c separated the perfect set +property from the Kechris-Louveau-Woodin dichotomy by adding a selective ultrafilter +over L(R) [DT98]. +176See Theorem 6.45 +177See Subsection 6.3. +178See Theorem 6.35. +179The first three nodes of the second row can be equivalently replaced by ODDκ +κ(κκ), ODDκ +κ(κκ, Dκ) +and ODDd +κ(Π0 +1(κ)), respectively, by Lemma 3.5 and Subsection 6.3. + +104 +PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI +Figure 4 in Subsection 5.5 summarizes implications and separations between several +variants of the open dihypergraph dichotomy. Which of these implications can be re- +versed? A striking difference between the countable and uncountable settings is that one +can choose the homomorphisms to be injective in the latter case, assuming ♦i +κ holds.180 +We ask whether this assumption can be eliminated. +Problem 7.7. Does ODDH +κ ⇒ ODDIH +κ hold for all uncountable κ and all relatively box- +open κ-dihypergraphs H? +Equivalently, one can ask whether the existence of a continuous homomorphism from +Hκκ to a relatively box-open κ-dihypergraph H implies the existence of an injective map +with the same properties. This implication holds for d-dihypergraphs for d < κ by The- +orem 5.26. +Since ♦i +κ holds if κ is inaccessible or a successor cardinal ≥ω2, the first +interesting case is ω1. To separate ODDH +ω1 from ODDIH +ω1 for some relatively box-open H, +one would need to consider models where CH holds but ♦ω1 fails. To our knowledge, the +first such model was constructed by Jensen [DJ74]. However, a simpler construction is +possible using recent work of Aspero and Mota [AM17]. Do these models separate the +above two principles for some H? If one can show that ODDIκ +κ(Dκ) implies ♦i +κ for all κ, +then any model of CH + ¬♦ω1 + ODDω1 +ω1(Dκ) is as required. A related question is whether +ODDIH +κ ⇒ ODDHH +κ is consistent for all relatively box-open κ-dihypergraphs H. +Fuchino, Juh´asz and Yoshinobu informed us of the following well-known problem con- +cerning the search for higher analogues of the open graph axiom. +Problem 7.8. Is it consistent that OGAκ holds for all subsets X of κκ for some uncount- +able regular cardinal κ with κ<κ = κ? +Note that our results provide a choiceless model of OGAκ. In fact, Theorem 1.4 shows +that such a model arises as the L(κOrd) of an extension obtained by L´evy-collapsing +an inaccessible cardinal to κ+. One may try to approach Problem 7.8 by forcing over a +choiceless model of OGAκ. 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DIHYPERGRAPH DICHOTOMY FOR GENERALIZED BAIRE SPACES AND ITS APPLICATIONS PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The open graph dichotomy for a subset X of the Baire space ωω states that any open graph on X either admits a coloring in countably many colors or contains a perfect complete subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' It is a strong version of the open coloring axiom for X that was introduced by Todorˇcevi´c and Feng to study definable sets of reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We first show that its recent infinite dimensional generalization by Carroy, Miller and Soukup holds for all subsets of the Baire space in Solovay’s model, extending a theorem of Feng from dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Our main theorem lifts this result to generalized Baire spaces κκ in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (1) For any regular infinite cardinal κ, the following holds after a L´evy collapse of an inaccessible cardinal λ > κ to κ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that H is a κ-dimensional box-open directed hypergraph on a subset of κκ such that H is definable from a κ-sequence of ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Then either H admits a coloring in κ many colors or there exists a continuous homomorphism from a canonical large directed hypergraph to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) If λ is a Mahlo cardinal, then the previous extends to all relatively box-open directed hypergraphs on any subset of κκ that is definable from a κ-sequence of ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We derive several applications to definable subsets of generalized Baire spaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' among them variants of the Hurewicz dichotomy that characterizes subsets of Kσ sets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' an asymmetric version of the Baire property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' an analogue of the Kechris-Louveau-Woodin dichotomy that characterizes when two disjoint sets can be separated by an Fσ set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' the determinacy of V¨a¨an¨anen’s perfect set game for all subsets of κκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' and an analogue of the Jayne-Rogers theorem that characterizes the functions which are σ-continuous with closed pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Some of these applications lift results of Carroy, Miller and Soukup from the countable setting and extend results of V¨a¨an¨anen, L¨ucke, Motto Ros and the authors in the uncountable setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (Primary) 03E15, (Secondary) 03E35, 03E55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Generalized Baire space, directed hypergraph, dichotomy, graph coloring, open graph dichotomy, large cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We are grateful to Stevo Todorˇcevi´c for comments and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We further thank Rapha¨el Carroy and Ben Miller for answering a query regarding their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The first-listed author was partially supported by EPSRC grant number EP/V009001/1 and FWF grant number I4039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk�lodowska-Curie grant agreement No 794020 of the first- listed author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The second-listed author was partially supported by NKFIH grant number K129211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For the purpose of open access, the authors have applied a ‘Creative Commons Attribution’ (CC BY) public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='13274v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='LO] 30 Jan 2023 2 PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI Contents 1 Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 3 2 Preliminaries .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 13 3 Dihypergraphs and homomorphisms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 86 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='5 The asymmetric Baire property .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 92 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='6 The Jayne-Rogers theorem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 95 7 Open problems .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 101 References .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 104 THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Introduction It is natural to wonder whether Ramsey’s theorem for n-tuples of natural numbers [Ram30] can be extended to the set of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Sierpinski’s counterexample is a partition of pairs of reals into two pieces such that no uncountable homogeneous set exists [Sie33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Since his example is not constructive, one can ask if a version of Ramsey’s theorem holds for simple partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Galvin gave a partial answer to this by proving that for any partition of pairs of reals into two open pieces there exists an uncountable (in fact, perfect) homogeneous set [Gal68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Blass generalized this to partitions of n-tuples into finitely many Borel pieces by weakening the conclusion [Bla81]: instead of homogeneity, one requires that all pairs in the set belong to at most (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The next step was to generalize these results to spaces other than the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Abraham, Rubin and Shelah formulated a strengthening of Galvin’s theorem for countably based metric spaces of size ℵ1 [ARS85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Abstracting from many applications, Todorˇcevi´c in- troduced the open coloring axiom OGA [Tod89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='1 For a topological space X, OGA(X) states that any open graph G on X either admits a coloring2 in countably many colors or else contains an uncountable complete subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Furthermore, OGA states that OGA(X) holds for all countably based metric spaces X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' OGA follows from the proper forcing ax- iom PFA [Tod89] and has many applications [Bek91,TF95,Vel92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Feng and Todorˇcevi´c studied a stronger version of OGA(X), the open graph dichotomy OGD(X), that is useful for applications to definable sets of reals [Fen93, TF95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Here the uncountable complete subgraph has to be perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' OGD(X) is consistent for all definable sets X of reals and in fact, it is provable for all analytic sets [Fen93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3 Note that the previous statements only mention open graphs because they fail for closed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The verbatim analogues of the open graph axiom and dichotomy fail in dimension 3 and higher: an open hypergraph on the Cantor space might neither admit a countable coloring nor have an uncountable complete subhypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' However, one can replace the latter by a continuous homomorphic image of a canonical large hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Carroy, Miller and Soukup proved an infinite dimensional version of OGD for directed hypergraphs4 on analytic sets and extended this to all sets of reals assuming the axiom of determinacy AD [CMS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='5 A similar idea is implicit in work of Aviles and Todorˇcevi´c [AT11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='6 The main motivation of this dichotomy is to provide new proofs and strong versions of several 1It is also often called the open coloring axiom OCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Note that a graph G on a set X can be identified with a partition or coloring of pairs in X, for example by coloring edges in G red and edges in the complement of G blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Galvin’s theorem is then equivalent to the statement that for any clopen graph G on the reals, there exists either a perfect independent set or a perfect complete subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 2Recall that a coloring of a graph G assigns different colors to adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 3See [Fen93, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 676-7] for an alternative approach due to Todorˇcevi´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 4A directed hypergraph, or dihypergraph, of dimension d is a set of nonconstant functions with do- main d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 5It follows that this dichotomy is consistent with ZFC for all directed hypergraphs which are definable from a countable sequence of ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' To see this, take a model of AD + DC and add a well-ordering of the reals by a σ-closed homogeneous forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 6We would like to thank Stevo Todorˇcevi´c for pointing out that the idea for an infinite dimensional version of OGD(X) is implicit in the proof of [AT11, Theorem 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 4 PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI seemingly unrelated results in descriptive set theory [CMS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For instance, the Hurewicz dichotomy characterizes the circumstances in which an analytic sets is contained in a Kσ set [Hur28,SR75,Kec77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This dichotomy has also been studied by theoretical computer scientists [dB18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A dichotomy due to Kechris, Louveau and Woodin describes when an analytic set can be separated from another set by an Fσ set [KLW87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A celebrated theorem of Jayne and Rogers characterizes functions that can be obtained as a countable union of continuous functions on closed sets [JR82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The topological properties of the generalized Baire space κκ of functions on a regular uncountable cardinal κ resemble that of the Baire space ωω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In fact, larger classes of spaces including κκ have been identified as analogues to countably based complete metric spaces at uncountable cardinals [CS16, ARS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The study of descriptive set theory for these spaces is motivated, inter alia, by connections with model theory and classification theory [MV93, FHK14, Mor22a, Mor22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='7 There is ample background literature on generalized Baire spaces, beginning with results on the structure of closed, analytic and coanalytic subsets [V¨a¨a91,MV93,L¨uc12,LS15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' It is known that some properties of definable subsets of generalized Baire spaces are closely linked to statements in combinatorial set theory, for instance to the existence of Kurepa trees [LS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Extensions of several classical dichotomies to the uncountable setting pave the way for a structure theory of definable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For instance, the Hurewicz dichotomy for analytic sets is consistent by joint work of the first-listed author with L¨ucke and Motto Ros [LMS16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Assuming the existence of an inaccessible cardinal, it is also consistent for all definable sets [LMS18], and so is the perfect set property PSP for definable sets [Sch17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Moreover, the analogue of OGD for analytic sets is consistent relative to an inaccessible cardinal by a result of the second- listed author [Szir18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' While we do not study analogues of the original OGA here, there is a search for principles at cardinals beyond ω1 that can take its place [MV21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Our aim is to extend the above results about the open graph dichotomy OGD and its infinite dimensional variant to uncountable cardinals and use this to provide several applications to definable subsets of generalized Baire spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We next introduce the relevant variants of the open dihypergraph dichotomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We assume that κ is a regular infinite cardinal with κ<κ = κ throughout this paper, unless otherwise mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The κ-Baire space is the set κκ of functions from κ to κ equipped with the bounded topology (or <κ-box topology), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', the topology generated by the base {Ns : s ∈ <κκ}, where Ns = {x ∈ κκ : s ⊆ x} for each s ∈ <κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='8 The κ-Cantor space is κ2 with the subspace topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A graph is a symmetric irreflexive binary relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A graph on a topological space X is open if it is an open subset of the product space X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Consider the following analogue of Todorˇcevi´c’s [Tod89] open graph axiom for regular uncountable cardinals κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For any class C, OGAκ(C) states that the following holds for all subsets X ∈ C of κκ: OGAκ(X): If G is an open graph on X, then either G admits a κ-coloring or G has a complete subgraph of size κ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 7See the survey [V¨a¨a95] for early developements in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 8The ω-Baire space is the Baire space ωω, since the bounded topology equals the product topology for κ = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ 5 The following analogue of Feng’s open graph dichotomy [Fen93] is a stronger version of this axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For any class C, OGDκ(C) states that the following holds for all subsets X ∈ C of κκ: OGDκ(X): If G is an open graph on X, then either G admits a κ-coloring or G has a κ-perfect complete subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='9 Note that G has a κ-perfect complete subgraph if and only if there exists a continuous homomorphism from the complete graph Kκ2 to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Thus OGDκ(X) can be formulated for arbitrary topological spaces as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In the countable case,10 Carroy, Miller and Soukup introduced the box-open d-dimensional dihypergraph dichotomy as an analogue of OGDω(X) in higher dimensions 2 ≤ d ≤ ω [CMS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We now consider its analogue for regular infinite cardinals κ with κ<κ = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let X be any set and d a set of size at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A d-dihypergraph11 on X is a subset H of dX consisting of non-constant sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Recall that a homomorphism is a map that takes hyperedges to hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='12 A subset of X is H-independent if it contains no hyperedges, and a κ-coloring is a partition of X into κ many H-independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let Hκd denote the following d-dihypergraph on κd: Hκd := � t∈<κd � i∈d Nt⌢⟨i⟩ = {x ∈ d(κd) : ∃t ∈ <κd ∀i ∈ d t⌢⟨i⟩ ⊆ xi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='. t0 t1 tα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='. x0 x1 xα <κd tα = t⌢⟨α⟩ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Hκd We will understand d as a discrete topological space and equip κd with the <κ-box topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Ordinals are equipped with the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If X is a subset of the κ-Baire space, then dX is equipped with the box-topology unless otherwise mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 9I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', G has a complete subgraph whose domain is a κ-perfect subset of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' See Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 10I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', for κ = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 11I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', a d-dimensional directed hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 12See Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2 for the precise definitions of the concepts discussed in this paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 6 PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For any d-dihypergraph H on κκ,13 let ODDH κ denote the following statement: ODDH κ : Either H admits a κ-coloring, or there exists a continuous homo- morphism f : κd → κκ from Hκd to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For classes C and D, let ODDd κ(C, D) denote statement that ODDH↾X κ holds whenever (a) X is a subset of κκ in C and (b) H is a box-open14 d-dihypergraph on κκ in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If D is the class of all sets or all d-dihypergraphs on κκ, then we omit it from the notation and write ODDd κ(C) for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If C has a unique element X, then we write X instead of C, and similarly for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For example, ODDd κ(X) states that ODDH↾X κ holds for all box-open d-dihypergraphs H on κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' It is equivalent that ODDH κ holds for all d-dihypergraphs H on X that are box-open on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='15 Following [CMS20], we call ODDd κ(X) the box-open d-dimensional dihypergraph dichotomy for X, or open dihypergraph dichotomy for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The open graph dichotomy OGDκ(X) is equivalent to ODD2 κ(X), and therefore it follows from ODDd κ(X) for any d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='16 The preliminary Sections 2 and 3 provide definitions and basic results on directed hypergraphs, trees, forcing, homomorphisms and order preserving maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The following theorem, which is the main result of this paper, is proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let Dκ denote the class of all sets that are definable from a κ-sequence of ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose κ < λ are regular infinite cardinals and 2 ≤ d ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For any Col(κ, <λ)-generic filter G over V , the following statements hold in V [G]:17 (1) ODDd κ(Dκ, Dκ) if λ is inaccessible in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) ODDd κ(Dκ) if λ is Mahlo in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='18 A crucial idea in the proof of the uncountable case is to construct a forcing which adds a perfect tree of generic filters over an intermediate model that induces a homomorphism of dihypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This technical part of the proof is done in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Note that ODDd κ(Dκ, Dκ) ⇒ ODDd κ(Dκ) holds for any d < κ, since any box-open d- dihypergraph on κκ is in Dκ by virtue of the topology having a base of size κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='19 It thus suffices to assume in (2) that λ is inaccessible in V if d < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' It is open whether the Mahlo cardinal is neccessary for (2) if d = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' However, the inaccessible cardinal is necessary 13The definition of ODDH κ also makes sense for dihypergraphs H on arbitrary topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 14I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e, H is an open subset of d(κκ) in the box topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 15In general, we have to work with box-open dihypergraphs H on κκ instead of box-open dihypergraphs on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For instance, ODDH↾X κ might hold only for definable box-open dihypergraphs H, but H↾X is not necessarily definable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This is relevant for several applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 16See Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 17(1) is equivalent to (1) in the abstract by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3, and (2) is equivalent to (2) in the abstract by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 18Recall that a cardinal λ is Mahlo if the set of inaccessible cardinals ν < λ is stationary in λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 19See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ 7 for (1), since the statement implies the perfect set property PSPκ(X) for all closed sets X ⊆ κκ and thus the existence of an inaccessible cardinal above κ in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='20 We also obtain a global version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4 (1) for all regular infinite cardinals in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' It remains open whether (2) admits a global version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In Section 5, we investigate alternatives to the open dihypergraph dichotomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We observe that ODDd κ(X) is equivalent to a dichotomy for d-hypergraphs with the open graph dichotomy OGDκ(X) as a special case for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We then affirm the optimality of the formulation of ODDH κ in various aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' While Hκd shows that for any d ≥ 3, the existence of a continuous homomorphism cannot be replaced by the existence of a large complete subhypergraph,21 it is natural to ask for homomorphisms with additional properties such as injectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' While this cannot be done for d = κ = ω,22 it is possible in the uncountable setting assuming a weak version of ♦κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The next result follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Assume ♦i κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='23 If H is a box-open d-dihypergraph on a subset of κκ where 2 ≤ d ≤ κ, then ODDH κ is equivalent to the following statement: ODDIH κ : Either H admits a κ-coloring or there exists an injective contin- uous homomorphism from Hκd to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For d < κ, or alternatively with additional assumptions on H, the continuous homo- morphism in ODDH κ can be chosen to be a homeomorphism onto a closed subset even without assuming ♦i κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This is demonstrated in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='24 It is open whether the seemingly weak assumption ♦i κ in the previous theorem can be removed for d = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='25 It holds for all inaccessible cardinals κ and all successor cardinals κ ≥ ω2 with κ<κ = κ by a result of Shelah [She10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='26 Moreover, if κ is a regular uncountable cardinal, then ♦i κ holds in Col(κ, <λ)-generic extensions V [G] where λ > κ is inaccessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Therefore ODDκ κ can be replaced by ODDIκ κ in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='27 Section 6 studies a number of applications of the open dihypergraph dichotomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We now describe the main results of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The missing notation will be introduced in the respective subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The Hurewicz dichotomy characterizes analytic subsets of Kσ sets as those that do not contain a topological copy of the Baire space [Hur28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Kechris and Saint Raymond proved this independently for more complex sets under appropri- ate determinacy assumptions [SR75, Kec77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In the uncountable setting, the Hurewicz dichotomy has a topological version that is more similar to the classical one [LMS16], 20See [Jec71, Sections 3&4] and [L¨uc12, Section 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The inaccessible cardinal is also neccessary in the countable case by [Jec03, Theorem 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 21See Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3 for stronger counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 22The dihypergraph in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='29 is a counterexample by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='30 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 23♦i κ and its relationship with some other versions of ♦κ is discussed in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Note that ♦i κ implies that κ is uncountable by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 24See Definitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='24, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='37 and Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='26, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 25See Problem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='7 26See Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 27See Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 8 PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI as well as a combinatorial version using superperfect trees [LMS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The next result is proved in Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='8 using a variant of a dihypergraph from [CMS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ODDκ κ(X, Dκ) implies the following analogues of the Hurewicz dichotomy: (1) THDκ(X): Either X contains a closed homeomorphic copy of κκ, or X can be covered by κ many κ-compact subsets of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) HDκ(X): Either X contains a κ-superperfect subset, or X can be covered by the sets of branches of at most κ many <κ-splitting subtrees of <κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Kechris, Louveau and Woodin proved a dichotomy that characterizes when an analytic set of reals can be separated from a disjoint set by an Fσ set [KLW87, Theorem 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Their theorem strengthens the Hurewicz dichotomy for analytic sets in the sense that one can derive the latter by a short argument using compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='28 Let Rκ denote the set of elements of κκ that take the value 0 unboundedly often and Qκ the set of those that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Based on an argument in [CMS20], we prove a stronger version of the next theorem where the sets X and Y are not necessarily disjoint in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ODDκ κ(X) implies the following analogue of the Kechris-Louveau-Woodin dichotomy for all subsets Y of κκ disjoint from X: KLWκ(X, Y ): Either there is a Σ0 2(κ) set A separating X from Y ,29 or there is a homeomorphism f between κ2 and a closed subset of κκ that reduces (Rκ, Qκ) to (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='30 We study dihypergraphs with domain κκ in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' There is no immediate reason why the restriction ODDκ κ(κκ) to these dihypergraphs should be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' While it implies ODDκ κ(X) for subsets X of κκ that are continuous images of κκ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='5, for uncountable κ not all closed subsets of κκ are of this form [LS15, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' However, an argument resembling the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='7 shows the next implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' It follows that ODDκ κ(κκ, Dκ) has the same consistency strength as an inaccessible cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We will further see that ODDκ κ(κκ) suffices for some other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (1) ODDκ κ(κκ) ⇒ ODDκ κ(Σ1 1(κ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) ODDκ κ(κκ, Dκ) ⇒ ODDκ κ(Σ1 1(κ), Dκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' By Cantor-Bendixson analysis, any set of reals can be decomposed as the disjoint union of a crowded set and a countable scattered set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='31 V¨a¨an¨anen showed that an analogue of this statement for all closed subsets of κκ is consistent32 but not provable [V¨a¨a91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' To 28See [Kec95, Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='22 & Corollary 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='23] and note that this argument does not generalize to higher cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 29I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', X ⊆ A and A ∩ Y = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 30I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', f(Rκ) ⊆ X and f(Qκ) ⊆ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 31Recall that a set of reals X is crowded if it has no isolated points, and if and only if its closure is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' X is scattered if each nonempty subspace contains an isolated point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 32V¨a¨an¨anen used a measurable cardinal and Galgon reduced this to an inaccessible cardinal [Gal16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The second-listed author showed that in fact, the restriction to closed sets is equivalent to the perfect set property PSPκ(X) for closed sets [Szir18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ 9 this end, he generalized the notions of scattered and crowded sets to subsets of κκ via a game Vκ(X) of length κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='33 We extend V¨a¨an¨anen’s result to all subsets of κκ in the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This is proved in Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='31 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ODDκ κ(κκ) implies the following version of the Cantor-Bendixson decom- position for all subsets X of κκ: CB2 κ(X): X equals the disjoint union of a κ-scattered set of size κ and a κ-crowded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In particular, Vκ(X) is determined for all subsets X of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Moreover, ODDκ κ(κκ, Dκ) implies the previous statements for all definable subsets of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' While there is a verbatim analogue of the Baire property at higher cardinals, it is not as useful as in the countable setting since even simple sets such as the club filter do not satisfy it [HS01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The asymmetric Baire property ABPκ(X) introduced in [Sch17] is a more general condition that is equivalent to the determinacy of the Banach-Mazur game of length κ for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In the countable setting, ABPω(Dω) is thus equivalent to the Baire property for sets in Dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The next result is proved in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ODDκ κ(X, Dκ) implies the asymmetric Baire property ABPκ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The proof of this theorem relies on a characterization of the Baire property via ho- momorphisms of dihypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A new consequence in the countable setting is that ODDω ω(X, Dω) implies the Baire Property for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A celebrated of Jayne and Rogers characterizes ∆0 2-measurable functions on the reals as those that are σ-continuous with closed pieces [JR82, Theorem 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A simpler proof of this theorem was subsequently discovered by Semmes and Motto Ros [RS10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Its possible generalizations are a subject of intense study [Ros13, GKN21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Recently, Carroy, Miller and Soukup found a new proof that derives the theorem from the open dihypergraph dihotomy [CMS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Their proof allows us to generalize the Jayne-Rogers theorem to the uncountable setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ODDκ κ(Dκ, Dκ) implies the following analogue of the Jayne-Rogers the- orem for all X ∈ Dκ: JRκ(Dκ): Let X ∈ Dκ be a subset of κκ and let f : X → κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Then f is ∆0 2(κ)-measurable if and only if it is a union of κ many continuous functions on closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The above two versions of the Hurewicz dichotomy, the asymmetric Baire property and the Jayne-Rogers theorem for definable subsets of κκ are consistent relative to an inaccessible cardinal by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The first two statements in particular provide new proofs of the main results of [Sch17, LMS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The Kechris-Louveau-Woodin dichotomy KLWκ(Dκ),34 the determinacy of V¨a¨an¨anen’s perfect set game and in fact CB2 κ(X) for arbitrary subsets X of κκ are consistent relative to a Mahlo cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The latter extends 33More exactly, V¨a¨an¨anen worked with local versions Vξ(X, x) for x ∈ κκ of this game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 34I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', KLWκ(X, Y ) for all definable X and arbitrary subsets Y of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 10 PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI a result of V¨a¨an¨anen [V¨a¨a91] from closed sets to arbitrary subsets of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Moreover, it lowers the upper bound for the consistency strength of a dichotomy studied in [SzV17] from a measurable to a Mahlo cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' α, β, γ, δ denote ordinals and κ, λ, µ, ν cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' κ always denotes an infinite cardinal with κ<κ = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lim and Succ denote the classes of limit and successor ordinals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If α = β + 1, then α − 1 denotes β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' An α-sequence is a function f : α → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ⟨c⟩α denotes the constant sequence of length α with value c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For sequences sα for α < β, � α<β sα denotes their concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For any set d, α < κ and any sequence x = ⟨xi : i ∈ d⟩ ∈ d(κκ), let x↾α := ⟨xi↾α : i ∈ d⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Given a function f : X → Y , write f(U) for the pointwise image of a subset U of X under f, and write f−1(W) for the preimage of a subset W of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For any set d, let fd : dX → dY denote the function defined by letting fd(⟨xi : i ∈ d⟩) := ⟨f(xi) : i ∈ d⟩ for all ⟨xi : i ∈ d⟩ ∈ dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' idX denotes the identity function on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A class X is definable from a set y if X = {x : ϕ(x, y)} for some first order formula ϕ(v0, v1) with two free variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let Dκ denote the class of those sets which are definable from a κ-sequence of ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' By a definable set, we always mean an element of Dκ when κ is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We use the following notation for definable subsets of the κ-Baire space: If ϕ(v0, v1) is a first order formula with two free variables and y is a set, write Xκ ϕ,y := {x ∈ κκ : ϕ(x, y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We further write Xϕ,y for Xκ ϕ,y if κ is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Basic open subsets of κd are denoted Nt := {x ∈ κd: t ⊆ x} for any d ≤ κ and t ∈ <κd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The closure of a subset X of κd is denoted X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The κ-Borel subsets of κd are defined by closing the set of basic open sets under unions of length κ and complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The κ-Borel hierarchy begins with Σ0 1(κ) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', open) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For any γ with 1 < γ < κ+, Σ0 γ(κ) sets are of the form � α<κ Aα, where each Aα is in Π0 β(κ) for some β < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For any γ with 0 < γ < κ+, Π0 γ(κ) sets are complements of Σ0 γ(κ) sets and ∆0 γ(κ) sets are both Σ0 γ(κ) and Π0 γ(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' κ-analytic or Σ1 1(κ) subsets of κd are of the form f(X), where f : κκ → κd is continuous and X is a closed subset of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A Π1 1(κ) set is a complement of a Σ1 1(κ) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A partial function f : κκ ⇀ κκ is called C-measurable with respect to a collection C of subsets of κκ if f−1(U) ∈ C for every open subset U of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Dihypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In this subsection, X denotes a set and d a set of size at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ∆d X denotes the diagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', the set of constant sequences in dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A d-dihypergraph35 on X is a subset H of dX \\ ∆d X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let Sym(d) denote the set of all permutations of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We write xπ := ⟨xπ(α) : α < d⟩ for any sequence x = ⟨xα : α < d⟩ and π ∈ Sym(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A d-hypergraph on X is a d-dihypergraph H on X that is closed under permutation of hyperedges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', xπ ∈ H for all x ∈ H and π ∈ Sym(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 35This is short for d-dimensional directed hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ 11 A digraph on X is then a 2-dihypergraph on X, while a graph on X is a 2-hypergraph on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', a symmetric irreflexive binary relation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) Kd X := dX − ∆d X is the complete d-hypergraph on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) KX = K2 X is the complete graph on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let H and I be d-dihypergraphs on X and Y , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) A homomorphism from H to I is a function f : X → Y such that fd(H) ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) A subset Z of X is H-independent if H ∩ dZ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (c) Let λ be a cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A function c : X → λ is a λ-coloring of H if c−1({α}) is H-independent for all α < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='36 Given a family of topological spaces ⟨Xi : i ∈ d⟩, the box-topology on � i∈d Xi is the topology generated by sets of the form � i∈d Ui, where Ui is an open subset of Xi for each i ∈ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If X is a subset of κκ, then we always work with the box topology on dX unless otherwise mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In the following, let H be a d-dihypergraph on a topological space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We call H box- open if it is open as a subset of dX with the box-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='37 For finite d, we will simply say that H is open on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) The domain domH of H is the set of all x ∈ X such that x is an element of some hyperedge of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='38 (b) H is relatively box-open if it is a box-open dihypergraph on its domain dom H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Note that H is box-open on X if and only if H is relatively open and domH is an open subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Trees and order preserving maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose s, t ∈ ≤κκ and A ⊆ ≤κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let lh(t) := dom(t) and lh(A) := sup{lh(u) : u ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let t↓ := {s ∈ <κκ : s ⊊ t} and succ(t) := {u ∈ <κκ : t ⊊ u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let s ∧ t denote the maximal r ∈ ≤κκ with r ⊆ s and r ⊆ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' s and t are called compatible, denoted s∥t, if s ⊆ t or t ⊆ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='39 s and t are called incompatible, denoted s⊥t, if they are not compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Note that s ∧ t is the node where s and t split if s ⊥ t, and s ∧ t = s if s ⊆ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For incompatible s and t, let ∆(s, t) := min{α < κ : s(α) ̸= t(α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Then ∆(s, t) = lh(s ∧ t) and s ∧ t = s↾∆(s, t) = t↾∆(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A subtree of <κκ is a subset that is closed under forming initial segments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', it is downwards closed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If A is a subset of ≤κκ, let T(A) := {t ∈ <κκ : ∃a ∈ A t ⊆ a} denote the tree of initial segments t ∈ <κκ of elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If T is a subtree of <κκ, then [T] := {x ∈ κκ: ∀α < κ x↾α ∈ T} denotes the set of its branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Note that [T(X)] is the closure of any subset X of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 36Equivalently, if c is a homomorphism from H to Kd λ 37We say H is box-closed if it is closed as a subset of Kd X with the box-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Note that H is box-closed if and only if H ∪ ∆d X is a closed subset of the space dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 38More precisely, domH = � i∈d pi(H), where pi denotes projection onto the ith coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 39For general posets P, this is called comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Conditions p, q ∈ P are called compatible if there exists a common extension r ≤ p, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 12 PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose T is a subtree of <κκ and t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) t is splitting in T is there exist u ⊥ v in T with t ⊆ u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) T is cofinally splitting if for each t ∈ T, there exists a splitting node u ∈ T with t ⊆ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (c) T is <κ-closed if for any strictly increasing sequence t = ⟨ti : i < α⟩ in T with α < κ, there exists an upper bound t ∈ T for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (d) T is κ-perfect if it is cofinally splitting and <κ-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (e) T is pruned if every node t ∈ T extends to a branch x ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (f) A closed subset X of κκ is κ-perfect if T(X) is κ-perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The κ-perfect set property (κ-PSP) holds for a subset X of κκ if either |X| ≤ κ or X contains a κ-perfect subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' PSPκ(C) states that the κ-PSP holds for all subsets X ∈ C of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that P and Q are posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A function ι: P → Q is called strict order preserving if p < q implies ι(p) < ι(q) for all p, q ∈ P and strict order reversing if p < q implies ι(p) > ι(q) for all p, q ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let ι be a partial function from <κκ to <κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) ι is ⊥-preserving if s ⊥ t implies ι(s) ⊥ ι(t) for all s, t ∈ dom(ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) Suppose that ι is strict order preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ι is called continuous if ι(t) = � s⊊t ι(s) for all t ∈ dom(ι) with lh(t) ∈ Lim and t↓ ⊆ dom(ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that ι is a strict order preserving partial function from <κκ to <κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) Let [ι] denote the partial function from κκ to κκ where dom([ι]) consists of those x ∈ κκ with x↾α ∈ dom(ι) for unboundedly many α < κ, and for all x ∈ dom([ι]) [ι](x) = � α<κ ι(x↾α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) Let T(ι) := T(ran(ι)) denote the tree of initial segments of elements of ran(ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We will usually assume that dom(ι) is a subtree of <κκ, in which case dom([ι]) = [dom(ι)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' It is clear from the definitions that T(ι) = T(ran([ι])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Thus, [T(ι)] is the closure of ran([ι]) in κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let ι be a strict order preserving partial function from <κκ to <κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (1) [ι] is a continuous function from dom([ι]) to κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) If ι is ⊥-preserving, then [ι] is a homeomorphism between dom([ι]) and ran([ι]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Since [ι] is strict order preserving, we have [ι] � Nt ∩ dom([ι]) � ⊆ Nι(t) for all t ∈ dom(ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Item (1) follows easily from this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' To show (2), Suppose that ι is ⊥-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Then Nι(s) ∩ Nι(t) = ∅ for all s, t ∈ dom(ι) such that s ⊥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Therefore [ι] is injective, and [ι] � Nt ∩ dom([ι]) � = Nι(t) ∩ ran([ι]) for all t ∈ dom(ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This implies that [ι] is a homeomoprhism between dom([ι]) and ran([ι]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' □ THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' A forcing P = (P, ≤P, 1P) is a triple such that ≤P is a pre-order (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', a reflexive transitive relation) on P and 1P is a largest element of P with respect to ≤P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We confuse P with its domain P, and we usually write ≤, ⊥, and ⊩ instead of ≤P, ⊥P, 1P and ⊩P, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For all p, q ∈ P, we let p ∥ q (or sometimes p ∥P q) denote the statement that p and q are compatible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=', that p ̸⊥ q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We let B(P) denote the Boolean completion of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='40 If P is separative, we assume that P is a dense subset of B(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) An atom in a forcing P is a condition p ∈ P with no incompatible extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Moreover, a forcing P is non-atomic if it has no atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) A forcing P is homogeneous if for all p, q ∈ P, there is an automorphism π: P → P such that π(p) and q are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose P, Q are forcings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) A dense embedding ι : P → Q is a homomorphism with respect to ≤, ⊥ and 1 such that ι(P) is a dense subset of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) Two forcings P and Q are equivalent (P ≃ Q) if there exist dense embeddings ι : P → R and ν : Q → R into some forcing R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (c) Let ι : P → Q be a dense embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We define a P-name σι for each Q-name σ by recursion on the rank as σι := � (τ ι, p) : p ∈ P, ∃q ∈ Q � ι(p) ≤ q ∧ (σ, q) ∈ τ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' It is easy to check that in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='11 (c), if G is a P-generic filter over V and H is the upwards closure of ι(G) in Q, then we have (σι)G = σH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We recall the following standard facts (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' [Kun13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Two forcings are equivalent if and only if they have isomorphic Boolean completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose ι : P → Q is a dense embedding between forcings P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Given a P-generic filter G over V , the upwards closure H of ι(G) in Q is a Q-generic filter over V , and G = ι−1(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Conversely, suppose H is a Q-generic filter over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Then G = ι−1(H) is a P-generic filter over V , and H is equal to the upwards closure of G in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In both of the above cases, we have V [G] = V [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The following definition of the forcing for adding Cohen subsets is non-standard, but is essential in several arguments below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that κ is a regular uncountable cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) Add(κ, 1) is defined as the forcing Add(κ, 1) := {p : α → κ | α < κ}, ordered by reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) Add(κ, ξ) is defined as the <κ-support product � i<ξ Add(κ, 1) for any ordinal ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We use the following convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For a given y ∈ κκ, we sometimes confuse y with the set y↓ = {t ∈ <κκ : t ⊊ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For example, we say that y is Add(κ, 1)-generic if and only if y↓ is an Add(κ, 1)-generic filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Furthermore, if y is Add(κ, 1)-generic and σ is an 40Note that B(P) is unique up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 14 PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI Add(κ, 1)-name, let σy := σy↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We also let � i<ξ yi denote � i<ξ yi↓ whenever ⟨yi : i < ξ⟩ is a sequence of elements of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We will often use the standard facts about adding Cohen subsets and collapse forcings which are found in Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='14 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that κ is an uncountable cardinal such that κ<κ = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If P is a non-atomic <κ-closed forcing of size κ, then P has a dense subset which is isomorphic to the dense subforcing Add∗(κ, 1) = {p ∈ Add(κ, 1) : dom p ∈ Succ} of Add(κ, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In particular, P is equivalent to Add(κ, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We omit the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='13, since is straightforward and well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='14 ([Fuc08, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let κ be a regular cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that ν > κ is a cardinal with ν<κ = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let P be a separative <κ-closed forcing of size ν which forces that ν has size κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Then P has a dense subset which is isomorphic to the dense subforcing Col∗(κ, ν) = {p ∈ Col(κ, ν) : dom p ∈ Succ} of Col(κ, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In particular, P is equivalent to Col(κ, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='14 can be proven using an adaptation of the proof of [Jec03, Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='7] which can be found in the proof of [Fuc08, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We will use the following notation for subforcings of the L´evy collapse Col(κ, < λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that κ < λ are cardinals, α ≤ λ and I ⊆ λ is not an ordinal (to avoid a conflict with the notation for the standard collapse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Then let Pα := Col(κ, <α) and PI = Col(κ, I) = {p ∈ Col(κ, < λ) : dom p ⊆ I × κ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The notation PI = Col(κ, I) will be used for intervals I, for which we use the standard notation (α, γ) = {β ∈ Ord : α < β < γ} [α, γ) = {β ∈ Ord : α ≤ β < γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In particular, let Pα := P[α,λ) for all α < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If G is a Pλ-generic filter over V , we write Gα := G ∩ Pα, GI := G ∩ PI and Gα := G ∩ Pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We will also often use the following consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let λ > κ be an inaccessible cardinal and γ < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If P is a <κ-closed forcing of size <λ, then the forcings P × Pλ and P[γ,λ) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='15 is a variant of [Fuc08, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (More specifically, [Fuc08, Corol- lary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4] asserts that the conclusion of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='15 holds when γ = 0, under the weaker assumption that λ > κ is a cardinal such that for all cardinals µ < λ, we have µ<κ < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=') Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='15 follows easily from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='14 using a straightforward analogue of the proof of [Fuc08, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4] (see also [Fuc08, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ 15 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that P, Q are forcings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) A complete embedding i: P → Q is a homomorphism with respect to ≤, ⊥ and 1 with the property that for every q ∈ Q, there is a condition p ∈ P (called a reduction of q to P) such that for every r ≤ p in P, i(r) is compatible with q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) P is a complete subforcing of Q (P ⋖ Q) if P is a subforcing of Q and the inclusion map idP : P → Q is a complete embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (c) Suppose that i: P → Q is a complete embedding and G is P-generic over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The quotient forcing Q/G for G in Q is defined as the subforcing Q/G := {q ∈ Q : ∀p ∈ G i(p) ∥ q} of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Moreover, we fix a P-name Q/P for for the quotient forcing for ˙G in P, where ˙G is the canonical P-name for the P-generic filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We also refer to Q/P as (a name for) the quotient forcing for P in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We recall the following standard facts (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' [Kun13] and Exercises (C7), (C8), (D4) and (D5) in [Kun80, Chapter VII]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose P and Q are forcings and i : P → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (i) If i is a homomorphism with respect to ≤, ⊥ and 1, then i is a complete embedding if and only if for all maximal antichains A of P, i(A) is a maximal antichain of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If P and Q are complete Boolean algebras, then i is a complete embedding (in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='16) if and only if i is an injective complete homomorphism of Boolean algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (ii) Any complete embedding i : P → Q defines a complete embedding j : B(P) → B(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If P and Q are separative, we may assume that i ⊆ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Specifically, if P is a complete subforcing of Q and Q is separative, we may assume that B(P) ⊆ B(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (iii) Suppose i is a complete embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let G be a P-generic filter over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If D is a dense subset of Q, then D ∩ Q/G is a dense subset of Q/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Therefore all Q/G-generic filters H over V [G] are also Q-generic filters over V , and G = i−1(H) holds for all such H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Conversely, if H is Q-generic over V and G = i−1(H), then G is a P-generic filter over V and H is Q/G-generic over V [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Furthermore, under either of the above assumptions, we have V [H]Q = V [G][H]Q/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Given complete Boolean algebras P and Q and a complete embedding i : P → Q, the retraction associated to i is the map πi : Q → P defined by letting πi(q) := �P{p ∈ P : i(p) ≥ q} for all q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The following lemma is standard, but we include a proof for the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose i : P → Q is a complete embedding between the complete Boolean algebras P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (1) For all q ∈ Q, πi(q) is the largest reduction of q to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) If G is a P-generic filter over V , then q ∈ Q/G if and only if πi(q) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 16 PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' First, observe that i(p) ⊥ q if and only if p ⊥ πi(q) holds for all p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Indeed, given p ∈ P, we have i(p) ⊥ q iff i(p) ∧ q = 0 iff ¬i(p) ≥ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' By the definition of πi and since i is a Boolean homomorphism, the last statement is equivalent to ¬p ≥ πi(q), and is therefore also equivalent to p ∧ πi(q) = 0 and to p ⊥ πi(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The above observation easily implies that πi(q) is a reduction of q to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' To see that πi(q) is the largest reduction of q to P, suppose that p ∈ P and p ̸≤ πi(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let r := p ∧ ¬πi(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Then p ≥ r ≥ 0P and r ⊥ πi(q), and therefore i(r) ⊥ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Thus, r witnesses that p is not a reduction of q to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The above observation also implies that if G is a P-generic filter over V and q ∈ Q, then q ∈ Q/G holds if and only if p ⊥ πi(q) holds for all p ∈ P, and therefore if and only if πi(q) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' □ As usual, by a name σ for a subset of a set x, we mean a name σ such that ⊩ σ ⊆ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' By a name σ for an element of κκ, we mean a name σ such that ⊩ σ ∈ κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' By a name σ for a new object, we mean a name σ such that ⊩ σ /∈ ˇV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In general, if ϕ(v) is any formula with one variable v, then by a name σ for an object with property ϕ, we mean a name σ such that ⊩ ϕ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We will use the following notation when working with quotient forcings induced by names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose Q is a complete Boolean algebra and σ is a Q-name for a subset of a set x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let B(σ) := BQ(σ) denote the complete Boolean subalgebra of Q that is completely generated by the set of Boolean values {�y ∈ σ�Q : y ∈ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The following lemma is standard, but we include a proof for the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that Q is a complete Boolean algebra and σ is a Q-name for a subset of a set x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' If H is a Q-generic filter over V , then V � σH� = V � BQ(σ) ∩ H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let A := {�y ∈ σ�Q : y ∈ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' By [Jec03, Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='40], we have V � BQ(σ) ∩ H � = V � A ∩ H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The fact that V � A ∩ H � = V � σH� follows from the observation that for all y ∈ x, we have y ∈ σH iff �y ∈ σ�Q ∈ H iff �y ∈ σ�Q ∈ A ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' □ In the next definition and lemmas, suppose that Q ∈ V is a forcing, q ∈ Q and σ ∈ V is a Q-name for an element of κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (a) σ[q] := �{t ∈ <κκ : q ⊩ t ⊆ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) σ(q) := {(α, β) : q ⊩ (α, β) ∈ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (c) The tree T σ,q of possible values for σ below q is defined as follows: T σ,q := {t ∈ <κκ : ∃r ≤ q t ⊆ σ[r]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For clarity, we at times write σ[q,Q], σ(q,Q) and T σ,q Q instead of σ[q], σ(q) and T σ,q, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The next two lemmas list basic properties of σ[q] and T σ,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The first one follows imme- diately from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' THE OPEN DIHYPERGRAPH DICHOTOMY FOR κκ 17 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (1) σ[q] ⊆ σq, and σ[q] = σ(q) if and only if dom(σ(q)) is an ordinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) σ[q]↾α = σ(q)↾α for all ordinals α with α ⊆ dom(σ(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (1) If q ̸⊩Q σ ∈ ˇV , then σ[q] ∈ <κκ equals the stem of T σ,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) q ⊩V Q σ ∈ [T σ,q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (3) If S is a subtree of <κκ with q ⊩V Q σ ∈ [S], then T σ,q ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (4) If M ⊇ V is a transitive model of ZFC with (<κκ)V = (<κκ)M, then (T σ,q)V = (T σ,q)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (1)-(3) are immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (4) holds since the formula “q ⊩ t ⊆ σ” is absolute between M and V , as it can be defined by a recursion which uses only absolute concepts [Kun13, Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Dihypergraphs and homomorphisms Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='1 contains several claims on the dichotomy ODDd κ(X, H) which will be used in some later arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2, we give an equivalent characterization of the existence continuous homomorphisms from Hκd to box-open dihypergraphs H via certain strict order preserving maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This characterization will be an important ingredient throughout the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We also characterize H-independence at the level of subtrees of <κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Basic facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Throughout this subsection, we assume that 2 ≤ d ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We first observe that the two options in the definition of ODDH κ are mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let H be a d-dihypergraph on a topological space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose that there is a homomorphism f : κd → X from Hκd to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (1) H does not have a κ-coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) H↾f(Nt) does not have a κ-coloring for any t ∈ <κd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Note that (1) implies that |X| > κ and (2) implies that |f(Nt)| > κ for all t ∈ <κd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' The proof of (1) is a straightforward analogue of the proof for the κ = ω case in [CMS20, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We give the details for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Suppose c′ : X → κ is a κ-coloring of H, and let c := c′ ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Since f is a homomorphism from Hκd to H, c is a κ-coloring of Hκd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We recursively define a continuous increasing sequence ⟨tα : α < κ⟩ such that tα ∈ αd and c−1({α})∩Ntα+1 = ∅ for each α < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' We use the Hκd-independence of c−1({α}) at stage α + 1 of the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Then � α<κ tα is an element of κd which is not in c−1({α}) for any α < d, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) follows from (1), since the map ft : κd → f(Nt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' x �→ f(t⌢x) is a homomorphism from Hκd to H↾f(Nt) for any given t ∈ κd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' □ We did not need to assume that f is continuous or that H is box-open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Therefore, if ODDH κ holds and there exists any homomorphism from Hκd to H, then there already exists a continuous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This is analogous to the fact that for any subset X of κκ that 18 PHILIPP SCHLICHT AND DOROTTYA SZIR´AKI satisfies the perfect set property, the existence of an injective function from κ2 to X already implies the existence of a continuous injective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Let 2 ≤ d < κ and X ⊆ κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (1) If U is a box-open subset of d(κκ) then U ∈ Dκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (2) ODDd κ(X) is equivalent to ODDd κ(X, Dκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For (1), note that U is given by a sequence x: κ → d(<κκ) since the base of the topology has size |d(<κκ)| = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Since x can be coded by an element of κ2, we have U ∈ Dκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Moreover, (2) follows from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' □ The next two lemmas give reformulations of ODDd κ(Dκ) and ODDd κ(Dκ, Dκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Recall that H is a relatively box-open dihypergraph if it is box-open on its domain domH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ODDd κ(Dκ) is equivalent to the statement that ODDH κ holds for all relatively box open d-dihypergraphs H with domH ∈ Dκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This follows from the observation that a dihypergraph H is relatively box-open with domH ∈ Dκ if and only if H = H′↾X for some subset X ∈ Dκ of κκ and some box-open d-dihypergraph H′ on κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' To see the direction from right to left, note that domH′↾X is a relatively open subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' ODDd κ(Dκ, Dκ) is equivalent to the statement that ODDH κ holds for all relatively box-open d-dihypergraphs H ∈ Dκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' This follows from the equivalence of the following statements for any d ≤ κ and any d-dihypergraph H on κκ: (a) H ∈ Dκ is relatively box-open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' (b) H = H′↾X for some subset X ∈ Dκ of κκ and some box-open d-dihypergraph H′ ∈ Dκ on κκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFQT4oBgHgl3EQfODbS/content/2301.13274v1.pdf'} +page_content=' For the implication (a)⇒(b), note that X := domH ∈ Dκ and H′ := �{ � α n, although it is not needed. Note that ∆ measures the Euclidean +distance between µ(1) and µ(2) and thus represents their separation. +The objective we consider is to minimize the total data size D = np needed to correctly classify +1 +arXiv:2301.00344v1 [math.ST] 1 Jan 2023 + +the individuals in the sample as a function of the “average quality” γ of the features: +γ := ∆2/p, where ∆2 := +p +� +k=1 +(µ(1) +k +− µ(2) +k )2 and µ(i) = (µ(i) +1 , . . . , µ(i) +p ) ∈ Rp, i = 1, 2. +(1) +Suppose we are given a data matrix X ∈ Rn×p with samples from two populations C1, C2, such that +∀i ∈ Cg, E(Xij) = µ(g) +j +g = 1, 2, ∀j = 1, . . . , p. +(2) +Our goal in the present work is to estimate the group membership vector ¯x ∈ {−1, 1}n such that +¯xi = 1 for i ∈ C1 and ¯xi = −1 for i ∈ C2, +(3) +where the sizes of clusters |Cj| =: nj, ∀j may not be the same. Our ultimate goal is to estimate the +solution to the discrete optimization problem: +maximize xT ¯Ax +subject to x ∈ {−1, 1}n +(4) +where ¯A is a static reference matrix to be specified. It was previously shown that, in expectation, +among all balanced cuts in the complete graph formed among n vertices (sample points), the cut of +maximum weight corresponds to the correct partition of the n points according to their distributions +in the balanced case (n1 = n2 = n/2). Here the weight of a cut is the sum of weights across all +edges in the cut, and the edge weight equals the Hamming distance between the bit vectors of +the two endpoints [11, 45]. Under suitable conditions, the statement above also holds with high +probability (w.h.p.). +In other words, in the context of population clustering, it has been previously shown one can use +a random instance of the integer quadratic program: +maximize xT Ax +subject to +x ∈ {−1, 1}n +(5) +to identify the correct partition of nodes according to their population of origin w.h.p., so long as +the data size D is sufficiently large and the separation metric is at the order of ∆2 = Ω(log n). The +analyses focused on the high dimensional setting, where p ≫ n [11, 45]. Here A = (aij) is an n × n +symmetric matrix where for 1 ≤ i, j ≤ n, aij = aji denotes the edge weight between nodes i and +j, computed from the individuals’ bit vectors. This result is structural, rather than algorithmic. +The integer quadratic program (4) (or (5)) is NP-hard. In a groundbreaking paper [21], Goemans +and Williamson show that one can use semidefinite program (SDP) as relaxation to solve these +approximately. +In this paper, we propose a semidefinite relaxation framework, inspired by [22], where we design +and analyze computational efficient algorithms to partition data into two groups approximately +according to their population of origin. More generally, one may consider semidefinite relaxations +for the following sub-gaussian mixture model with k centers (implicitly, with rank-k mean matrix +embedded), where +Xi = µ(ψi) + Zi +(6) +where Z1, . . . , Zn ∈ Rp are independent, sub-gaussian, mean-zero, random vectors and ψi : i → +{1, . . . , k} assigns node i to a group Cj with the mean µ(j) ∈ Rp for some j ∈ [k]. Here we denote by +2 + +[k] the set of integers {1, . . . , k}. Here, each row vector of X is a p-dimensional sub-gaussian random +vector and we assume rows are independent. We will consider a flexible model of parametrization +for Zj, j ∈ [n] in Section 2. In particular, we allow each population to have distinct covariance +structures, with diagonal matrices as special cases. The analysis framework for the semidefinite +relaxation by Gu´edon and Vershynin [22] was set in the context of community detection in sparse +networks, where A represents the adjacency matrix of a random graph. In other words, they study +the semidefinite relaxation of the integer program (5), where an n×n symmetric random adjacency +matrix A (observed) is used to replace the hidden static matrix ¯A in the original problem (4) such +that E(A) = ¯A. +The innovative proof strategy of [22] is to apply the Grothendieck’s inequality for the random +error A − ¯A rather than the original matrix A as considered in the earlier literature. We call this +approach the global analysis, following [13]. With proper adjustments, we apply this methodology +to our settings and prove the first main Theorem 2.5 regarding the partial recovery of the group +memberships based on n sequences of p features following the mean model (2). The important +distinction is: here, we replace the random adjacency matrix A arising from stochastic block models +as considered in [22] with an instance of symmetric matrix A, cf. (9), computed from the centered +data matrix which we now elaborate. Let diag(A) and offd(A) be the diagonal and the off-diagonal +part of matrix A respectively. +Estimators. We propose the following estimators. As in many statistical problems, one simple +but crucial step is to first obtain the centered data. Let 1n = [1, . . . , 1] ∈ Rn denote a vector of all +1s. Let X be a data matrix with row vectors X1, . . . , Xn as defined in (2). Denote by +Y += +X − 1n ⊗ �µn = X − P1X, +where +P1 = 1 +n1n1T +n +and +(7) +�µn += +1 +n +� +i +Xi is the average over n random vectors in Rp. +(8) +Loosely speaking, this procedure is called “global centering” in the statistical literature, for example, +see [24]. To estimate the group membership vector ¯x ∈ {−1, 1}n, we use the following adjusted A: +A +:= +Y Y T − λ(En − In), +where λ = +2 +n(n − 1) +� +i 0 +: +E exp(X2/t2) ≤ 2}. A random vector W ∈ Rm is called sub-gaussian if the one- +dimensional marginals ⟨ W, h ⟩ are sub-gaussian random variables for all h ∈ Rm: (1) W is called +isotropic if for every h ∈ Rm, E | ⟨ W, h ⟩ |2 = ∥h∥2 +2, where ∥h∥2 +2 = �m +j=1 h2 +j; (2) W is ψ2 with a +constant C0 if for every h ∈ Rm, ∥ ⟨ W, h ⟩ ∥ψ2 ≤ C0 ∥h∥2. The sub-gaussian norm of W ∈ Rm is +denoted by +∥W∥ψ2 := +sup +h∈Sm−1 ∥ ⟨ W, h ⟩ ∥ψ2 . +(13) +Throughout this paper, we use Z = (Zij) to denote the positive semidefinite matrix in SDP objective +functions. We use Z = (zij) to denote the mean-zero random matrix with independent, mean-zero, +sub-gaussian row vectors Z1, . . . , Zn ∈ Rp as considered in (6), where for a constant C0 , +∀j = 1, . . . , n, ∥ ⟨ Zj, x ⟩ ∥ψ2 +≤ +C0 ∥ ⟨ Zj, x ⟩ ∥L2 for any x ∈ Rp +(14) +where ∥ ⟨ Zj, x ⟩ ∥2 +L2 +:= +E ⟨ Zj, x ⟩ 2 = xT E(ZjZT +j )x =: xT Cov(Zj)x. +(15) +Examples of random vectors with sub-gaussian marginals include the multivariate normal random +vectors Zj ∼ N(0, Σ) with covariance Σ ≻ 0, and vectors with independent Bernoulli random +variables, where the mean parameters pi +j := E(Xij) for all i ∈ [n] and j ∈ [p] are assumed to be +bounded away from 0 or 1; See for example [11, 9]. For a symmetric matrix A, let λmax(A) and +λmin(A) be the largest and the smallest eigenvalue of A respectively. The operator norm ∥A∥2 is +defined to be +� +λmax(AT A). +Signal-to-noise ratios and sample lower bounds. Our work is inspired by the two threads of +work in combinatorial optimization and in community detection, and particularly by [22] to revisit +the max-cut problem (5) and to formulate the SDP (11). Our focus is on the sample size lower +bound, similar to the earlier work of the author [11, 45]. Moreover, we adopt the following notion +of signal-to-noise ratio when the sub-gaussian random vectors Zi in (6) are isotropic: +SNR isotropic: +s2 = (∆2/C2 +0) ∧ (npγ2/C4 +0) +(16) +where C0 is ψ2-constant of the high dimensional vectors Zi ∈ Rp; This notion of SNR appears +in [20], which can be properly adjusted when coordinates in Zi are dependent in view of (14) and +(15): +SNR anisotropic: +s2 = +∆2 +C2 +0 maxj ∥Cov(Zj)∥2 +∧ +npγ2 +C4 +0 maxj ∥Cov(Zj)∥2 +2 +. +(17) +We can rewrite the separation condition that is implicit in Theorem 2.5 as follows: +∆2 = pγ ≥ C2 +0 max +i +∥Cov(Zi)∥2 +� 1 +ξ2 ∨ +� p +nξ2 +� +for some 0 < ξ < 1/2. +(18) +4 + +We obtain in Theorem 2.5 misclassification error that is inversely proportional to the square root +of the SNR parameter s2 as in (16) (resp. (17)) for isotropic Zi (resp. for Zi, i ∈ [n] with covariance +structures), assuming that it is lower bounded. In the settings of Theorem 2.5 and Lemma 2.4, we +are able to prove that the error decays exponentially with respect to the SNR s2 in Theorem 2.7. +The implication of such an exponentially decaying error bound is: when s2 = Ω(log n), perfect +recovery of the cluster structure is accomplished. This result is in the same spirit as that in [20]; +See also [39, 16, 17] and references therein. Due to its significant length, we defer its proof to +another paper. We compare with [16, 17, 20] in the sequel. Also closely related is the work of [9]. +In more details, spectral algorithms in [9] partition samples based on the top few eigenvectors of the +gram matrix XXT , following an idea which goes back at least to [18]. In [9], the two parameters n, p +are assumed to be roughly at the same order, hence not allowing a full range of tradeoffs between +the two dimensions as considered in the present work; cf (18). The spectral analysis in this paper +is based on Y Y T , which will directly improve the results in [9] in the sense that we remove the +lower bound on n, concerning spectral clustering for k = 2; cf. Theorem 4.1. Such a lower bound +on n was deemed to be unnecessary given the empirical evidence in [9]; See “summary and future +direction” in [9]. +1.1 +Contributions +In summary, we make the following theoretical contributions in this paper: (a) We construct the +estimators in (11), which crucially exploit the geometric properties of the two mean vectors, as we +show in Section 3; (b) Moreover, we use Y Y T (and the corresponding A (9)) instead of the gram +matrix XXT as considered in [9], as the input to our optimization algorithms, ensuring both com- +putational efficiency and statistical convergence, even in the low SNR case (s2 = o(log n)); (c) This +approach allows a transparent and unified global and local analysis framework for the semidefinite +programming optimization problem (11), as given in Theorems 2.5 and 2.7 respectively; (d) With +the new results on concentration of measure bounds on +��Y Y T − EY Y T ��, we can simultaneously +analyze the SDP (11) as well as spectral algorithms based on the leading eigenvector of Y Y T . Here +and in the sequel, we use ∥·∥ to indicate either an operator or a cut norm; cf. Definition 3.1. +In Section 4, we make further connections to the existing semidefinite relaxations of the k-means +clustering problems, which include the baseline spectral algorithm based on the singular value +decomposition (SVD) of Y Y T . This allows even faster computation. We analyze a simple spectral +algorithm in Theorem 4.1 through the Davis-Kahan Perturbation Theorem, where we obtain error +bounds similar to Theorem 2.5. There, we further justify the global centering approach taken in the +current paper. We compare numerically the two algorithms, namely, based on SDP and spectral +clustering respectively and show they indeed have similar trends as predicted by the signal-to-noise +ratio parameter. +1.2 +Notations and organizations +Let e1, . . . , en be the canonical basis of Rn. For a set J ⊂ {1, . . . , n}, denote EJ = span{ej : j ∈ +J}. Let P1 = 1n1T +n/n and En = nP1. Denote by En×m ⊂ Rn×m a matrix of all ones. For a +vector v ∈ Rn, we use vJ to denote the subvector (vj)j∈J. For a vector x, ∥x∥∞ := maxj |xj|, +∥x∥1 := � +j |xj|, and ∥x∥2 := +�� +j x2 +j; diag(x) denotes the diagonal matrix whose main diagonal +entries are the entries of x. For a matrix B ∈ Rn×n, tr(B) = �n +i=1 Bii. For a matrix A = (aij) +5 + +of size n × m, let vec { A } be formed by concatenating columns of matrix A into a long vector of +size nm; we use ∥A∥1 = �n +i=1 +�m +j=1 |aij| denote the ℓ1 norm of vec { A }, and ∥A∥F = (� +i,j a2 +ij)1/2 +the ℓ2 norm of vec { A }, which is also known as the matrix Frobenius norm. For a matrix A, let +∥A∥∞ = maxi +�n +j=1 |aij| denote the maximum absolute row sum; Let ∥A∥max = maxi,j |aij| denote +the component-wise max norm. For two numbers a, b, a ∧ b := min(a, b), and a ∨ b := max(a, b). +We write a ≍ b if ca ≤ b ≤ Ca for some positive absolute constants c, C which are independent +of n, p, and γ. We write f = O(h) or f ≪ h if |f| ≤ Ch for some absolute constant C < ∞ and +f = Ω(h) or f ≫ h if h = O(f). We write f = o(h) if f/h → 0 as n → ∞, where the parameter n +will be the size of the matrix under consideration. In this paper, C, C1, C2, C4, c, c′, c1, etc, denote +various absolute positive constants which may change line by line. +The rest of the paper is organized as follows. In Section 2, we present the main theoretical results +of the paper. +In Section 3, we present the proof outline for Theorem 2.5 on the semidefinite +program (11) and concentration of measure bounds on B in Theorem 3.5. In Section 4, we discuss +various forms of semidefinite relaxations that have been considered in the literature, and highlight +the connections and main differences with the current work. +Section 5 gives an outline of the +arguments for proving Theorem 3.5, highlighting concentration bounds on +��Y Y T − EY Y T �� in +Theorems 5.3 and 5.2. In Section 6, we present main ideas in proving Theorem 5.2 with regards to +the operator and cut norm using independent design. In Section 7, we discuss correlated design and +their concentration of measure bounds concerning Theorem 5.3, one of the most technical results of +this paper. Section 8 shows numerical results that validate our theoretical predictions. We conclude +in Section 9. We defer all technical proofs to the supplementary material. +2 +Theory +We will first construct a matrix Y such that we subtract the sample mean �µn ∈ Rp as computed +from (8) from each row vector Xi of the data matrix. A straight-forward calculation leads to the +expression of the reference matrix R in view of (20), and hence E(Y )E(Y )T = R for R as in (21). +Definition 2.1. (The estimators) Let Y be as in (7) and X as in (2). Denote by Zi = Xi−E(Xi) +and +�µn − E�µn +:= +1 +n +n +� +i=1 +Zi = 1 +n +n +� +i=1 +Xi − E(Xi), +(19) +where µn is as in (8). +Clearly, by linearity of expectation, E�µn = w1µ(1) + w2µ(2), where wj = |Cj|/n for j = 1, 2. Hence +we have for wi = |Ci| /n, i = 1, 2 +E(Yi) +:= +� w2(µ(1) − µ(2)) +if i ∈ C1; +w1(µ(2) − µ(1)) +if i ∈ C2. +(20) +Definition 2.2. (The reference matrix) Denote by n1 = w1n and n2 = w2n. For Y as defined +in (7) (cf. Definition 2.1), and ∆2 = pγ as in (1), we have +R = E(Y )E(Y )T +=: +pγ +� +w2 +2En1 +−w1w2En1×n2 +−w1w2En2×n1 +w2 +1En2 +� +. +(21) +6 + +Definition 2.3. (Data generative process.) Suppose that random matrix W = (wjk) ∈ Rn×m +has W1, . . . , Wn ∈ Rm as row vectors, where Wj, j ∈ [n] are independent, mean-zero, isotropic +sub-gaussian random vectors with independent entries satisfying +∀j ∈ [n], +Cov(Wj) := E(WjW T +j ) = Im, E[wjk] = 0, ∀j, k, +and +max +jk ∥wjk∥ψ2 ≤ C0. +(22) +Suppose that we have for row vectors of Z ∈ Rn×p, ∀j = 1, . . . , n, +ZT +j += +W T +j HT +j +where Hj ∈ Rp×m, 0 < ∥Hj∥2 < ∞, +and Hj is allowed to repeat, for example, across rows from the same cluster Ci for some i = 1, 2. +Throughout this paper, we assume that m ≥ p to simplify our exposition, although this is not +necessary. First, Lemma 2.4 characterizes the two-group design matrix variance and covariance +structures to be considered in Theorems 2.5 and 2.7. It is understood that when Hi is a symmetric +square matrix, it can be taken as the unique square root of positive semidefinite covariance matrix, +denoted by Cov(Zj) := HiHT +i =: Σi ⪰ 0 for all j ∈ Ci. +Lemma 2.4. (two-group sub-gaussian mixture model) Denote by X the two-group design +matrix as considered in (2). +Let W1, . . . , Wn ∈ Rm be independent, mean-zero, isotropic, sub- +gaussian random vectors satisfying (22). +Let Zj = Xj − EXj = HiWj, for all j ∈ Ci, where +Hi ∈ Rp×m, and 0 < ∥Hi∥2 < ∞, for i ∈ {1, 2}. Then Z1, . . . , Zn are independent sub-gaussian +random vectors with Cov(Zi) satisfying (14) and (15), where +∀j ∈ Ci, +Cov(Zj) := E(ZjZT +j ) += +E(HiWjW T +j HT +i ) = HiHT +i and Vi := ∥Hi∥2 +F = tr(Σi).(23) +2.1 +Main results +Throughout this paper, we use nmin := nwmin and nmax := nwmax to represent the size of the small- +est and the largest clusters respectively. Denote by wmin := minj=1,2 wj, where wj = nj/n. We +first make the following assumptions (A1) and (A2), assuming random matrix Z has independent +sub-gaussian entries, matching the separation (and SNR) condition (25). As a baseline, we state in +Theorem 2.5 our first main result under (A1) and (A2). However, the conclusions of Theorem 2.5 +hold for the general two-group model so long as (A2) holds, upon adjusting (25). +(A1) Let Z = X − EX = (zij). +Let Zi = Xi − EXi, i = 1, . . . , n be independent, mean- +zero, sub-gaussian random vectors with independent coordinates such that for all i, j, ∥zij∥ψ2 := +∥Xij − EXij∥ψ2 ≤ C0. +(A2) The two distributions have variance profile discrepancy bounded in the following sense: +|V1 − V2| +≤ +1 +3ξnpγ +for some +1 > 2ξ = Ω(1/nmin), where +V1 += +E ⟨ Zj, Zj ⟩ +∀j ∈ C1 and V2 = E ⟨ Zj, Zj ⟩ +∀j ∈ C2. +(24) +Theorem 2.5. Let 1 > δ = Ω(1/n). Let Cj ⊂ [n] denote the group membership, with |Cj| = nj +and � +j nj = n. Suppose that for j ∈ Ci, EXj = µ(i), where i = 1, 2. Let �Z be a solution of the +SDP (11). Suppose that (A1) and (A2) hold and for some absolute constants C, C′, +pγ = ∆2 ≥ C′C2 +0 +ξ2 +and pn ≥ CC4 +0 +ξ2γ2 , +where ξ is the same as in (24). +(25) +7 + +Then with probability at least 1 − 2 exp(−cn), we have +��� �Z − ¯x¯xT ��� +1 /n2 +=: +δ ≤ 2KGξ/w2 +min and +��� �Z − ¯x¯xT ��� +2 +F /n2 ≤ 4KGξ/w2 +min, +(26) +where ¯x is as in (3). The same error bounds (26) also hold for the more general two-group sub- +gaussian mixture model as considered in Lemma 2.4, upon adjusting (25), so that (18) holds. +Discussions. We give a proof outline of Theorem 2.5 in Section 3 for completeness. Our proof +covers both isotropic and anisotropic cases. See [39, 20] for justifications of (A2). Our analysis +shows the surprising result that Theorem 2.5 does not depend on the clusters being balanced, nor +does it require identical variance profiles, so long as (A2) holds. Let us also choose a convex subset +Mopt: +Mopt = {Z : Z ⪰ 0, diag(Z) = In} ⊂ M+ +G. +(27) +Our proof follows the sequence of arguments in [22], which were specified for the stochastic block +model. However, when adapting to our setting, we crucially use the sub-gaussian concentration +of measure bounds as given in Theorems 3.5 and 5.2, as well as verifying a non-trivial global +curvature of the excess risk +⟨ R, Z∗ − �Z ⟩ +for the feasible set Mopt at the maximizer Z∗ = +arg maxZ∈Mopt ⟨ R, Z ⟩ = ¯x¯xT , cf. Lemma 3.7. In order to control the misclassification error using +the global approach, the parameters (δ, ξ) must satisfy the following: in view of (25) and (26), +ξ2 ≍ C2 +0 +pγ ∨ C4 +0 +npγ2 = 1/s2 +and +δ ≤ 2KGξ +w2 +min +. +(28) +Here the parameter 0 < ξ2 < 1/4 is understood to be chosen to be inversely proportional to the +SNR parameter s2, so that with probability at least 1 − exp(−cn), +��Y Y T − EY Y T �� +2 ≤ C(C2 +0(√pn ∨ n) ∨ C0n√pγ) ≍ ξnpγ +as we will show in Theorems 5.2 and 5.3. +Clearly, the larger separation ∆2, the larger sample size n, and the larger s2, the easier it is for +(A2) to be satisfied, since by definition and (25), +ξnpγ ≥ n/(ξ) ∨ 1 +ξγ ≍ +√ +s2(n ∨ 1 +γ ) +Hence so far, the misclassification error rate δ ≍ ξ/w2 +min is bounded to be inversely proportional to +the square root of s2. More explicitly, we have Corollary 2.6. +Corollary 2.6. (Clustering with o(n) misclassified vertices) Let �x denote the eigenvector of �Z +corresponding to the largest eigenvalue, with ∥�x∥2 = √n. Then in both settings of Theorem 2.5, we +have with probability at least 1 − 2 exp(−cn), +min +α=±1 ∥α�x − ¯x∥2 +2 ≤ δn = δ ∥¯x∥2 +2 , +where δ ≤ 32KGξ/w2 +min; +(29) +Moreover, the signs of the coefficients of �x correctly estimate the partition of the vertices into the +two clusters, up to at most δn misclassified vertices. +8 + +Next, we present in Theorem 2.7 (resp. Corollary 2.8) an error bound (31) (resp. (32)), which +decays exponentially in the SNR parameter s2 as defined in (17). The settings as considered in +Theorem 2.7 include that of Theorem 2.5 as a special case, which we elaborate in Section 2.2. We +prove Theorem 2.7 in a concurrent paper. Corollaries 2.6 and 2.8 follow from the Davis-Kahan +Theorem, Theorems 2.5 and 2.7 respectively, which we prove in the supplementary Section B. +Theorem 2.7. Let W1, . . . , Wn ∈ Rm be independent, mean-zero, isotropic, sub-gaussian random +vectors satisfying (22). Suppose the conditions in Theorem 2.5 hold, except that instead of (A1), +we assume that the noise matrix Z = X − E(X) is generated according to Definition 2.3: +∀j ∈ Ci, +Zj = HiWj for i ∈ {1, 2} +and Hi ∈ Rp×m, +where 0 < ∥Hi∥2 < ∞. Suppose that for some absolute constant C, C1, +pγ ≥ CC2 +0 maxj ∥Cov(Zj)∥2 +w4 +min +and +np ≥ C1C4 +0 maxj ∥Cov(Zj)∥2 +2 +γ2w4 +min +. +(30) +Let s2 be as defined in (17). Then with probability at least 1 − 2 exp(−c1n) − c2/n2, +��� �Z − ¯x¯xT ��� +1 /n2 ≤ exp(−c0s2w4 +min) +(31) +for �Z as in (11), for some absolute constants c, c0, c1. +Corollary 2.8. (Exponential decay in s2) Denote by θSDP = ∠(�x, ¯x), the angle between �x and ¯x, +where recall ¯xj = 1 if j ∈ C1 and ¯xj = −1 if j ∈ C2, and �x is as in Corollary 2.6. In the settings of +Theorem 2.7, with probability at least 1 − 2 exp(−cn) − 2/n2, for some absolute constants c, c0, c1, +sin(θSDP) +≤ +2 +��� �Z − ¯x¯xT ��� +2 /n ≤ exp(−c1s2w4 +min) +and +min +α=±1 +��(α�x − ¯x)/√n +�� +2 +≤ +23/2 ��� �Z − ¯x¯xT ��� +2/n ≤ 4 exp(−c0s2w4 +min/2) +(32) +2.2 +Covariance estimation +Remarks on covariance being diagonal. In Theorem 2.5, each noise vector Zj, ∀j ∈ [n] has +independent, mean-zero, sub-gaussian coordinates with uniformly bounded ψ2 norms. Suppose that +we generate two clusters according to Lemma 2.4, with diagonal H1 and H2 respectively, where +HjHT +j = diag(σ2 +1j, . . . , σ2 +pj) +∀j ∈ {1, 2}, +Let σmax := maxi ∥Hi∥2 = (maxj,k E(z2 +jk))1/2. Then for each row vector in Ci, we have +Cov(Zj) += +HiHT +i +and hence +Vi := E ⟨ Zj, Zj ⟩ = tr(HiHT +i ) = +p +� +k=1 +σ2 +ki +by Lemma 2.4, where Vi is the common variance profile for nodes j ∈ Ci. +Now, we have by +independence of coordinates of Zj and by definition of (13) +∥Zj∥ψ2 +:= +sup +h∈Sp−1 ∥ ⟨ Zj, h ⟩ ∥ψ2 ≤ C max +k≤p ∥zjk∥ψ2 ≤ CC0(max +i +∥Hi∥2) +9 + +where Sp−1 denotes the sphere in Rp, and we use (22) and the fact that +max +k≤p ∥zjk∥ψ2 ≤ σmax(max +j,k ∥wjk∥ψ2) ≤ σmaxC0 since ∥wjk∥ψ2 ≤ C0, ∀j, k. +Then clearly, maxj ∥Cov(Zj)∥2 = maxi +��HiHT +i +�� +2 = σ2 +max and hence (30) implies that (25) holds; cf. +Remarks on more general covariance. When we allow each population to have distinct co- +variance structures following Theorem 2.7, we have for some universal constant C, and for all +j ∈ Ci, +∥Zj∥ψ2 := +sup +h∈Sp−1 ∥ ⟨ Zj, h ⟩ ∥ψ2 +≤ +∥Wj∥ψ2 ∥Hi∥2 ≤ CC0 max +i +∥Hi∥2 +since +(33) +∀h ∈ Sp−1, +∥ ⟨ Zj, h ⟩ ∥ψ2 += +∥ ⟨ HiWj, h ⟩ ∥ψ2 ≤ ∥Wj∥ψ2 +��HT +i h +�� +2 +(34) +where ∥Wj∥ψ2 ≤ CC0 by definition of (22). Without loss of generality (w.l.o.g.), one may assume +that C0 = 1, as one can adjust Hi to control the upper bound in (33) through ∥Hi∥2. As we will +show in Theorems 6.3 and 7.2, with probability at least 1 − 2 exp(−cn), for Z as in Definition 2.3, +1 +p +��ZZT − EZZT �� +2 +≤ +C′(C0 max +i +∥Hi∥2)2 +��n +p ∨ n +p +� +, +(35) +for absolute constants c, C′. We discuss the concentration of measure bounds on +��Y Y T − EY Y T ��, +using (35) in Sections 5.1 and 7; cf. Lemmas 5.4 and 5.5. +2.3 +Related work +In the present work, we use semidefinite relaxation of the graph cut problem (5), which was origi- +nally formulated in [11, 45] in the context of population clustering. The biological context for this +problem is we are given DNA information from n individuals from k populations of origin and we +wish to classify each individual into the correct category. DNA contains a series of markers called +SNPs, each of which has two variants (alleles). Given the population of origin of an individual, +the genotypes can be reasonably assumed to be generated by drawing alleles independently from +the appropriate distribution. In the theoretical computer science literature, earlier work focused +on learning from mixture of well-separated Gaussians (component distributions), where one aims +to classify each sample according to which component distribution it comes from; See for exam- +ple [14, 5, 41, 3, 26, 28]. In earlier works [14, 5], the separation requirement depends on the number +of dimensions of each distribution; this has recently been reduced to be independent of p, the di- +mensionality of the distribution for certain classes of distributions [3, 27]. While our aim is different +from those results, where n > p is almost universal and we focus on cases p > n, we do have one +common axis for comparison, the ℓ2-distance between any two centers of the distributions as stated +in (36), which is essentially optimal. +Suppose (25) holds so that the ℓ2-separation and total data size satisfy +∆2 := pγ = �Ω(1/(ξ2)) and pn = �Ω(1/(ξ2γ2)), +where 1/n < ξ ≤ cw2 +min +(36) +and the �Ω(·) symbol only hides ψ2-constants for the high dimensional sub-gaussian random vectors +Zi ∈ Rp in (6). Our results show that even when n is small, by increasing p so that the total +10 + +sample size satisfies (36), we ensure partial recovery of cluster structures using the SDP (11) or +the spectral algorithm as described in Theorem 4.1. Previously, such results were only known to +exist for balanced max-cut algorithms [45, 11], where �Ω(·) symbol in (36) may also hide logarithmic +factors. Results in [45, 11] were among the first such results towards understanding rigorously and +intuitively why their proposed algorithms and previous methods [34, 37] work with low sample +settings when p ≫ n and np satisfies (36). These earlier results still need the SNR to be at the +order of s2 = O(log n); Moreover these results were structural as no polynomial time algorithms +were given for finding the max-cut. +The main contribution of the present work is: we use the proposed SDP (11) and the related +spectral algorithms to find the partition, and prove quantitively tighter bounds than those in [45, 11] +by removing these logarithmic factors. Recently, this barrier has also been broken down by the +sequence of work [39, 16, 20], which we elaborate in Section 4, cf. Variation 3. For example, [16, 17] +have also established exponentially decaying error bounds with respect to an appropriately defined +SNR, which focuses on balanced clusters and requires an extra √log n factor in (37) in the second +component: +In [16], cf. eq.(8): +∆2 = pγ += +Ω +� +1 + +� +p log n +n +� +or +(37) +In [17], cf. eq.(13): +∆2 = pγ += +Ω +� +(1 ∨ p +n) + +� +p log n +n +� +(38) +As a result, in (38), a lower bound on the sample size is imposed: n ≥ 1/γ in case p > n, and +moreover, the size of the matrix np ≥ log n/γ2, similar to the bounds in [9]; cf. Theorem 1.2 therein. +We refer to [11, 9] for references to earlier results on spectral clustering and graph partitioning. +We also refer to [26, 38, 22, 1, 7, 12, 8, 20, 29, 17, 30, 2, 32] and references therein for related +work on the Stochastic Block Models (SBM), mixture of (sub)Gaussians and clustering in more +general metric spaces. Our proof technique may be of independent interests, since centering the +data matrix so that each column has empirical mean 0 is an idea broadly deployed in statistical +data analysis. +3 +The (oracle) estimators and the global analysis +Exposition in this subsection follows that of [22], which we include for self-containment. First we +state Grothendieck’s inequality following [22]. The concept of cut-norm plays a major role in the +work of Frieze and Kannan [19] on efficient approximation algorithms for dense graph and matrix +problems. The cut norm is also crucial for the arguments in [22] to go through. +Definition 3.1. (Matrix cut norm) For a matrix A = (aij), we denote by ∥(aij)∥∞→1 its +ℓ∞ → ℓ1 norm, which is +∥(aij)∥∞→1 += +max +∥s∥∞≤1 ∥As∥1 = +max +s,t∈{−1,1}n ⟨ A, stT ⟩ +This norm is equivalent to the matrix cut norm defined as: for A ∈ Rm×n, +∥A∥□ += +max +I⊂[m],J⊂[n] +������ +� +i∈I +� +j∈J +ai,j +������ +11 + +and hence +∥A∥∞→1 += +max +x,y∈{−1,1}n +n +� +i=1 +n +� +j=1 +aijxiyi ≤ ∥x∥2 ∥y∥2 ∥A∥2 ≤ n ∥A∥2 +Theorem 3.2. (Grothendieck’s inequality) Consider an n × n matrix of real numbers B = (bij). +Assume that, for any numbers si, tj ∈ {−1, 1}, we have +������ +� +i,j +bijsitj +������ += +�� ⟨ B, stT ⟩ +�� ≤ 1 +(39) +Then for all vectors Si, Vi ∈ Bn +2 , we have +���� +i,j bij ⟨ Si, Vj ⟩ +��� = +�� ⟨ B, SV T ⟩ +�� ≤ KG, where KG is +an absolute constant referred to as the Grothendieck’s constant: +KG ≤ +π +2 ln(1 + +√ +2) ≤ 1.783. +(40) +Here Bn +2 = {x ∈ Rn : ∥x∥2 ≤ 1} denotes the unit ball for Euclidean norm. Consider the following +two sets of matrices: +M1 := +� +stT : s, t ∈ {−1, 1}n� +, +MG := +� +SV T : +all rows Si, Vj ∈ Bn +2 +� +. +Clearly, M1 ⊂ MG. As a consequence, Grothendieck’s inequality can be stated as follows: +∀B ∈ Rn×n +max +Z∈MG | ⟨ B, Z ⟩ | ≤ KG max +Z∈M1 | ⟨ B, Z ⟩ | . +(41) +Clearly, the RHS (41) can be related to the cut norm in Definition 3.1: +max +Z∈M1 | ⟨ B, Z ⟩ | = +max +s,t∈{−1,1}n ⟨ B, stT ⟩ = ∥B∥∞→1 +(42) +To keep the discussion sufficiently general, following [22], we first let Mopt be any subset of the +Grothendieck’s set M+ +G defined in (43): +M+ +G := {Z : Z ⪰ 0, diag(Z) ⪯ In} ⊂ MG ⊂ [−1, 1]n×n. +(43) +Lemma 3.3 elaborates on the relationship between �Z for any given B (random or deterministic), +and Z∗ with respect to the objective function using R, as defined in (44). Let +�Z := arg max +Z∈Mopt +⟨ B, Z ⟩ +and +Z∗ := arg max +Z∈Mopt +⟨ R, Z ⟩ +(44) +Lemma 3.3. (Lemma 3.3 [22]) Let Mopt be any subset of M+ +G ⊂ [−1, 1]n×n as defined in (43). +Then for �Z and Z∗ as defined in (44), +⟨ R, Z∗ ⟩ − 2KG ∥B − R∥∞→1 ≤ ⟨ R, �Z ⟩ ≤ ⟨ R, Z∗ ⟩ +(45) +where the Grothendieck’s constant KG is the same as defined in (40). Then +0 ≤ ⟨ R, Z∗ − �Z ⟩ +≤ +2KG ∥B − R∥∞→1 moreover, we have +(46) +sup +Z∈Mopt +| ⟨ B − R, Z − Z∗ ⟩ | +≤ +2KG ∥B − R∥∞→1 +(47) +12 + +Lemma 3.3 shows that �Z as defined in (44) for the original problem for a given B provides an almost +optimal solution to the reference problem if the original matrix B and the reference matrix R are +close. Lemma 3.3 motivates the consideration of the oracle B as defined in (48) in Section 3 and �Z +as in (44). Lemma 3.3 appears as Lemma 3.3 in [22]. We include the proof in the supplementary +Section C for self-containment. +3.1 +The oracle estimators +The overall goal of convex relaxation is to: (a) estimate the solution of the discrete optimization +problem (4) with an appropriately chosen reference matrix R such that solving the integer quadratic +problem (4) (with R replacing ¯A) will recover the cluster exactly; (b) Moreover, the convex set M+ +G +(resp. Mopt) is chosen such that the semidefinite relaxation of the static problem (4) is tight. This +means that when we replace A (resp. A′) with R = E(Y )E(Y )T in SDP (11) (resp. SDP2 (12)), we +obtain a solution Z∗ = ¯x¯xT , which can then be used to recover the clusters exactly; cf. Lemma 3.6. +Note that unlike the settings of [22], EA ̸= R, resulting in a bias; However, a remedy is to trans- +form (11) into an equivalent Oracle SDP formulation to bridge the gap between Y Y T and the +reference matrix R which we now define: recall Mopt = {Z : Z ⪰ 0, diag(Z) = In} ⊂ M+ +G, +OracleSDP : +maximize +⟨ B, Z ⟩ +subject to +Z ∈ Mopt where +(48) +B +:= +A − EτIn +where +τ = 1 +n +n +� +i=1 +⟨ Yi, Yi ⟩ +(49) +and A is as in (11). +Moreover, on Mopt, the adjustment term EτIn plays no role in optimization, since the extra trace +term ∝ +⟨ In, Z ⟩ += tr(Z) is a constant function of Z across the feasible set Mopt. +However, +the diagonal term EτIn is added in (49) so that the bias ∥EB − R∥ is small. To conclude, the +optimization goal (11) is equivalent to (48) in view of Proposition 3.4; cf (51). In words, optimizing +the original SDP (11) over the larger constraint set M+ +G is equivalent to maximizing ⟨ B, Z ⟩ over +Z ∈ Mopt as shown in (51), where we replace the symmetric matrix A with B. +Proposition 3.4. The optimal solutions �Z as in (11) must have their diagonals set to In. Thus, +the set of optimal solutions �Z in (11) coincide with those on the convex subset Mopt as in (27), +arg max +Z∈M+ +G +⟨ A, Z ⟩ += +arg max +Z∈Mopt +⟨ A, Z ⟩ +(50) += +arg max +Z∈Mopt +( ⟨ A, Z ⟩ − Eτ ⟨ In, Z ⟩ ). +(51) +We prove Proposition 3.4 in the supplementary Section C.2. We emphasize that our algorithm +solves the SDP (11) rather than the oracle SDP (48). However, formulating the oracle SDP (48) +helps us with the global analysis, in controlling ∥EB − R∥, as we now show in Theorem 3.5. +Theorem 3.5. (R is the leading term) Suppose the conditions in Theorem 2.5 hold. Then with +probability at least 1 − 2 exp(−cn), we have +∥B − R∥2 +≤ +ξnpγ and +∥B − R∥∞→1 ≤ ξn2pγ +13 + +Discussions. Notice that B is not attainable, since we do not know Eτ; however, this is irrelevant, +since in the proposed algorithm (11), we are able to readily compute A using the centered data (or +their gram matrix). Theorem 3.5 is useful in proving Theorem 2.5 in view of Lemma 3.3; A proof +sketch for Theorem 3.5 appears in Section 5 and the complete proof appears in the supplementary +Section D. The effectiveness of the SDP procedure (11) crucially depends on controlling the bias +term ∥EB − R∥∞→1 as well as the concentration of measure bounds on ∥B − EB∥∞→1, which in +turn depend on Lemma 5.1, Theorems 5.2 and 5.3 respectively. As we will show in the proof of +Theorem 3.5, the bias term +EB − R += +EY Y T − E(Y )E(Y )T − Eλ(En − In) − EτIn +is substantially smaller than EA − R in the operator and cut norm, under assumption (A2). More- +over, the concentration of measure bounds on +��Y Y T − EY Y T �� imply that, up to a constant factor, +the same bounds also hold for ∥B − EB∥. Controlling both leads to the conclusion in Theorem 3.5. +In Theorem 4.1, we prove convergence results on bounding the angle and ℓ2 distance between the +leading eigenvectors of R and B (resp. Y Y T ) respectively. Indeed, computing the operator and +cut norm for B − R is one of the key technical steps in the current work, unifying Theorems 4.1 +and 2.5. +3.2 +Proof of Theorem 2.5 +Lemma 3.6 shows that the outer product of group membership vector, namely, Z∗ will maximize +⟨ R, Z ⟩ among all Z ∈ [−1, 1]n×n, and naturally among all Z ∈ Mopt. The final result we need is +to verify a non-trivial global curvature of the excess risk ⟨ R, Z∗ − �Z ⟩ for the feasible set Mopt +at the maximizer Z∗, which is given in Lemma 3.7. We then combine Lemmas 3.3 and 3.7, and +Theorem 3.5 to obtain the final error bound for ∥ �Z − Z∗∥ in the ℓ1 or Frobenius norm. Recall +∥A∥1 = � +i,j |aij|. +Lemma 3.6. ( Optimizer of the reference objective function) Let R be as defined in Defi- +nition 2.2. Let Mopt ⊆ M+ +G ⊂ [−1, 1]n×n be as defined in (27). Then +Z∗ = arg max +Z∈Mopt +⟨ R, Z ⟩ = +� +En1 +−En1×n2 +−En2×n1 +En2 +� += ¯x¯xT +(52) +The proof of Lemma 3.7 follows from ideas in Lemma 6.2 [22] and is deferred to the supplementary +Section C.4. As a result, we can apply the Grothendieck’s inequality for the random error B − R +(cf. Lemma 3.3) to obtain an upper bound on ⟨ R, Z∗ − �Z ⟩ uniformly for all �Z ∈ Mopt, where Z∗ +is as defined in (52). Putting things together, we can prove Theorem 2.5. +Lemma 3.7. Let R be as defined in Definition 2.2 and Z∗ be as in (52). For every Z ∈ Mopt, +⟨ R, Z∗ − Z ⟩ ≥ pγw2 +min ∥Z − Z∗∥1 . +(53) +Proof of Theorem 2.5. +We will first conclude from Theorem 3.5 and Lemma 3.3 that the +maximizer of the actual objective function �Z = arg maxZ∈Mopt ⟨ B, Z ⟩ , must be close to Z∗ as in +(52) in terms of the ℓ1 distance. Under the conditions of Lemmas 3.3 and 3.7, +��� �Z − Z∗��� +1 /n2 +≤ +⟨ R, Z∗ − �Z ⟩ +n2pγw2 +min +≤ 2KG ∥B − R∥∞→1 +n2pγw2 +min +≤ 2KGξ +w2 +min +=: δ +14 + +where by Theorem 3.5, ∥B − R∥∞→1 ≤ ξn2pγ. Thus +���Z∗ − �Z +��� +2 +F +≤ +���Z∗ − �Z +��� +max +���Z∗ − �Z +��� +1 ≤ 2δ +where all entries of �Z, Z∗ belong to [−1, 1] and hence +���Z∗ − �Z +��� +max ≤ 2. +□ +4 +Semidefinite programming relaxation for clustering +Denote by X ∈ Rn×p the data matrix with row vectors Xi as in (54). +The k-means criterion +of a partition C = {C1, . . . , Ck} of sample points {1, . . . , n} is based on the total sum-of-squared +Euclidean distances from each point Xi ∈ Rp to its assigned cluster centroid cj, namely, +f(X, C, k) := +k +� +j=1 +� +i∈Cj +∥Xi − cj∥2 +2 +where cj := +1 +|Cj| +� +ℓ∈Cj +Xℓ ∈ Rp +(54) +Getting a global solution to (54) through an integer programming formulation as in [36, 35], is +NP-hard and it is NP-hard for k = 2 [15, 4]. Various semidefinite relaxations of the objective +function have been considered in different contexts. We refer to [44, 35, 6, 25, 29, 31, 39, 16, 20, 17] +and references therein for a more complete picture. Let Ψn denote the linear space of real n by n +symmetric matrices. +Representation of the partition. The work by [44, 36, 35] show that minimizing the k-means +objective f(X, C, k) is equivalent to solving the following maximization problem: +maximize +⟨ �Sn, Z ⟩ +s.t. Z ∈ Pk +(55) +where �Sn = XXT and the constraint set Pk is defined as in (56): +Pk = {B ∈ Ψn : B ≥ 0, B2 = B, B1n = 1n, tr(B) = k} +(56) +where B ≥ 0 means that all elements of B are nonnegative. +Hence matrices in Pk are block +diagonal, symmetric, nonnegative projection matrices with 1n as an eigenvector. The following +matrix set Φn,k is a compact convex subset of Ψn, for any k ∈ [n]: +Φn,k = {Z ∈ Ψn : In ⪰ Z ⪰ 0, tr(Z) = k} +(57) +Variation 1. Peng and Wei [35] first replace the requirement that Z2 = Z, namely, Z is a projection +matrix, with the relaxed condition that all eigenvalues of Z must stay in [0, 1]: In ⪰ Z ⪰ 0. Now +consider the following semidefinite relaxation of (55), +maximize +⟨ �Sn, Z ⟩ s.t. Z ∈ Mk where Mk = {Z ∈ Φn,k : Z ≥ 0, Z1n = 1n} +(58) +The key differences between this and the SDP (11) are: (a) In the convex set Mopt (27), we do not +enforce that all entries are nonnegative, namely, Zij ≥ 0, ∀i, j; This allows faster computation; (b) +In order to derive concentration of measure bounds that are sufficiently tight, we make a natural, +yet important data processing step in the current work, where we center the data according to their +15 + +column means following Definition 2.1 before computing A as in (9); (c) Given this centering step, +we do not need to enforce Z1n = 1n. See Variation 2 for details. +Variation 2. To speed up computation, one can drop the nonnegative constraint on elements of +Z in (58) [44, 35]. The following semidefinite relaxation is also considered in [35]: +maximize +⟨ �Sn, Z ⟩ +s.t. Z1n = 1n, Z ∈ Φn,k +for Φn,k as in (57). +(59) +Moreover, Peng and Wei [35] show that the set of feasible solutions to (59) have immediate connec- +tions to the SVD of Y Y T , via the following reduction step, closely related to our proposal. When +Z is a feasible solution to (59), 1n/√n is the unit-norm leading eigenvector of Z and one can define +Z1 +:= +Z − 1 +n1n1T +n +and hence +Z1 := (I − P1)Z = (I − P1)Z(I − P1). +(60) +Then tr(Z1) = tr(Z) − 1 = k − 1 and Z1 ∈ Φn,k−1. Hence (59) is reduced to +maximize +⟨ Y Y T , Z1 ⟩ +s.t. In ⪰ Z1 ⪰ 0, tr(Z1) = k − 1 +(61) +since Y Y T = (I − P1)�Sn(I − P1). Let λ1 ≥ . . . ≥ λn−1 be the largest (n − 1) eigenvalues of Y Y T +in descending order. The optimal solution to (61) can be achieved if and only if ⟨ Y Y T , Z1 ⟩ = +�k−1 +i=1 λi; see for example [33]. Then the algorithm for solving (61) and correspondingly (59) is +given as follows [35]: +(a) Use singular value decomposition method to compute the first k −1 largest eigenvalues of Y Y T , +and their corresponding eigenvectors v1, . . . , vk−1; (b) Set +Z1 = +k−1 +� +j=1 +vjvT +j ; and return Z = 1 +n1n1T +n + Z1 as a solution to (59). +Now for k = 2, we have Z1 = v1vT +1 . In Theorem 4.1, we show convergence for the angle as well as the +ℓ2 distance between the two vectors v1 and ¯v1, where v1 and ¯v1 are the leading eigenvectors of Y Y T +and the reference matrix R respectively. Theorem 4.1 demonstrates another excellent application +of our estimation procedure and concentration of measure bounds, namely, Theorem 3.5. +Theorem 4.1. (SVD: imbalanced case) Denote by v1 the leading unit-norm eigenvector of Y Y T , +which also coincides with that of A (9) and B (49). Let ¯v1 be the leading unit-norm eigenvector of +R as in (21): +¯v1 = [w21n1, −w11n2]/√w2w1n = [ +� +w2/w11n1, − +� +w1/w21n2]/√n, +(62) +where ⟨ ¯v1, 1n ⟩ = 0. Then under the conditions in Theorem 3.5, we have with probability at least +1 − 2 exp(−cn), for some absolute constants c, c0, c1, c2, +sin(θ1) +:= +sin(∠(v1, ¯v1)) ≤ 2 ∥B − R∥2 +w1w2npγ +≤ +2ξ +w1w2 +(63) +min +α=±1 ∥αv1 − ¯v1∥2 +2 +≤ +δ′, where δ′ = 8ξ2/(w2 +1w2 +2) ≤ c2ξ2/w2 +min; +(64) +where θ1 = ∠(v1, ¯v1) denotes the angle between the two vectors v1 and ¯v1. +16 + +Corollary 4.2. (Clustering with o(n/s2) misclassified vertices) Suppose that w1, w2 ∈ (0, 1) are +bounded away from 0, 1. Under the conditions in Theorem 4.1, we have with probability at least +1 − 2 exp(−cn), for some absolute constants c, the signs of the coefficients of v1 correctly estimate +the partition of the vertices into two clusters, up to at most O(ξ2n) misclassified vertices. +Discussions. We prove Theorem 4.1 and its corollary in the supplementary Section E. The signs +of the coefficients of v1 correctly estimate the partition of the vertices, up to at most δ′n ≍ ξ2n +misclassified vertices, where recall ξ2 ≍ 1/s2 (28). Hence the misclassification error is bounded to +be inversely proportional to the SNR parameter s2; cf. (28). This should be compared with (29), +where we show in Theorem 2.5 that we have up to at most δn ≍ ξn misclassified vertices, which +is improved to O(n exp(−c0s2w4 +min)) in Theorem 2.7. Moreover, one can sort the values of v1 and +find the nearly optimal partition according to the k-means criterion; See Section 8 for Algorithm +2 and numerical examples. +Variation 3. The main issue with the k-means relaxation is that the solutions tend to put sample +points into groups of the same sizes, and moreover, the diagonal matrix Γ can cause a bias, where +Γ = (E[ ⟨ Zi, Zj ⟩ ])i,j = diag([tr(Cov(Z1)), . . . , tr(Cov(Zn))]), +especially when V1, V2 differ from each other; See the supplementary Section H for bias analysis. +In [39, 20, 10], they propose a preliminary estimator of Γ, denoted by �Γ, and consider +�Z ∈ arg max +Z∈Mk ⟨ XXT − �Γ, Z ⟩ +where Mk is as in (58) +(65) +instead of the original Peng-Wei SDP relaxation (58). Although our general results in Theorem 2.7 +coincide with that of [20] for k = 2, we emphasize that we prove these bounds for the SDP (11), +which is motivated by the graph partition problem (5), while they establish such bounds for the +semidefinite relaxation based on the k-means criterion (54) directly, following [35]. There, cf. (58), +and (65), the matrix Z is not only constrained to be positive semidefinite but also with non-negative +entries. As mentioned, the advantage of dropping the nonnegative constraints on elements of Z in +(11) is to speed up the computation. +Hence another main advantage of our SDP and spectral formulation is that we do not need to have +a separate estimator for tr(Σj), where Σj, j = 1, 2 denote the covariance matrices of sub-gaussian +random vectors Zj, j ∈ [n], so long as (A2) holds. When it does not, one may consider adopting +similar ideas. We emphasize that part of our probabilistic bounds, namely, Theorems 6.3 and 7.2, +already work for the general k-means clustering problem. +5 +Outline of the arguments for proving Theorem 3.5 +We emphasize that results in this section apply to both settings under consideration: design matrix +with independent entries or with independent anisotropic sub-gaussian rows. This allows us to +prove Theorem 3.5 for both cases. Let Y be as in Definition 2.1. By definition of (9) and (49), +A − EA = B − EB +:= +Y Y T − EY Y T − (λ − Eλ)(En − In) +(66) +hence ∥B − R∥∞→1 += +∥B − EB + EB − R∥∞→1 +≤ +∥B − EB∥∞→1 + ∥EB − R∥∞→1 +(67) +17 + +We have by the triangle inequality, (66), (67) and the supplementary Lemma D.1, for +∥B − R∥2 +≤ +2 ∥Ψ∥2 + ∥EB − R∥2 +where Ψ := Y Y T − E(Y Y T ), +(68) +and +∥B − R∥∞→1 +≤ +∥Ψ∥∞→1 + n ∥Ψ∥2 + ∥EB − R∥∞→1 +Lemma 5.1 states that the bias EB − R is substantially reduced for B as in (49), thanks to the +adjustment term EτIn, and even more so when clusters have similar variance profiles in the sense +that (24) is bounded. Theorem 3.5 follows immediately from Lemma 5.1 and Theorem 5.2 (resp. +5.3), where we bound +��Y Y T − E(Y Y T ) +�� for design matrix with independent entries (resp. with +independent anisotropic sub-gaussian rows). All results except for Theorems 5.2 and 5.3 are stated +as deterministic bounds. Let c7, c8, C2, C3, . . . be some absolute constants. All constants such as +1/6, 2/3, . . . are arbitrarily chosen. +Lemma 5.1. Suppose (A2) holds. Suppose that ξ ≥ +1 +2n(4 ∨ +1 +wmin ) and n ≥ 4. Then we have +∥EB − R∥2 +≤ +2 +3ξnpγ and +∥EB − R∥∞→1 ≤ 2 +3ξn2pγ +Finally, when V1 = V2, we have ∥EB − R∥∞→1 ≤ n ∥EB − R∥2 ≤ pnγ/3. +Theorem 5.2. (Design with independent entries) In the initial settings as specified in The- +orem 2.5, suppose that (A1), (A2) and (25) hold with maxj,k ∥zjk∥ψ2 := ∥Xjk − EXjk∥ψ2 ≤ C0. +Then, with probability at least 1 − 2 exp(−c7n), +��Y Y T − E(Y Y T ) +�� +2 +≤ +C2C2 +0(√pn ∨ n) + C3C0n√pγ ≤ 1 +6ξnpγ +Theorem 5.3. (Anisotropic design matrix.) Let Y be as in Definition 2.1. Suppose all con- +ditions in Theorem 2.5 and Lemma 2.4 hold. Suppose (18) holds. Then with probability at least +1 − 2 exp(−c8n), +��Y Y T − E(Y Y T ) +�� +2 +≤ +1 +12ξnpγ + C4(C0 max +i +∥Hi∥2)2(√pn ∨ n) ≤ 1 +6ξnpγ. +where C0 is the same as in (18) and (33). +We prove Lemma 5.1 in the supplementary Section H.2, where balanced cases are shown to be +slightly more tightly bounded; cf Lemma H.7 therein. +We prove Theorems 5.2 and 5.3 in the +supplementary Section F.1 and Section 7 respectively. It is understood that for both theorems, +we also obtain +��Y Y T − E(Y Y T ) +�� +∞→1 to be within a factor of O(n)∥Y Y T − E(Y Y T )∥2. We prove +Theorem 3.5 in the supplementary Section D. +5.1 +Reduction +In this section, we present a unified framework for bounding +��Y Y T − EY Y T �� +2. First, +Y Y T − E(Y Y T ) = Y Y T − E(Y )E(Y )T + E(Y )E(Y )T − E(Y Y T ) += +E(Y )(Y − E(Y ))T + (Y − E(Y ))(E(Y ))T + �ΣY − ΣY +(69) +18 + +where �ΣY = (Y − E(Y ))(Y − E(Y ))T and +�ΣY − ΣY += +(Y − E(Y ))(Y − E(Y ))T + E(Y )E(Y )T − E(Y Y T ), +from which we obtain from the well known relationship on covariance matrix +ΣY := E +� +(Y − E(Y ))(Y − E(Y ))T � += E(Y Y T ) − E(Y )E(Y )T . +We now state in Lemma 5.4 a reduction principle for bounding the first component in (69): To +control +∥MY ∥ = +��E(Y )(Y − E(Y ))T + (Y − E(Y ))(E(Y ))T �� , +we need to bound the projection of each mean-zero random vector Zj, ∀j ∈ [n], along the direction +of v := µ(1) −µ(2). In other words, a particular direction for which we compute the one-dimensional +marginals, is the direction between µ(1) and µ(2). +Lemma 5.4. (Reduction: a deterministic comparison lemma) Let Zj, j ∈ [n] be row vectors +of X − EX and �µn be as defined in (8). For xi ∈ {−1, 1}, +n +� +i=1 +xi ⟨ Yi − EYi, µ(1) − µ(2) ⟩ +≤ +2(n − 1) +n +n +� +i=1 +��� ⟨ Zi, µ(1) − µ(2) ⟩ +��� +(70) +Then we have for MY := E(Y )(Y − E(Y ))T + (Y − E(Y ))(E(Y ))T , +∥MY ∥∞→1 +≤ +8w1w2(n − 1) +n +� +i=1 +��� ⟨ Zi, µ(1) − µ(2) ⟩ +��� +and +∥MY ∥2 +≤ +4√n√w1w2 +sup +q∈Sn−1 +����� +� +i +qi ⟨ Zi, µ(1) − µ(2) ⟩ +����� +Upon obtaining (20), Lemma 5.4 is deterministic and does not depend on covariance structure of Z. +On the other hand, controlling the second component in (69) amounts to the problem of covariance +estimation given the mean matrix E(Y ); Lemma 5.5 is again deterministic, where we show that +controlling the operator (and cut) norm of �ΣY −ΣY is reduced to controlling that for ZZT −E(ZZT ). +We prove Lemmas 5.4 and 5.5 in the supplementary Sections F.2 and F.3 respectively. +Lemma 5.5. Suppose that Y and Z are matrices as defined in Definition 2.1. The following holds: +�ΣY − ΣY = (I − P1)(ZZT − E(ZZT ))(I − P1) +cf. Proposition F.1 in the supplementary material. Then +����ΣY − ΣY +��� +2 ≤ +��ZZT − E(ZZT ) +�� +2. +6 +Proof outline of Theorem 5.2 +We provide a proof outline for Theorem 5.2 in this section. We will bound these two components (69) +in Lemma 6.1 and Theorem 6.3 respectively. Lemma 6.1 follows from Lemma 5.4 and the sub- +gaussian concentration of measure bounds in Lemma 6.2. We will only state the operator norm +bound in Theorem 6.3, with the understanding that cut norm of a matrix is within O(n) factor +of the operator norm on the same matrix. We defer the proof of Theorems 5.2 and 6.3 to the +supplementary Sections F.1 and G.3 respectively. The proof for Lemmas 6.1 and 6.2 appear in the +supplementary Section G. Let c, c′, c1, c5, C3, C4, . . . be absolute constants. +19 + +Lemma 6.1. (Projection: probabilistic view) Suppose conditions in Theorem 5.2 hold. Then +we have with probability at least 1 − 2 exp(−cn), +��E(Y )(Y − E(Y ))T + (Y − E(Y ))(E(Y ))T �� +2 +≤ +2C3C0n√pγ and +��E(Y )(Y − E(Y ))T + (Y − E(Y ))(E(Y ))T �� +∞→1 +≤ +C4C0n(n − 1)√pγ +Lemma 6.2 follows from the sub-gaussian tail bound, since the one-dimensional marginals of Zj, ∀j ∈ +[n] are sub-gaussian with bounded ψ2 norms. Denote by +µ := +µ(1) − µ(2) +��µ(1) − µ(2)�� +2 += µ(1) − µ(2) +√pγ +∈ Sp−1 +(71) +Lemma 6.2. (Projection for sub-gaussian random vectors) In the settings of Theorem 5.2, +suppose (A1) holds and C0 = maxi,j ∥zij∥ψ2. Then for any t > 0, and any u = (u1, . . . , un) ∈ +{−1, 1}n +P +� n +� +i=1 +ui ⟨ Zi, µ ⟩ ≥ t +� +≤ +2 exp +� +−ct2/(C2 +0n) +� +; +(72) +and for any q ∈ Sn−1, P +� n +� +i=1 +⟨ qiZi, µ ⟩ ≥ t +� +≤ +2 exp +� +−c′t2/C2 +0 +� +(73) +Theorem 6.3. In the settings of Theorem 5.2, we have with probability at least 1 − 2 exp(−c6n), +����ΣY − E�ΣY +��� +2 += +��E(ZZT ) − E(ZZT ) +�� +2 ≤ C2C2 +0(√pn + n) ≤ 1 +12ξnpγ. +7 +Proof outline for Theorem 5.3 +We provide an outline for Theorem 5.3 in this section. First, we state Lemma 7.1, where we extend +Lemma 6.1 to the anisotropic cases. The anisotropic version of Lemma 6.2 is presented in the +supplementary Lemma I.1. The model under consideration in Theorem 7.2 is understood to be a +special case of Theorem 7.3. Theorem 5.3 follows from Theorem 7.2 and Lemma 7.1 immediately, +and the probability statements hold upon adjusting the constants. +We defer all proofs to the +supplementary Section I. Let c, c′, C2, C3, C4, . . . be absolute constants. +Lemma 7.1. (Projection: probabilistic view) Let µ be as in (71) and MY be as in Lemma 5.4. +Suppose all conditions in Theorem 5.3 hold. Then with probability at least 1−2 exp(−c′n), we have +∥MY ∥∞→1 +≤ +C4(C0 max +i +∥Riµ∥2)n(n − 1)√pγ ≤ 1 +12ξn(n − 1)pγ +∥MY ∥2 +≤ +2C3(C0 max +i +∥Riµ∥2)n√pγ ≤ 1 +12ξnpγ +Theorem 7.2. In the settings of Theorem 5.3, we have with probability at least 1 − 2 exp(−c6n), +����ΣY − E�ΣY +��� +2 +≤ +��ZZT − EZZT �� +2 ≤ C2(C0 max +i +∥Hi∥2)2(√np ∨ n). +(74) +20 + +Theorem 7.3. (Hanson-Wright inequality for anisotropic sub-gaussian vectors.) +Let +H1, . . . , Hn be deterministic p × m matrices, where we assume that m ≥ p. Let ZT +1 , . . . , ZT +n ∈ Rp be +row vectors of Z. We generate Z according to Definition 2.3. +Then we have for t > 0, for any A = (aij) ∈ Rn×n, +P +� +� +������ +n +� +i=1 +n +� +j̸=i +⟨ Zi, Zj ⟩ aij +������ +> t +� +� +≤ +2 exp +� +−c min +� +t2 +(C0 maxi ∥Hi∥2)4p ∥A∥2 +F +, +t +(C0 maxi ∥Hi∥2)2 ∥A∥2 +�� +(75) +where maxi ∥Zi∥ψ2 ≤ CC0 maxi ∥Hi∥2 in the sense of (33). +Remarks on covariance estimation. Essentially, (74) matches the optimal bounds on covariance +estimation, where the mean-zero random matrix consists of independent columns Zj, j = 1, . . . , p +that are isotropic, sub-gaussian random vectors in Rn, or columns which can be transformed +to be isotropic through a common covariance matrix. +See, for example, Theorems 4.6.1 and +4.7.1 [42]. +The difference between (74) and such known results are: +(a) we do not assume +that columns are independent; (b) we do not require anisotropic row vectors to share identi- +cal covariance matrices. +More generally, we allow the (sample by sample) covariance matrix +ΣX := EZZT = diag([tr(H1HT +1 ), . . . , tr(HnHT +n )]), through Definition 2.3; and hence we are es- +timating a diagonal matrix with p dependent features, where we assume that E(X) is given. We +state the operator norm bound in Theorem 7.2, where it is understood that (74) holds under the +general covariance model as considered in Definition 2.3 and Theorem 7.3. We prove Theorem 7.3 +in the supplementary Section I.4. The proof might be of independent interests. Such generalization +is useful since we may consider the more general k-component mixture problems, as elaborated in +Section 4. See also Exercise 6.2.7 [42] for a related result. +8 +Experiments +In this section, we use simulation to illustrate the effectiveness and convergence properties of the +two estimators. We use a similar setup as the one used in [9]. We generate data that is a mixture +of two populations. Data matrix X ∈ {0, 1}n×p consists of independent Bernoulli random variables, +where the mean parameters E(Xij) := qj +ψ(i) for all i ∈ [n] and j ∈ [p], where ψ(i) ∈ {1, 2} assigns +nodes i to a group C1 or C2 for each i ∈ [n]. Let |C1| = w1n and |C2| = w2n. We conduct experiments +for both balanced (w1 = w2) and imbalanced cases. The entrywise expected values are chosen as +follows: for half of the p features, the mean parameters qj +1 > qj +2, and for the other half, qj +1 < qj +2 +such that ∀j, qj +1, qj +2 ∈ {1+α +2 ++ ϵ +2, 1−α +2 ++ ϵ +2}. We set ϵ = 0.1α and α = 0.04. Hence γ = α2 = 0.0016, +1 +γ2 = 390, 625, and 1 +γ = 625. We implement Algorithm 1: the SDP as described in (11), and +classify according to signs of �x as prescribed by Corollary 2.6; and Algorithm 2: the Peng-Wei +spectral method following [35]. +21 + +Algorithm 2: Spectral method for k-means clustering (Peng-Wei) [35]: +Input: Centered data matrix Y ∈ Rn×p, k = 2 +Output: A group assignment vector P +Step 1. Use SVD to obtain the leading eigenvector v1 of Y Y T and let v := v1; +Step 2. Let S be the vector of sorted values of v in descending order. For each +index j in [n], compute the two means c1, c2, one for each of the two groups, +namely, C1 = SL := {S1, . . . , Sj} and C2 = SR := {Sj+1, . . . , Sn} to the left +(inclusive) and the right of this index; +Step 3. Compute the total sum-of-squared Euclidean distances from each point +within a particular group to the respective mean, according to (54); Let t be the +index that gives the minimum total distance, and its corresponding value be St; +Step 4. Set Pi = 1 if vi ≥ St, and Pi = −1 if vi < St. +Success rate and misclassification rate. For each experiment, we run 100 trials; and for each +trial, we first generate a data matrix Xn×p according to the mixture of two Bernoulli distributions +with parameters described above, and then feed Y (7) to the two estimators for classification. We +measure success rate and misclassification rate based on P, the output assignment vector. Success +rate is computed as the number of correctly classified individuals divided by the sample size n. +Hence misclassification rate is 1− success rate. Each data point corresponds to the average of 100 +trials. Fig. 1 shows the average success rates (over 100 trials) as n increases for different values of +p for the balanced case. +We observe that SDP has higher average success rate for each setting of (n, p) when npγ2 > 1.5, +despite the exhaustive search in Algorithm 2; For npγ2 < 1.5, the rates are closer. We also see from +the plot that when p < 1/γ = 625, for example, when p = 500, the success rate remains flat across +n. Note that a success rate of 50% is equivalent to a total failure. In contrast, when n is smaller +than 1/γ, as we increase p, we can always classify with a high success rate. In general, npγ2 > 1 +is indeed necessary to obtain a success rate larger than 60%, when p ≥ 1/γ. When n < 625, npγ2 +plays the role of the SNR, since npγ2 < pγ; This remains the case throughout our experiments. +Angle and ℓ2 convergence. Here we take a closer look at the trends of �x and �Z, the solution +to SDP (11) as n increases, and of v1, the leading eigenvector of Y Y T . In the second experiment, +we set p ∈ {20000, 50000, 80000}, and increase n. In the left column of Fig. 2, which is for the +imbalanced case of w1 = 0.7, we plot θSDP := ∠(�x, ¯x) between �x and its reference vector ¯x as +defined in Theorem 2.5 and Corollary 2.6. +For Algorithm 2, θ1 := ∠(v1, ¯v1) between v1 and its reference ¯v1, where ¯v1 is as defined in Theo- +rem 4.1. In this case, the angle ∠(¯v1, ¯x) between the two reference vectors is about 22 degrees (blue +horizontal dashed line). We observe that as n increases, for both algorithms, the angles θSDP and +θ1 decrease, but θSDP drops much faster and decreases to 0 when n > 200 for p = 80, 000. We also +show the angle φ = ∠(�x, v1) between the two leading eigenvectors �x and v1, which largely remains +flat across all n. +In the right column of Fig. 2, we plot sin(θ1) for Algorithm 2, and for SDP, we plot sin(θSDP), ∥Z∗− +�Z∥2/n, and ∥Z∗ − �Z∥F /n, where Z∗ = ¯x¯xT . We see that for Algorithm 1, all three metrics decrease +as n increases, following an exponential decay in n as predicted by our theory in Theorem 2.7 and +Corollary 2.8, where in each plot, p, γ are being fixed. The gaps between the three curves for SDP +22 + +0 +200 +400 +600 +800 +50 +60 +70 +80 +90 +100 +w1 = 0.5 +n +Succ Rate % +p= 500 +p= 625 +p= 750 +p= 1000 +p= 1200 +p= 1500 +p= 2000 +p= 2800 +p= 3500 +p= 5000 +p= 10000 +p= 20000 +p= 80000 +Figure 1: Balanced case w1 = 0.5. We plot the success rate for various values of dimension p +ranging from 500 to 80000, as n increases. Here γ = 0.0016 and 1/γ = 625. For two lines with the +same marker, the black solid line is for SDP solution �Z (and partition based on �x), and the blue +dashed line is for Algorithm 2. Red lines (with no markers) highlight the success rates at different +levels of npγ2 ranging from 0.5 to 3.5, from bottom to top with a step of 0.5. The solid red line +is for npγ2 = 1. In general, npγ2 > 1 is necessary to see a success rate larger than 60%, when +p ≥ 625 = 1/γ. +shrink when p, n increase. For Algorithm 2, sin(θ1) also decreases as n increases, but at a slower +rate of 1/n, again as predicted by Theorem 4.1 and Corollary 4.2. +9 +Conclusion +Exploring the tradeoffs of n and p that are sufficient for classification, when sample size n is small, +is both of theoretical interests and practical value. A recent line of work establishes approximate +recovery guarantees of the SDPs in the low-SNR regime for sub-gaussian mixture models; see +[16, 20] among others. The present work aims to further illuminate the geometric and probabilistic +features for this problem, while allowing cluster sizes and variance profiles to vary across the +two populations. Although we use the population clustering problem as a motivating example, +our concentration of measure analyses in Section 7, upon adaptation, will work for the general +settings (6) as well. In particular, we study SDP relaxation as well as a simple spectral algorithm, +23 + +100 +200 +300 +400 +500 +600 +0 +20 +40 +60 +80 +p= 20000 w1= 0.7 +n +Angle +θSDP +θ1 +φ +0 +100 +200 +300 +400 +500 +600 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p= 20000 w1= 0.7 +n +value +sin(θSDP) +sin(θ1) +SDP Frob/n +SDP Op/n +0 +100 +200 +300 +400 +500 +600 +0 +20 +40 +60 +80 +p= 50000 w1= 0.7 +n +Angle +θSDP +θ1 +φ +0 +100 +200 +300 +400 +500 +600 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p= 50000 w1= 0.7 +n +value +sin(θSDP) +sin(θ1) +SDP Frob/n +SDP Op/n +0 +100 +200 +300 +400 +500 +600 +0 +20 +40 +60 +80 +p= 80000 w1= 0.7 +n +Angle +θSDP +θ1 +φ +0 +100 +200 +300 +400 +500 +600 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p= 80000 w1= 0.7 +n +value +sin(θSDP) +sin(θ1) +SDP Frob/n +SDP Op/n +Figure 2: Imbalanced case w1 = 0.7, p ∈ {20000, 50000, 80000}. Left column shows the angle θSDP +(resp. θ1) between the leading eigenvector �x of SDP solution �Z (resp. v1 of Y Y T ) and ¯x (resp. +¯v1). As n increases, θSDP decreases faster than θ1, especially for larger values of p. Horizontal +dashed (straight) line is the static angle ∠(¯v1, ¯x); The dashed curve around it is for the random +φ = ∠(�x, v1). Each vertical bar shows one standard deviation over 100 trials. Right column plots +sin(θSDP), ∥Z∗ − �Z∥F /n, ∥Z∗ − �Z∥2/n for SDP, and sin(θ1) for Algorithm 2. +24 + +which are efficiently solvable in both theoretical and practical senses, and provide a unified analysis +of the two most commonly studied procedures in the literature. By doing so, we gained new insight +that the leading eigenvectors not only contain sufficient information for clustering but it is also +feasible to use algorithmic techniques to identify group memberships effectively once the SNR is +bounded below by a constant. +Acknowledgement +I would like to thank Alan Frieze for reading a crude draft of this manuscript, and Mark Rudelson +for many helpful discussions. I thank my family for their support, especially during the pandemic. +A +Organization +We prove Corollaries 2.6 and 2.8 in Section B. Proofs for lemmas appearing in Section 3 appear in +Section C. We prove Theorem 3.5 in Section D. Proof of Theorem 4.1 appears in Section E. Proofs +of Theorem 5.2 appears in Section F.1. Section G contains the concentration of measure analysis +with regards to the random matrix Y Y T − E(Y Y T ), leading to Theorem 5.2. In Section H, we +prove the corresponding result for Lemma 5.1. Section I contains the concentration of measure +analysis for anisotropic random vectors, leading to the conclusion of Theorem 5.3. In Section I.4, +we prove Theorem 7.3, the Hanson-Wright inequality for anisotropic sub-gaussian vectors, which +may be of independent interests. +B +Proof of Corollaries 2.6 and 2.8 +Theorem B.1 is a well-known result in perturbation theory. See [9] for a proof. See also Theorem +4.5.5 [42] and Corollary 3 in [43]. +Theorem B.1. (Davis-Kahan) For A and M being two symmetric matrices and E = M −A. Let +λ1(A) ≥ λ2(A) ≥ . . . ≥ λn(A) be eigenvalues of A, with orthonormal eigenvectors v1, v2, . . . , vn and +let λ1(M) ≥ λ2(M) ≥ . . . ≥ λn(M) be eigenvalues of M and w1, w2, . . . , wn be the corresponding +orthonormal eigenvectors of M, with θi = ∠(vi, wi). Then +θi ∼ sin(θi) ≤ +2 ∥E∥2 +gap(i, A) where gap(i, A) = min +j̸=i |λi(A) − λj(A)| . +(76) +Proof of Corollary 2.6. +See proof of Corollary 1.2 [22] for the first result, which follows from +Davis-Kahan Theorem and is a direct consequence of the error bound (26), while noting that the +largest eigenvalue of ¯x¯xT is n while all others are 0, and hence the spectral gap in the sense of +Theorem B.1 equals n; In more details, we have by Theorem 2.5 and Corollary 3 [43], +min +α=±1 +��(α�x − ¯x)/√n +��2 +2 +≤ +23 ��� �Z − ¯x¯xT ��� +2 +2 +gap(1, ¯x¯xT )2 +≤ 23 ��� �Z − ¯x¯xT ��� +2 +F /n2 +≤ +234KGξ/w2 +min +□ +25 + +Proof of Corollary 2.8. +The angle between �x/√n and ¯x/√n can be expressed as +cos(θSDP) = cos(∠(�x, ¯x)) += +⟨ �x, ¯x ⟩ /n +(77) +The upper bound on sin(θSDP) follows from Theorems 2.7 and B.1; Moreover, by Davis-Kahan +Theorem, cf. Corollary 3 [43], we have with probability at least 1 − 2 exp(−cn) − 2/n2, +min +α=±1 +��(α�x − ¯x)/√n +�� +2 +≤ +23/2 ��� �Z − ¯x¯xT ��� +2 +gap(1, ¯x¯xT ) +≤ 23/2 ��� �Z − ¯x¯xT ��� +F /n +≤ +23/2(2 +��� �Z − ¯x¯xT ��� +1)1/2/n ≤ 4 exp(−c0s2w4 +min/2) +where the last inequality holds upon adjusting that constants. The corollary thus holds. +□ +C +Proofs for results in Section 3 +Combining (43), (41) and (42), we have the following Fact C.1. +Fact C.1. (Grothendieck’s inequality, PSD) Every matrix B ∈ Rn×n satisfies +max +Z∈M+ +G +| ⟨ B, Z ⟩ | ≤ KG ∥B∥∞→1 . +C.1 +Proof of Lemma 3.3 +The upper bound in (45) is trivial by definition of Z∗, �Z ∈ Mopt and uses the fact that Z∗ := +arg maxZ∈Mopt ⟨ R, Z ⟩ ; The lower bound depends on Fact (C.1), which implies that +∀Z ∈ Mopt, +| ⟨ B − R, Z ⟩ | ≤ KG ∥B − R∥∞→1 =: ε +(78) +Now to prove the lower bound in(45), we will first replace R by B using (78), +⟨ R, �Z ⟩ +≥ +⟨ B, �Z ⟩ − ε +≥ +⟨ B, Z∗ ⟩ − ε ≥ ⟨ R, Z∗ ⟩ − 2ε +where the second inequality uses the fact that �Z := arg maxZ∈Mopt ⟨ B, Z ⟩ , and �Z, Z∗ ∈ Mopt by +definition (44), while the last inequality holds by (78), since Z∗ ∈ Mopt and hence +| ⟨ B − R, Z∗ ⟩ | +≤ +KG ∥B − R∥∞→1 +(79) +Hence (46) holds. Finally, we prove (47); By (78), (79), and the triangle inequality, we have for all +Z ∈ Mopt, +| ⟨ B − R, Z − Z∗ ⟩ | +≤ +2KG ∥B − R∥∞→1 +from which (47) follows. +□ +26 + +C.2 +Proof of Proposition 3.4 +Recall +Mopt +:= +{Z : Z ⪰ 0, diag(Z) = In} ⊂ M+ +G ⊂ [−1, 1]n×n; +Notice that for the second term in (9), we have ⟨ (En − In), Z ⟩ = ⟨ (En − In), offd(Z) ⟩ in the +objective function (11), which does not depend on diag(Z); Hence, to maximize +⟨ A, Z ⟩ += +⟨ A, offd(Z) ⟩ + ⟨ A, diag(Z) ⟩ += +⟨ offd(A), offd(Z) ⟩ + ⟨ diag(Y Y T ), diag(Z) ⟩ , +one must set the diagonal Zjj ∈ [0, 1] to be 1, since diag(Y Y T ) ≥ 0. Moreover, increasing diag(Z) +will only make it easier to satisfy Z ⪰ 0 and hence to maximize ⟨ offd(A), offd(Z) ⟩ . Thus, the +set of optimizers �Z as in (12) must satisfy diag( �Z) = In. Thus (50) holds by definition of Mopt as +above. Moreover, (51) holds due to the fact that ⟨ In, Z ⟩ = tr(Z) = n for all Z in the feasible set +Mopt. +□ +C.3 +Proof of Lemma 3.6 +One can check that the maximizer of ⟨ R, Z ⟩ on the larger set [−1, 1]n×n, which contains the +feasible set Mopt, is Z∗. Clearly diag(Z∗) = In. Since Z∗ = ¯x¯xT ⪰ 0 belongs to the smaller set +Mopt ⊂ M+ +G , it must be the maximizer of ⟨ R, Z ⟩ on that set as well. +□ +C.4 +Proof of Lemma 3.7 +We will prove that (53) holds for all Z ∈ [−1, 1]n ⊃ Mopt. Recall we have +R = E(Y )E(Y )T +=: +���µ(1) − µ(2)��� +2 +2 +� +w2 +21n1 ⊗ 1n1 +−w1w21n1 ⊗ 1n2 +−w1w21n2 ⊗ 1N1 +w2 +11n2 ⊗ 1n2 +� += +pγ +� +w2 +2En1 +−w1w2En1×n2 +−w1w2En2×n1 +w2 +1En2 +� +=: pγ +� A +B +BT +C +� +Now all entries of Z∗, Z belong to [−1, 1]. Clearly, for the upper left and lower right diagonal +blocks, denoted by D = {A, C}, we have Z∗ − Z ≥ 0, since all entries of Z∗ on these blocks are 1s. +Similarly, for the off-diagonal blocks {B, BT }, we have Z − Z∗ ≥ 0 since all entries of Z∗ on these +blocks are −1s. Thus we have +1 +pγ ⟨ R, Z∗ − Z ⟩ += +� +(i,j)∈A +w2 +2(Z∗ − Z)ij + +� +(i,j)∈C +w2 +1(Z∗ − Z)ij − +� +(i,j)∈B,BT +w1w2(Z∗ − Z)ij +≥ +(w2 +2 ∧ w2 +1 ∧ w1w2) +� � +(i,j)∈D +(Z∗ − Z) + +� +(i,j)∈{B,BT } +(Z − Z∗)ij +� +≥ +min +j=1,2 w2 +j ∥Z − Z∗∥1 +where we use the fact that +� +(i,j)∈D +(Z∗ − Z)ij + +� +(i,j)∈{B,BT } +(Z − Z∗)ij += +∥Z − Z∗∥1 +The lemma is thus proved. +□ +27 + +D +Proof of Theorem 3.5 +We first state Lemma D.1. +Lemma D.1. (Deterministic bounds) Let λ and τ be as defined in (10) using matrix Y as specified +in Definition 2.1. By definition of τ and λ, we have +(n − 1) |λ − Eλ| += +|τ − Eτ| ≤ +��Y Y T − E(Y Y T ) +�� +2 +(80) +Proof. Now (80) holds since 2 +�n +2 +� +|λ − Eλ| = |n(τ − Eτ)| ≤ n +��Y Y T − E(Y Y T ) +�� +2 where +����� +n +� +i=1 +( ⟨ Yi, Yi ⟩ − E ⟨ Yi, Yi ⟩ ) +����� =: n |(τ − Eτ)| +≤ +n max +i +| ⟨ Yi, Yi ⟩ − E ⟨ Yi, Yi ⟩ | ≤ n +��Y Y T − E(Y Y T ) +�� +2 +□ +Proof of Theorem 3.5. +We have by Theorem 5.2 (resp. Theorem 5.3), with probability at least +1 − 2 exp(−cn), +∥B − EB∥∞→1 +≤ +��Y Y T − E(Y Y T ) +�� +∞→1 + |λ − Eλ| ∥En − In∥∞→1 +≤ +��Y Y T − E(Y Y T ) +�� +∞→1 + n +��Y Y T − E(Y Y T ) +�� +2 ≤ 1 +3ξn2pγ +where we use the fact that ∥En − In∥∞→1 = n(n − 1) and by Lemma D.1, +(n − 1) |λ − Eλ| = |τ − Eτ| = 1 +n +��tr(Y Y T − E(Y Y T )) +�� ≤ +��Y Y T − E(Y Y T ) +�� +2 +Now ∥En − In∥2 ≤ ∥En − In∥∞ = (n − 1); and thus similarly, +∥B − EB∥2 +≤ +��Y Y T − E(Y Y T ) +�� +2 + |λ − Eλ| ∥En − In∥2 +≤ +2 +��Y Y T − E(Y Y T ) +�� +2 ≤ 1 +3ξnpγ +Theorem 3.5 then holds by the triangle inequality; cf. (67) and (68), in view of Lemma 5.1. See +also Lemma H.7. +□ +E +Proof of Theorem 4.1 +It is known that for any real symmetric matrix, there exist a set of n orthonormal eigenvectors. +First we state Fact E.1. Fact E.1 is also not surprising, since the sum of all off-diagonal entries of +A is 0. +Fact E.1. Suppose that we observe one instance of the gram matrix �Sn := XXT . Then +Y Y T += +(I − P1)XXT (I − P1) +(81) +where +⟨ Y Y T , En ⟩ = 1T +nY Y T 1n = 0. +28 + +Moreover, by construction, we have for A as defined in (9), +⟨ A, En ⟩ += +1T +nY Y T 1n − λ ⟨ (En − In), En ⟩ = −λn(n − 1) = tr(Y Y T ) +where +λ += +1 +n(n − 1) +� +i̸=j +⟨ Yi, Yj ⟩ = − +1 +n(n − 1)tr(Y Y T ) = − +τ +n − 1 +In other words, we have ⟨ A, P1 ⟩ = ⟨ A, 1n1T +n/n ⟩ = tr(Y Y T )/n =: τ +Recall that R is rank one with λmax(R) = tr(R) = w1w2npγ while ¯xR¯x/n = (4w1w2)w1w2npγ ≤ +1 +4npγ. Hence ¯x/√n coincides with ¯v1 when w1 = w2 = 1/2. Hence for gap(1, R), as defined in +Theorem B.1, +gap(1, R) = λmax(R) = w1w2npγ. +We check the claim that the leading eigenvector of B coincides with that of Y Y T in Fact E.2. +Clearly, +cos(θ1) = cos(∠(v1, ¯v1)) += +⟨ v1, ¯v1 ⟩ +(82) +and hence +θ1 = arccos( ⟨ v1, ¯v1 ⟩ ). +Hence we can use the first eigenvector of Y Y T to partition the two groups of points in Rp. To +obtain an upper bound on sin(θ1), we apply the Davis-Kahan perturbation bound as follows. Since +v1, ¯v1 are the leading eigenvectors of B and R respectively, (63) holds by Theorems 3.5 and B.1: +sin(θ1) +:= +sin(∠(v1, ¯v1)) ≤ 2 ∥B − R∥2 +λmax(R) += 2 ∥B − R∥2 +w1w2npγ +≤ +2ξ +w1w2 +. +Moreover, we have (64) holds by Corollary 3 [43]: since +min +α=±1 ∥αv1 − ¯v1∥2 +2 +≤ +� +23/2 ∥B − R∥2 +w1w2npγ +�2 +≤ +23ξ2 +(w1w2)2 =: δ′, where δ′ = 8ξ2/(w2 +1w2 +2) ≤ c2ξ2/w2 +min; +The theorem thus holds. +□ +It remains to state Fact E.2. +Fact E.2. Let Y Y T = �n−1 +j=1 λjvjvT +j . Denote by �A = Y Y T − λEn, then +�A +:= +Y Y T − λEn = +n−1 +� +j=1 +λjvjvT +j + +nτ +n − 111T /n ⪰ 0; +(83) +29 + +The leading eigenvector of �A (resp. A and B) will coincide with that of Y Y T with +λmax( �A) += +λmax(Y Y T ) ≥ nτ/(n − 1) +(84) +where strict inequality holds if and only if not all eigenvalues of Y Y T are identical. +Thus the +symmetric matrices A = �A + λIn and B = A + EτIn also share the same leading eigenvector v1 +with Y Y T , so long as not all eigenvalues of Y Y T are identical, with λmax(A) ≥ τ. +Proof. Clearly, the additional terms involving En and In are either orthogonal to eigenvectors +v1, . . . , vn−1 of Y Y T , or act as an identity map on the subspace spanned by {v1, . . . , vn−1}. Now +(83) holds since ⟨ vj, 1n ⟩ = 0 for all j and hence {v1, . . . , vn−1, 1n/√n} forms the set of orthonormal +eigenvectors for �A (resp. A and B); and moreover, in view of Fact E.1, +−λEn += +nτ +n − 1P1 = +nτ +n − 11n1T +n/n, +where +λ = − +τ +n − 1, +Since we have at most n − 1 non-zero eigenvalues and they sum up to be tr(Y Y T ), we have +λmax(Y Y T ) +≥ +tr(Y Y T )/(n − 1) = nτ/(n − 1) +where strict inequality holds when these eigenvalues are not all identical. +Finally, (84) holds since λ1( �A) := λmax(Y Y T ) in view of the eigen-decomposition (83) and the +displayed equation immediately above. Now for A = Y Y T −λ(En −In), we have tr(A) = tr(Y Y T ), +and hence λmax(A) ≥ τ. Moreover, the extra terms ∝ In in A (resp. B) will not change the order +of the sequence of eigenvalues for B (resp. A) with respect to that established for �A; Hence all +symmetric matrices B, �A, and A share the same leading eigenvector v1 with Y Y T . +□ +F +Proofs for results in Section 5 +Proposition F.1 holds regardless of the weights or the number of mixture components. +Proposition F.1. (Covariance projection: general mixture models) Let Y = X − P1X be +as defined in Definition 2.1. Let Z = X − EX. We first rewrite �ΣY = (Y − E(Y ))(Y − E(Y ))T as +follows: +�ΣY +:= +(I − P1)ZZT (I − P1) = M1 − (M2 − M3), +(85) +where M1 +:= +�ΣX = (X − E(X))(X − E(X))T = ZZT , +and P1 = 1 +n1n1T +n +(86) +M2 += +ZZT P1 + P1ZZT , and M3 = P1ZZT P1, +(87) +and ΣY +:= +E�ΣY := (I − P1)E(ZZT )(I − P1) +(88) +Then we have (69), since +Y Y T − E(Y )E(Y )T += +�ΣY + E(Y )(Y − E(Y ))T + (Y − E(Y ))(E(Y ))T . +30 + +Proof. Recall 1(X) = 1 +n1n1T +nX =: P1X; Then +�ΣY := (Y − EY )(Y − EY )T += +(X − E(X) − (1(X) − E1(X)))(X − E(X) − (1(X) − E1(X)))T += +(X − P1X − (E(X) − P1EX))(X − P1X − (E(X) − P1EX))T += +� +(I − P1)(X − E(X) +�� +(I − P1)(X − E(X)) +�T +(89) += +(I − P1)(X − E(X))(X − E(X))T (I − P1) += +(I − P1)ZZT (I − P1) +The rest are obvious. +□ +F.1 +Proof of Theorem 5.2 +We use a shorthand notation for MY := E(Y )(Y − E(Y ))T + (Y − E(Y ))(E(Y ))T . Thus we +have by the triangle inequality, (69), (69) (Proposition F.1), Lemma 6.1, and Theorem 6.3, with +probability at least 1 − 2 exp(−c6n) − 2 exp(−cn), +��Y Y T − E(Y Y T ) +�� +2 +≤ +����ΣY − ΣY +��� +2 + ∥MY ∥2 ≤ +��ZZT − E(ZZT ) +�� +2 + ∥MY ∥2 +≤ +C2C2 +0(√pn ∨ n) + 2C3C0n√pγ ≤ 1 +6ξnpγ +��Y Y T − E(Y Y T ) +�� +∞→1 +≤ +n +��Y Y T − E(Y Y T ) +�� +2 ≤ 1 +6ξn2pγ +where the last inequality holds by (25), while adjusting the constants. +□ +F.2 +Proof of Lemma 5.4 +First, we verify (20): +∀i ∈ C1 EYi += +EXi − (w1µ(1) + w2µ(2)) = µ(1)(1 − w1) − w2µ(2) += +w2(µ(1) − µ(2)) +(90) +∀i ∈ C2 EYi += +EXi − (w1µ(1) + w2µ(2)) = µ(2)(1 − w2) − w1µ(1) += +w1(µ(2) − µ(1)) +(91) +Lemma F.2. Suppose all conditions in Lemma 5.4 hold. Then +sup +q∈Sn−1 +n +� +i=1 +qi ⟨ Yi − E(Yi), µ(1) − µ(2) ⟩ +≤ +2 sup +q∈Sn−1 +n +� +i=1 +qi ⟨ Zi, µ(1) − µ(2) ⟩ . +(92) +Proof. +n +� +i=1 +qi ⟨ Yi − E(Yi), µ(1) − µ(2) ⟩ = 1 +n +n +� +i=1 +qi ⟨ +� +j̸=i +(Zi − Zj), µ(1) − µ(2) ⟩ += +n − 1 +n +� n +� +i=1 +qi ⟨ Zi, µ(1) − µ(2) ⟩ +� ++ 1 +n +n +� +i=1 +qi ⟨ Zi − +n +� +j=1 +Zj, µ(1) − µ(2) ⟩ += +n +� +i=1 +qi ⟨ Zi, µ(1) − µ(2) ⟩ − 1 +n +n +� +i=1 +qi +n +� +j=1 +⟨ Zj, µ(1) − µ(2) ⟩ +31 + +where +������ +1 +n +n +� +i=1 +qi +n +� +j=1 +⟨ Zj, µ(1) − µ(2) ⟩ +������ +≤ +sup +q∈Sn−1 +1 +n ∥q∥1 +������ +n +� +j=1 +⟨ Zj, µ(1) − µ(2) ⟩ +������ +≤ +������ +n +� +j=1 +1 +√n ⟨ Zj, µ(1) − µ(2) ⟩ +������ +≤ +sup +q∈Sn−1 +n +� +i=1 +qi ⟨ Zi, µ(1) − µ(2) ⟩ +Thus (92) holds and the lemma is proved. +□ +Proof of Lemma 5.4 . +Due to the symmetry, we need to compute only +��(Y − E(Y ))E(Y )T �� += +��(Z − (1(X) − E1(X)))(E(X) − E1(X))T �� +First, we show that (70) holds. Now +Yi − EYi += +(Xi − EXi) − ((�µn − E�µn) = Zi − +� +1 +n +n +� +i=1 +(Xi − EXi) +� += +n − 1 +n +Zi − 1 +n +n +� +j̸=i +Zj = 1 +n +n +� +j̸=i +(Zi − Zj) +and for xi ∈ {−1, 1}, +n +� +i=1 +xi ⟨ Yi − EYi, µ(1) − µ(2) ⟩ = 1 +n +n +� +i=1 +xi +n +� +j̸=i +⟨ (Zi − Zj), µ(1) − µ(2) ⟩ +≤ +1 +n +n +� +i=1 +n +� +j̸=i +��� ⟨ (Zi − Zj), µ(1) − µ(2) ⟩ +��� ≤ 2(n − 1) +n +n +� +i=1 +��� ⟨ Zi, µ(1) − µ(2) ⟩ +��� +Then we have by definition of the cut norm, (90), (91), and (70), +��(Y − E(Y ))E(Y )T �� +∞→1 = +sup +x,y∈{−1,1}n +n +� +i=1 +xi· +� +�� +j∈C1 +yj ⟨ Yi − E(Yi), w2(µ(1) − µ(2)) ⟩ + +� +j∈C2 +yj ⟨ Yi − E(Yi), w1(µ(2) − µ(1)) ⟩ +� +� +≤ +sup +x,y∈{−1,1}n +n +� +i=1 +xi +� +� ⟨ Yi − E(Yi), µ(1) − µ(2) ⟩ ( +� +j∈C1 +w2yj − +� +j∈C2 +w1yj) +� +� +≤ +(w2 |C1| + w1 |C2|) +n +� +i=1 +��� ⟨ Yi − E(Yi), µ(1) − µ(2) ⟩ +��� += +2w2w1n +max +x∈{−1,1}n +n +� +i=1 +xi ⟨ Yi − E(Yi), µ(1) − µ(2) ⟩ +≤ +4w1w2(n − 1) +n +� +i=1 +��� ⟨ Zi, µ(1) − µ(2) ⟩ +��� +32 + +Similarly, we have by (90) and (91), +��(Y − E(Y ))E(Y )T �� +2 = +sup +q,h∈Sn−1 +n +� +i=1 +qi· +� +�� +j∈C1 +hj ⟨ Yi − E(Yi), w2(µ(1) − µ(2)) ⟩ + +� +j∈C2 +hj ⟨ Yi − E(Yi), w1(µ(2) − µ(1)) ⟩ +� +� +≤ +sup +q,h∈Sn−1 +� +� +n +� +i=1 +qi ⟨ Yi − E(Yi), µ(1) − µ(2) ⟩ ( +� +j∈C1 +w2hj − +� +j∈C2 +w1hj) +� +� =: Q +(93) +where by (93) and (92), and w1w2 ≤ 1/4, +Q +≤ +sup +q∈Sn−1 +����� +n +� +i=1 +qi ⟨ Yi − E(Yi), µ(1) − µ(2) ⟩ +����� · +sup +h∈Sn−1 +������ +� +j∈C1 +w2hj − +� +j∈C2 +w1hj +������ +≤ +2 sup +q∈Sn−1 +����� +n +� +i=1 +qi ⟨ Zi, µ(1) − µ(2) ⟩ +����� · √nw1w2 ≤ √n sup +q∈Sn−1 +����� ⟨ +� +i +qiZi, µ(1) − µ(2) ⟩ +����� +where for ∥hCi∥2 = +�� +j∈Ci h2 +j, i = 1, 2 and h ∈ Sn−1, +������ +� +j∈C1 +w2hj − +� +j∈C2 +w1hj +������ +≤ +w2 +� +j∈C1 +|hj| + w1 +� +j∈C2 +|hj| =: w2 ∥hC1∥1 + w1 ∥hC2∥1 +≤ +w2 +� +|C1| ∥hC1∥2 + w1 +� +|C2| ∥hC2∥2 +≤ +√w1w2n (√w2 ∥hC1∥2 + √w1 ∥hC2∥2) ≤ √w1w2n +where 1 = w1 + w2 ≥ 2√w1w2 and by Cauchy-Schwarz, we have for ¯ +w0 = (√w2, √w1) such that +∥ ¯ +w0∥2 = √w1 + w2 = 1 and z = (∥hC1∥2 , ∥hC2∥2) such that ∥z∥2 = 1, +⟨ ¯ +w0, z ⟩ = (√w2 ∥hC1∥2 + √w1 ∥hC2∥2) ≤ ∥w0∥2 ∥z∥2 = 1. +□ +F.3 +Proof of Lemma 5.5 +Reduction in the operator norm holds since by Proposition F.1, +�ΣY − ΣY +:= +(Y − EY )(Y − EY )T − E((Y − EY )(Y − EY )T ) += +(I − P1)(ZZT − E(ZZT ))(I − P1) +and clearly, +����ΣY − ΣY +��� +2 +≤ +∥I − P1∥2 +��ZZT − E(ZZT ) +�� +2 ∥I − P1∥2 +(94) +≤ +��ZZT − E(ZZT ) +�� +2 . +□ +33 + +G +Proofs for Section 6 on isotropic design +Under (A2), the row vectors Z1, Z2, . . . , Zn ∈ Rp of matrix Z = X − EX, are independent, sub- +gaussian vectors with sub-gaussian norm, cf. Lemma 3.4.2 [42]. To bridge the deterministic bounds +in Lemma 5.4 and the probabilistic statements in Lemma 6.1, we use the tail bounds in Lemma 6.2. +Combining Lemmas 5.4 and 6.2 proves Lemma 6.1. +G.1 +Proof of Lemma 6.1 +Let ε = 1/3. Let Πn be an ε-net of Sn−1 such that |Πn| ≤ (1 + 2/ε)n; We have by (73) and the +union bound, +P (E3) +:= +P +������ sup +q∈Πn +n +� +i=1 +qi ⟨ Zi, µ ⟩ +����� ≥ 1 +2C3C0 +√n +� +(95) +≤ +9n · 2 exp +� +−c6C2 +3n/4 +� +≤ 2 exp(−c1n) +for some absolute constants C3, c1 and µ as in (71). Thus we have on event Ec +3, by a standard +approximation argument, +sup +q∈Sn−1 +n +� +i=1 +qi ⟨ Zi, µ ⟩ +≤ +1 +1 − ε sup +q∈Πn +n +� +i=1 +qi ⟨ Zi, µ ⟩ ≤ C3C0 +√n +Similarly, we have by the union bound and (72), +P (E4) +:= +P +� +max +u∈{−1,1}n +����� +n +� +i=1 +ui ⟨ Zi, µ ⟩ +����� ≥ 1 +2C4C0n +� +(96) +≤ +2n exp +� +−c5 +(C4C0)2n2 +4C2 +0n +� +≤ 2 exp(−c′n); +Hence on Ec +4, the second inequality follows from (70). +sup +u∈{−1,1}n +n +� +i=1 +ui ⟨ Yi − EYi, µ ⟩ +≤ +2(n − 1) +n +sup +u∈{−1,1}n +n +� +i=1 +ui ⟨ Zi, µ ⟩ ≤ C4C0(n − 1) +We have by Lemma 5.4, on event Ec +3 ∩ Ec +4, +∥MY ∥2 +≤ +4√n√w1w2 +sup +q∈Sn−1 +����� +� +i +qi ⟨ Zi, µ(1) − µ(2) ⟩ +����� ≤ 2C3C0n√pγ +and +∥MY ∥∞→1 +≤ +8w1w2(n − 1) +sup +u∈{−1,1}n +n +� +i=1 +ui ⟨ Zi, µ(1) − µ(2) ⟩ ≤ C4C0n(n − 1)√pγ +Thus the lemma holds upon adjusting the constants. +□ +34 + +G.2 +Proof of Lemma 6.2 +Let µ be as in (71) and recall +max +i +∥Zi∥ψ2 ≤ CC0 +Moreover, by independence of Z1, . . . , Zn, we have for u = (u1, . . . , un) ∈ {−1, 1}n, +����� +n +� +i=1 +ui ⟨ Zi, µ ⟩ +����� +2 +ψ2 +≤ +C +n +� +i=1 +∥ ⟨ Zi, µ ⟩ ∥2 +ψ2 ≤ C +n +� +i=1 +∥Zi∥2 +ψ2 +where ∥ ⟨ Zj, µ ⟩ ∥ψ2 ≤ ∥Zj∥ψ2 by definition of (13), and for any q ∈ Sn−1 and t > 0, we have +����� +n +� +i=1 +qi ⟨ Zi, µ ⟩ +����� +2 +ψ2 +≤ C +n +� +i=1 +q2 +i ∥ ⟨ Zi, µ ⟩ ∥2 +ψ2 ≤ C max +i +∥Zi∥2 +ψ2 ≤ C′C2 +0 +Thus we have the following sub-gaussian tail bounds, for any u = (u1, . . . , un) ∈ {−1, 1}n and +t > 0, +P +� n +� +i=1 +ui ⟨ Zi, µ ⟩ ≥ t +� +≤ +2 exp +� +− +ct2 +�n +i=1 ∥Zi∥2 +ψ2 +� +≤ 2 exp +� +− ct2 +C2 +0n +� +and for any q ∈ Sn−1 and t > 0, +P +� n +� +i=1 +qi ⟨ Zi, µ ⟩ ≥ t +� +≤ +2 exp +� +− +ct2 +maxn +i=1 ∥Zi∥2 +ψ2 +� +≤ 2 exp +� +−c′t2 +C2 +0 +� +See Proposition 2.5.2 (i) [42]. Thus the lemma holds. +□ +G.3 +Proof sketch of Theorem 6.3 +First, notice that M1 = ZZT =: �ΣX is the empirical covariance matrix based on the original data +X. In order to prove the concentration of measure bounds for Theorem 6.3, we will first state the +operator norm bound on M1 − EM1 in Lemma G.1. +Let Cdiag, Coffd, C1, C2, c, c′, . . . be some absolute constants, which may change line by line. Denote +by E0 the following event: +E0 : +∃j ∈ [n] +���∥Zj∥2 +2 − E ∥Zj∥2 +2 +��� > CdiagC2 +0(√np ∨ n) +(94) +Lemma G.1. (M1 term: operator norm) Choose ε = 1/4 and construct an ε-net N whose +size is upper bounded by |N| ≤ ( 2 +ε + 1)n ≤ 9n. Recall that Z = X − EX. Fix ε = 1/4. Under the +conditions in Theorem 6.3, denote by E8 the following event: +event E8 : +� +� +� max +q,h∈N +n +� +i=1 +n +� +j̸=i +⟨ Zi, Zj ⟩ qihj > C1C2 +0(√np ∨ n) +� +� +� +35 + +As a consequence, on event Ec +0 ∩ Ec +8, we have +��ZZT − EZZT �� +2 +≤ +C2C2 +0(√np ∨ n) +where P (Ec +0 ∩ Ec +8) ≥ 1 − 2 exp(−c4n), upon adjusting the constants. +When we take Z as a sub-gaussian ensemble with independent entries, our bounds on ∥M1 − EM1∥ +(cut norm and operator norm) depend on the Bernstein’s type of inequalities and higher dimen- +sional Hanson-Wright inequalities. We will state Lemma G.4 in Section G.4, where we bound the +probability of event E0. We prove Lemma G.1 in Section G.5 using the standard net argument. +Neither weights, nor the number of mixture components, will affect such bounds. +G.4 +Bounds on independent sub-exponential random variables +We now derive the corresponding bounds using properties of sub-exponential random variables. +The sub-exponential (or ψ1) norm of random variable S, denoted by ∥S∥ψ1, is defined as +∥S∥ψ1 = inf{t > 0 : E exp(|S| /t) ≤ 2}. +(95) +A random variable Z is sub-gaussian if and only if S := Z2 is sub-exponential with ∥S∥ψ1 = ∥Z∥2 +ψ2. +The proof does not depend on the specific sizes |Cj| ∀j of clusters. Lemma G.2 concerns the sum of +independent sub-exponential random variables. We also state the Hanson-Wright inequality [40]. +Lemma G.2. (Bernstein’s inequality, cf. +Theorem 2.8.1 [42]) Let X1, . . . , Xn be independent, +mean-zero, sub-exponential random variables. Then for every t > 0, +P +� +� +������ +n +� +j=1 +Xj +������ +≥ t +� +� +≤ +2n exp +� +−c min +� +t2 +�n +j=1 ∥Xj∥2 +ψ1 +, +t +maxj ∥Xj∥ψ1 +�� +Theorem G.3. [40] Let X = (X1, . . . , Xm) ∈ Rm be a random vector with independent components +Xi which satisfy EXi = 0 and ∥Xi∥ψ2 ≤ C0. Let A be an m × m matrix. Then, for every t > 0, +P +���XT AX − E(XT AX) +�� > t +� +≤ 2 exp +� +−c min +� +t2 +C4 +0 ∥A∥2 +F +, +t +C2 +0 ∥A∥2 +�� +. +Lemma G.4. Let Z = (zjk) ∈ Rn×p be a random matrix whose entries are independent, mean-zero, +sub-gaussian random variables with maxj,k ∥zjk∥ψ2 ≤ C0. Then we have for tdiag = CdiagC2 +0(√np ∨ +n), +P (E0) +:= +P +� +∃j ∈ [n], +���∥Zj∥2 +2 − E ∥Zj∥2 +2 +��� > tdiag +� +≤ 2 exp(−c0n) +where {Zj, j ∈ [n]} are row vectors of matrix Z, and Cdiag and c0 are absolute constants. +Proof. Denote by +Sjk = z2 +jk − Ez2 +jk, +where zjk = Xjk − EXjk, ∀i ∈ [n], k ∈ [p] +(96) +36 + +It follows from (95) that Sjk is a mean-zero, sub-exponential random variable since +max +j,k ∥Sjk∥ψ1 +≤ +CC2 +0 +since ∀j, k, +��z2 +jk +�� +ψ1 = ∥zjk∥2 +ψ2 ≤ C2 +0, +and ∥Sjk∥ψ1 += +��z2 +jk − Ez2 +jk +�� +ψ1 ≤ C +��z2 +jk +�� +ψ1 = C ∥zjk∥2 +ψ2 ≤ CC2 +0 +See Exercise 2.7.10 [42]. Set t3 = CdiagC2 +0 +√np ∨ n. We have by Bernstein’s inequality Lemma G.2 +and the union bound, the following large deviation bound: +P +� +∃j ∈ [n], +���∥Zj∥2 +2 − E ∥Zj∥2 +2 +��� ≥ t3 +� +≤ +n +� +j=1 +P +������ +p +� +k=1 +Sjk +����� ≥ t3 +� +≤ +2n exp +� +−c min +� +(CdiagC2 +0 +√np ∨ n)2 +maxj∈[n] +�p +k=1 ∥Sjk∥2 +ψ1 +, CdiagC2 +0 +√np ∨ n +maxj,k ∥Sjk∥ψ1 +�� +≤ +2n exp +� +−c min +� +C2 +diagnp +p +, Cdiagn +�� +≤ 2 exp(−c0n) +(97) +where for all j ∈ [n], �p +k=1 ∥Sjk∥2 +ψ1 ≤ pC2C4 +0. The lemma thus holds. +□ +G.5 +Proof of Lemma G.1 +We use Theorem G.3 to bound the off-diagonal part. Recall maxj,k ∥zjk∥ψ2 ≤ C0. Let vec { Z } = +vec { X − EX } be formed by concatenating columns of matrix Z into a long vector of size np. +For a particular realization of q, h ∈ Sn−1, we construct a block-diagonal matrix �A(q, h), where +diag( �A) = 0, with p identical block-diagonal coefficient matrices A(k) +qh = offd(q ⊗ h), ∀k of size n × n +along the diagonal. Then +������ +n +� +i=1 +qi +n +� +j̸=i +hj ⟨ Zi, Zj ⟩ +������ +:= +���vec { Z }T �Aq,hvec { Z } +��� +and +∀q, h ∈ Sn−1, +��� �A(q, h) +��� +2 +F += +p +� +k=1 +���A(k) +qh +��� +2 +F = p ∥offd(q ⊗ h)∥2 +F ≤ p, +��� �A(q, h) +��� +2 += +∥offd(q ⊗ h)∥2 ≤ 1, +since +∥offd(q ⊗ h)∥2 +≤ +∥offd(q ⊗ h)∥F ≤ ∥q ⊗ h∥2 +F = tr(qhT hqT ) = 1. +Taking a union bound over all |N|2 pairs q, h ∈ N, the ε-net of Sn−1, we have by Theorem G.3, for +some sufficiently large constants C1 and c > 4 ln 9, +P (E8) := P +� +� max +q,h∈N +������ +n +� +i=1 +n +� +j̸=i +qihj ⟨ Zi, Zj ⟩ +������ +> C1C2 +0(√pn ∨ n) +� +� +≤ +|N|2 P +����vec { Z }T �A(q, h)vec { Z } +��� > C1C2 +0(√pn ∨ n) +� +≤ +2 × 92n exp +� +−c min +� +C2 +1pn/p, C1n +�� +≤ +2 exp (−cn + 2n ln 9) = 2 exp (−c3n) +37 + +A standard approximation argument shows that under Ec +8, we have +��offd(ZZT ) +�� +2 += +sup +q∈Sn−1 +n +� +i=1 +n +� +j̸=i +qiqj ⟨ Zi, Zj ⟩ +≤ +1 +(1 − 2ε) sup +q,h∈N +n +� +i=1 +n +� +j̸=i +qihj ⟨ Zi, Zj ⟩ ≤ 2C1C2 +0(√np ∨ n) +See for example Exercise 4.4.3 [42]. +The large deviation bound on the operator norm follows from the triangle inequality: on event +Ec +8 ∩ Ec +0, +��ZZT − E(ZZT ) +�� +2 +≤ +��diag(ZZT − E(ZZT )) +�� +2 + +��offd(ZZT ) +�� +2 ≤ C2C2 +0(√np ∨ n) +for some absolute constant C2. +□ +H +Bias terms +This section proves results needed for Theorem 3.5. We prove Lemmas 5.1 in Section H.2, where +we also state Lemma H.7. Combining (114), (115), and Proposition H.4, we obtain an expression +on EB − R. Recall R = E(Y )E(Y )T is as defined in (21). We have the following facts about R. +Fact H.1. When we sum over all entries in R, clearly, we have for R as defined in (21), 1T +nR1n = 0, +1T +noffd(R)1n += +1T +nR1n − tr(R) = −tr(R) = −npγw2w1, +where +(98) +1 +pγ tr(R) += +nw2 +2w1 + nw2 +1w2 = w1w2n +and hence +∥R∥2 = tr(R) = w1w2npγ +Lemma H.2. We have by Fact H.1, tr(R)/n = pγw1w2 and hence for n ≥ 4, +���� +tr(R) +n − 1 (In − En/n) +���� +2 +≤ +n +n − 1oγw1w2 ≤ pγ/3 +(99) +and hence +���� +tr(R) +n − 1 (In − En/n) +���� +∞→1 +≤ +n +n − 1npγw1w2 ≤ npγ/3 +(100) +where ∥In − En/n∥∞→1 ≤ n ∥In − En/n∥2 = 1 since In − En/n is a projection matrix. +H.1 +Some useful propositions +Next, we compute the mean values in Proposition H.3, and we obtain an expression on EB − R in +Proposition H.4. Proposition H.3 is proved in Section H.3. +Proposition H.3. (Covariance projection: +two groups) Let Nj = wjn for j ∈ {1, 2}. +W.l.o.g., suppose that the first N1 rows in X are in C1 and the following N2 rows are in C2. +Let M1, M2, M3 be defined as in Proposition F.1. Let V1 and V2 be the same as in (24): +V1 := E ⟨ Zj, Zj ⟩ +∀j ∈ C1 +and +V2 := E ⟨ Zj, Zj ⟩ +∀j ∈ C2 +(101) +38 + +Let Vm := w1V1 + w2V2 = Wn/n. Let �ΣY be as in (85). Let Wn be defined as in (102): +Wn +:= +E +n +� +i=1 +⟨ Zi, Zi ⟩ = +n +� +i=1 +p +� +k=1 +Ez2 +ik. +(102) +Then +EM1 += +� V1IN1 +0 +0 +V2IN2 +� +=: (w1V1 + w2V2)In + W0, +(103) +EM2 += +V1 + V2 +n +En + W2, +(104) +EM3 +:= +E ⟨ �µn − E�µn, �µn − E�µn ⟩ 1n ⊗ 1n = Wn +n2 En, +(105) +and +Wn/n2 +:= +(|C1| V1 + |C2| V2)/n2 = (w1V1 + w2V2)/n =: Vw +n , +(106) +where tr(EM1)/n = Vm and +W0 += +(V1 − V2) +� w2IN1 +0 +0 +−w1IN2 +� +and W2 := V1 − V2 +n +� EN1 +0 +0 +−EN2 +� +. +(107) +Now putting things together, we obtain the expression for covariance of Y : +ΣY := E(Y Y T ) − E(Y )E(Y )T = EM1 + EM3 − EM2 += +� V1IN1 +0 +0 +V2IN2 +� +− w2V1 + w1V2 +n +En − W2 +which simplifies to +ΣY = V (In − En/n) in case +V1 = V2 = V +We prove Proposition H.4 in Section H.4. Intuitively, W2 and W arise due to the imbalance in +variance profiles. +Proposition H.4. (Bias decomposition) Let M1, M2, M3, W0, W2, V1, V2... be the same as in +Propositions F.1 and H.3. Then +EB − R += +W0 − W − tr(R) +(n − 1)(In − En +n ) where +(108) +W +:= +W2 + (V1 − V2)(w2 − w1) +n +En +(109) +where for W2 defined in (107). Moreover, when V1 = V2, W0 = W = 0. +Next, we state the following fact about W2. +Fact H.5. Denote by +W := W2 − 1 +ntr(W2)In − 1T +noffd(W2)1n +n(n − 1) +(En − In) +39 + +Then W coincides with (109). Moreover, we have +W += +W2 − (V2 − V1)(w2 − w1) +n +En +:= +V1 − V2 +n +� +2w2EN1 +(w2 − w1)EN1×N2 +(w2 − w1)EN1×N2 +−2w1EN2 +� +Moreover, we have +∥W∥2 +≤ +|V1 − V2| (w1 ∨ w2) and +∥W∥∞→1 ≤ n |V1 − V2| (w1 ∨ w2) +(110) +Proof. By definition of W2 as in (107), we have +tr(W2) += +V2 − V1 +n +(w2n − w1n) = (V2 − V1)(w2 − w1) and +1T +nW21n += +V2 − V1 +n +(w2 +2n2 − w2 +1n2) = n(V2 − V1)(w2 +2 − w2 +1) += +n(V2 − V1)(w2 − w1) +Thus +1T +noffd(W2)1n +n(n − 1) += +1T +nW21n − tr(W2) +n(n − 1) += (V2 − V1)(w2 − w1) +n +; +Then +W += +W2 − 1 +ntr(W2)In − 1T +noffd(W2)1n +n(n − 1) +(En − In) += +W2 − (V2 − V1)(w2 − w1) +n +En. +Hence (109) holds. Moreover, by symmetry, ∥W∥2 ≤ ∥W∥∞ ≤ |V1 − V2| (w1 ∨ w2). +□ +H.2 +Proof of Lemma 5.1 +We have by Proposition H.4, +∥EB − R∥ += +����W0 − W − tr(R) +(n − 1)(In − En +n ) +���� +≤ +∥W0∥ + ∥W∥ + +���� +tr(R) +(n − 1)(In − En +n ) +���� +(111) +where the ∥·∥ is understood to be either the operator or the cut norm. Recall that +W0 +:= +EM1 − VwIn = (V1 − V2) +� w2Iw1n +0 +0 +−w1Iw2n +� +and hence 1T +nW01n += +tr(W0) = 0 and Vm := tr(EM1)/n = w1V1 + w2V2 +and hence W0 disappears if the two clusters have identical sum of variances: V1 = V2. Clearly, +∥W0∥2 +≤ +|V1 − V2| (w2 ∨ w1) +(112) +40 + +Combining (111), (112), (110), and (100), we have for wmin := w1 ∧ w2 and nξ ≥ +1 +2wmin +∥EB − R∥2 +≤ +∥W0∥2 + ∥W∥2 + +���� +tr(R) +(n − 1)(In − En +n ) +���� +2 +≤ +2 |V1 − V2| (w1 ∨ w2) + pγ/3 += +2 +3ξnpγ(1 − wmin) + pγ/3 ≤ 2 +3ξnpγ +where we use the fact that +2 +3ξpγnwmin ≥ pγ/3 +since +2ξnwmin ≥ 1 +Now the bound on the cut norm follows since +∥EB − R∥∞→1 +≤ +n ∥EB − R∥2 ≤ 2 +3ξn2pγ +The lemma thus holds for the general setting; when V1 = V2, we show the improved bounds in +Lemma H.7. +□ +Corollary H.6 follows from the proof of Lemma 5.1, which we state to prove a bound for the balanced +cases. The proof is given in Section H.5. +Corollary H.6. For general cases, we have by definition, +W2 +:= +V1 − V2 +n +� +EN1 +0EN1×N2 +0EN1×N2 +−EN2 +� +Moreover we have the following term which depends on the weights, +���� +(V2 − V1)(w2 − w1) +n +En +���� +∞→1 +≤ +n |(V2 − V1)(w2 − w1)| ≤ ξn2pγ |w2 − w1| +∥W2∥∞→1 +:= +n |V1 − V2| (w2 +1 + w2 +2) +and hence +∥W0 − W2∥∞→1 +≤ +n ∥W0 − W2∥2 < n |V1 − V2| ≤ ξpn2γ +Lemma H.7. (Reductions) Let W0, W2, V1, V2 be the same as in Proposition H.3. Recall that +R = E(Y )E(Y )T . When V1 = V2, we have +EB − R = − tr(R) +(n − 1)(In − En +n ) = −pγw2w1 +n +n − 1(In − En +n ) +and hence for n ≥ 4, +∥EB − R∥∞→1 = +����pγw2w1 +n +n − 1(In − En +n ) +���� +∞→1 +≤ npγ/3 +For balanced clusters, that is, when w1 = w2, we have +EB − R += +W0 − W2 − tr(R) +(n − 1)(In − En +n ) +41 + +and hence for n ≥ 4, +∥EB − R∥2 +≤ +pγ +3 + |V1 − V2| ≤ 1 +3(1 + o(1))ξpnγ +∥EB − R∥∞→1 +≤ +npγ +3 ++ |V1 − V2| n ≤ 1 +3(1 + o(1))ξpn2γ. +Proof. Recall +W0 − W2 += +(V1 − V2) +� w2IN1 − EN1/n +0 +0 +−(w1IN2 − EN2/n) +� +The case where V1 = V2 follows from Lemma H.2 and (108). We now show the balanced case where +w1 = w2. Under the conditions of Lemma 5.1, we have for ξ = Ω(1/n), by (99) and (108), +∥EB − R∥2 +≤ +∥W0 − W2∥2 + +���� +(V1 − V2)(w1 − w2) +n +En +���� +2 ++ +���� +tr(R) +n − 1 (In − En/n) +���� +2 +≤ +|V1 − V2| + pγ/3 ≤ 1 +3ξnpγ + pγ/3 +Similarly, we obtain +∥EB − R∥∞→1 +≤ +|V1 − V2| n + npγ/3 ≤ 1 +3ξn2pγ + npγ/3 +The proof follows from Corollary H.6 immediately. +□ +H.3 +Proof of Proposition H.3 +Denote by tr(EM1)/n = (w1V1 + w2V2) = Vm. +EM1 += +E +� +(X − E(X))(X − E(X))T � += EZZT += +� V1Iw1n +0 +0 +V2Iw2n +� +Moreover, upon subtracting the component of VwIn = 1 +ntr(EM1)In from EM1, we have W0: +EM1 − VwIn +:= +EM1 − 1 +ntr(EM1)In = (V1 − V2) +� w2Iw1n +0 +0 +−w1Iw2n +� +=: W0; +Next we evaluate EM2: for Z = X − EX +EM2 += +E +�� +Z(1(X) − E1(X))T + (1(X) − E1(X))ZT �� += +1 +n +� +2V1EN1 +(V1 + V2)EN1×N2 +(V1 + V2)EN1×N2 +2V2EN2 +� +and hence +EM2 − (V1 + V2) +n +En += +V1 − V2 +n +� +EN1 +0EN1×N2 +0EN1×N2 +−EN2 +� +=: W2 +42 + +For Wn as defined in (102), we have +E ⟨ �µn − E�µn, �µn − E�µn ⟩ +:= +1 +n2 +n +� +i=1 +� +k +Ez2 +ik = w1V1 + w2V2 +n += Vm +n +EM3 +:= +E ⟨ �µn − E�µn, �µn − E�µn ⟩ En = |C1| V1 + |C2| V2 +n2 +En = Wn +n2 En +Now putting things together, +E(Y Y T ) − E(Y )E(Y )T = EM1 + EM3 − EM2 += +� V1IN1 +0 +0 +V2IN2 +� ++ Vm +n En − (V1 + V2) +n +En − W2 +where +Vm +n − (V1 + V2) +n += +−V1(1 − w1) + V2(1 − w2) +n += −V1w2 + V2w1 +n +The proposition thus holds. +□ +H.4 +Proof of Proposition H.4 +First, we have by Proposition F.1, and R = E(Y )E(Y )T , +E(Y Y T ) = ΣY + R = EM1 − EM2 + EM3 + R +(113) +We have by Fact H.1 and (113), +Eτ += +1 +n +n +� +i=1 +E ⟨ Yi, Yi ⟩ = Etr(Y Y T )/n = tr(ΣY )/n + tr(R)/n += +1 +n (tr(EM1 + EM3) − tr(EM2)) + tr(R)/n +(114) +Eλ += +1 +n(n − 1) +n +� +i̸=j +E ⟨ Yi, Yj ⟩ = +1 +n(n − 1)1T +nE(offd(Y Y T ))1n += +1 +n(n − 1)1T +noffd(EM3 − EM2)1n − +tr(R) +n(n − 1) +(115) +where in (115) we use the fact that offd(EM1) = 0 by (104) and (98). Hence by definition of B and +R, we have +EB − R += +EM1 − EM2 + EM3 − EτIn − Eλ(En − In) += +EM1 − EM2 + EM3 − +� 1 +ntr(EM1 + EM3) − 1 +ntr(EM2) + tr(R) +n +� +In +− +� +1 +n(n − 1)1T +noffd(EM1 + EM3 − EM2)1n − +tr(R) +n(n − 1) +� +(En − In) +(116) +• Notice that EM3 = Vw +n En and hence its contribution to Eτ and Eλ is the same; Thus we have +tr(EM3) += +1 +(n − 1)1T +noffd(EM3)1n = Wn +n += Vm and by (105), +EM3 − 1 +ntr(EM3)In − 1T +noffd(EM3)1n +n(n − 1) +(En − In) = 0; +(117) +43 + +• EM1 is a diagonal matrix and hence offd(EM1) = 0. Now we have by (104), +EM1 − 1 +ntr(EM1)In = W0 and offd(EM1) = 0 +(118) +• For EM2, we decompose it into one component proportional to En: +¯ +M2 := V1+V2 +n +En and +another component W2 = EM2 − ¯ +M2. By Proposition H.3, we have +W2 += +EM2 − ¯ +M2 = EM2 − V1 + V2 +n +En +:= +V1 − V2 +n +� +EN1 +0EN1×N2 +0EN1×N2 +−EN2 +� +(119) +where by definition +¯ +M2 − tr( ¯ +M2) +n +In − 1T +noffd( ¯ +M2)1n +n(n − 1) +(En − In) = 0 +(120) +Thus we have by Fact H.5 and (120) +EM2 − 1 +ntr(EM2)In − 1T +noffd(EM2)1n +n(n − 1) +(En − In) += +W2 − 1 +ntr(W2)In − 1T +noffd(W2)1n +n(n − 1) +(En − In) =: W +(121) +Now by (114), (115), (116), (117), (118), (121), and Proposition H.3, +EB − R += +EM1 − EM2 + EM3 − EτIn − Eλ(En − In) +=: +W0 − W − tr(R) +n − 1(In − En/n) +(122) +where in step 2, we simplify all terms involving EM1 and EM2, and eliminate all terms involving +EM3. +□ +H.5 +Proof of Corollary H.6 +Now +W0 − W2 += +(V1 − V2) +� w2IN1 − EN1/n +0 +0 +−(w1IN2 − EN2/n) +� +. +Moreover, due to symmetry, for wj > 1/n, +∥W0 − W2∥2 +≤ +∥W0 − W2∥∞ := max +i +n +� +j=1 +|W0,ij − W2,ij| +≤ +|V1 − V2| ((w2 − 1/n) + (w1n − 1)/n) ∨ ((w1 − 1/n) + (w2n − 1)/n) < |V1 − V2| +where +((w2 − 1/n) + (w1n − 1)/n) ∨ ((w1 − 1/n) + (w2n − 1)/n) += +((w2 − 1/n) + (w1 − 1/n) ∨ ((w1 − 1/n) + (w2 − 1/n) = 1 − 2/n +44 + +Thus we have by the triangle inequality, +∥W0 − W2∥∞→1 +≤ +∥W2∥∞→1 + ∥W0∥∞→1 +≤ +|V1 − V2| n +� +|w2w1 + w1w2| + (w2 +1 + w2 +2) +� += +|V1 − V2| n ≤ 1 +2ξn2pγ +□ +I +Proofs for Section 7 +Proof of Theorem 5.3. +The proof of Theorem 5.3 follows that of Theorem 5.2 in Section F.1, +in view of Theorem 7.2 and Lemma 7.1. Finally, the probability statements hold by adjusting the +constants. +□ +I.1 +Preliminary results +Lemma I.1 follows from the sub-gaussian tail bound. We prove Lemma 7.1 in Section I.2. +Lemma I.1. (Projection for anisotropic sub-gaussian random vectors). Suppose all con- +ditions in Lemma 7.1 hold. Let µ be as defined in (71). Then +∥ ⟨ Zj, µ ⟩ ∥ψ2 +≤ +C0 ∥ ⟨ Zj, µ ⟩ ∥L2 := C0 +� +µT Cov(Zj)µ +(123) +where +� +µT Cov(Zj)µ += +� +µT HiHT +i µ = ∥Riµ∥2 +for each j ∈ Ci, i = 1, 2 +Thus for any t > 0, for some absolute constants c, c′, we have for each q ∈ Sn−1 and u = +(u1, . . . , un) ∈ {−1, 1}n the following tail bounds: +P +������ +n +� +i=1 +qi ⟨ Zi, µ ⟩ +����� ≥ t +� +≤ +2 exp +� +− +ct2 +(C0 maxi ∥Riµ∥2)2 +� +, +and +(124) +P +������ +n +� +i=1 +ui ⟨ Zi, µ ⟩ +����� ≥ t +� +≤ +2 exp +� +− +c′t2 +n(C0 maxi ∥Riµ∥2)2 +� +(125) +We prove Lemma I.2 in Section I.5. As a special case, we recover results in Lemma G.4. +Lemma I.2. Let Wj be a mean-zero, unit variance, sub-gaussian random vector with independent +entries, with maxj,k ∥wjk∥ψ2 ≤ C0. Let {Zj, j ∈ [n]} be row vectors of Z, where Zj = HiWj for +j ∈ Ci, i = 1, 2. Then for each j ∈ Ci, i = 1, 2, we have +P +����∥Zj∥2 +2 − E ∥Zj∥2 +2 +��� > t +� += +P +����∥HiWj∥2 +2 − ∥Hi∥2 +F +��� > t +� +≤ +2 exp +� +−c min +� +t2 +(C2 +0 ∥Hi∥2 ∥Hi∥F )2 , +t +(C0 ∥Hi∥2)2 +�� +Hence for rank p matrix Hi, we recover the result in (97) in case H = I, +P +����∥Zj∥2 +2 − E ∥Zj∥2 +2 +��� > t +� +≤ +2 exp +� +−c min +� +t2 +p(C0 ∥Hi∥2)4 , +t +(C0 ∥Hi∥2)2 +�� +45 + +Next we show that conclusion identical to those in Lemma G.1 holds, upon updating events E0 +and E8 for the operator norm for anisotropic random vectors Zj. Denote by E0 the event: for some +absolute constant Cdiag, +E0 := +� +∃j ∈ [n] +���∥Zj∥2 +2 − E ∥Zj∥2 +2 +��� > Cdiag(C0 max +i +∥Hi∥2)2(√np ∨ n) +� +Denote by E8 the following event: for some absolute constant C1, +E8 : +� +� +� max +q,h∈N +n +� +i=1 +n +� +j̸=i +⟨ Zi, Zj ⟩ qihj > C1(C0 max +i +∥Hi∥2)2(√np ∨ n) +� +� +� +where N is the ε-net of Sn−1 for ε < 1/4 as constructed in Lemma G.1. +I.2 +Proof of Lemma 7.1 +Let c, c′, C3, C4 be some absolute constants. Let MY := E(Y )(Y − E(Y ))T + (Y − E(Y ))E(Y )T . +Clearly, vectors Z1, Z2, . . . , Zn ∈ Rp are independent. +Let µ be as in (71). +Then, we have by +Lemma I.1, +P (E4) := P +� +max +u∈{−1,1}n +n +� +i=1 +ui ⟨ Zi, µ ⟩ +� +≥ 1 +2C4n(C0 max +i +∥Riµ∥2) +≤ +2n2 exp +� +−c′n2(C0 maxi ∥Riµ∥2)2 +C(C0 maxi ∥Riµ∥2)2n +� +≤ 2 exp(−c′n). +(126) +Thus we have on event Ec +4, +� +i +��� ⟨ Zi, µ(1) − µ(2) ⟩ +��� +< +1 +2C4(C0 max +i +∥Riµ∥2)n√pγ. +Construct an ε-net Πn of Sn−1, where ε = 1/3 and |Πn| ≤ (1+2/ε)n. For a suitably chosen constant +C3, we have by Lemma I.1, +P (E3) +:= +P +� +∃q ∈ Πn, +����� +n +� +i=1 +qi ⟨ Zi, µ ⟩ +����� ≥ 1 +2C3(C0 max +i +∥Riµ∥2)√n +� +≤ +9n2 exp +� +−c′n(C0 maxi ∥Riµ∥2)2 +C(C0 maxi ∥Riµ∥2)2 +� +≤ 2 exp(−c′n). +Moreover, by a standard approximation argument, we have on event Ec +3, +sup +q∈Sn−1 +n +� +i=1 +qi ⟨ Zi, µ ⟩ +≤ +1 +1 − ε sup +q∈Πn +n +� +i=1 +qi ⟨ Zi, µ ⟩ ≤ C3C0(max +i +∥Riµ∥2)√n. +We have by Lemma 5.4, on event Ec +4 ∩ Ec +3, +∥MY ∥∞→1 +≤ +8w1w2(n − 1) +n +� +i=1 +��� ⟨ Zi, µ(1) − µ(2) ⟩ +��� ≤ C4n(n − 1)(C0 max +i +∥Riµ∥2)√pγ +∥MY ∥2 +≤ +4√w1w2n sup +q∈Sn−1 +����� +n +� +i=1 +qi ⟨ Zi, µ(1) − µ(2) ⟩ +����� ≤ 2C3(C0 max +i +∥Riµ∥2)n√pγ. +□ +46 + +I.3 +Proof of Lemma I.1 +Proof. Denote by µ = (µ(1) − µ(2))/√pγ ∈ Sp−1. First, we have by definition, Zj = HiWj, for each +j ∈ Ci, i = 1, 2; cf. (13). Hence Zj is a sub-gaussian random vector with its marginal ψ2 norm +bounded in the sense of (14) and (15) with +∥ ⟨ Zj, µ ⟩ ∥ψ2 +:= +∥ ⟨ HiWj, µ ⟩ ∥ψ2 ≤ ∥Wj∥ψ2 ∥Riµ∥2 ≤ C0 ∥Riµ∥2 . +(127) +where Wj ∈ Rm is a mean-zero, isotropic, sub-gaussian random vector satisfying ∥Wj∥ψ2 ≤ C0; +Hence (123) holds and for all j ∈ Ci, +��� ⟨ Zj, µ(1) − µ(2) ⟩ +��� +ψ2 +≤ +C0 +√pγ ∥Riµ∥2 +and Ri = HT +i . +(128) +First, we have by independence of Zj, ∀j, +∀q ∈ Sn−1, +����� +n +� +i=1 +qi ⟨ Zi, µ(1) − µ(2) ⟩ +����� +2 +ψ2 +≤ +C +n +� +i=1 +q2 +i +��� ⟨ Zi, µ(1) − µ(2) ⟩ +��� +2 +ψ2 +≤ +Cpγ(C0 max +i +∥Riµ∥2)2 +and for ui ∈ {−1, 1}, +����� +n +� +i=1 +ui ⟨ Zi, µ(1) − µ(2) ⟩ +����� +2 +ψ2 +≤ +C +n +� +i=1 +��� ⟨ Zi, µ(1) − µ(2) ⟩ +��� +2 +ψ2 +≤ +Cnpγ(C0 max +i +∥Riµ∥2)2. +Then (124) and (125) follow from the sub-gaussian tail bound, for example, Propositions 2.6.1 and +2.5.2 (i) [42]. See also the proof for Lemma 6.2. +□ +I.4 +Proof of Theorem 7.3 +In the rest of this section, we prove Theorem 7.3. The proof may be of independent interests. We +generate Z according to Definition 2.3: +Zj += +HjWj ∈ Rp +and hence Z = +n +� +j=1 +ejW T +j HT +j = +n +� +j=1 +diag(ej)WHT +j +where W T +1 , . . . , W T +n ∈ Rm are independent, mean-zero, isotropic row vectors of W = (wjk), where +we assume that coordinates wjk are also independent with maxj,k ∥wjk∥ψ2 ≤ C0. Throughout this +section, let vec { Z } = vec { X − EX } be formed by concatenating columns of matrix Z into a long +vector of size np. Denote by ⊗ the tensor product. Recall e1, . . . , en are the canonical basis of Rn. +Proof of Theorem 7.3. +By Definition 2.3, +vec { Z } += +n +� +j=1 +vec +� +diag(ej)WHT +j +� += +n +� +j=1 +Hj ⊗ diag(ej)vec { W } +=: +Lvec { W } ∈ Rnp +where L := +n +� +j=1 +Hj ⊗ diag(ej) ∈ Rnp×mn +(129) +47 + +On the other hand, we have for W ∈ Rn×m, Zj = HjWj and hence +ZT += +[Z1, . . . , Zn] = +n +� +j=1 +Zj ⊗ eT +j = +n +� +j=1 +HjWT diag(ej) and hence +vec +� +ZT � += +vec +� +� +� +n +� +j=1 +HjWTdiag(ej) +� +� +� = +n +� +j=1 +vec +� +HjWTdiag(ej) +� += +n +� +j=1 +(diag(ej) ⊗ Hj)vec +� +WT � +=: Rvec +� +WT � +(130) +Then there exist some permutation matrices P, Q such that +L = +n +� +i=1 +Hi ⊗ diag(ei) += +P T � +n +� +i=1 +diag(ei) ⊗ Hi +� +Q =: P T RQ +(131) +where HjHT +j denotes the covariance matrix for each row vector Zj, j ∈ [n]. +We now show (131) with an explicit construction. It is well known that there exist permutation +matrices P, Q such that +vec +� +ZT � += +Pvec { Z } and vec +� +WT � += Qvec { W } +(132) +and hence by (129), vec +� +ZT � += +Pvec { Z } = PLvec { W } +On the other hand, we have by (130) and (132) +vec +� +ZT � += +Rvec +� +WT � += RQvec { W } . +This shows that for P T = P −1, +PL = RQ +and hence L = P T RQ +and hence (131) indeed holds. See Lemma 4.3.1 and Corollary 4.3.10 [23]. +First we rewrite the quadratic form as follows: for any matrix A = (aij) ∈ Rn, +������ +n +� +i=1 +n +� +j̸=i +⟨ Zi, Zj ⟩ aij +������ += +������ +n +� +i=1 +n +� +j̸=i +aij +p +� +k=1 +zikzjk +������ += +vec { Z }T �Avec { Z } = vec { W }T LT�ALvec { W } , +where �A is a block-diagonal matrix with diag( �A) = 0, and p identical blocks �Ak = offd(A), ∀k ∈ [p] +of size n × n along the main diagonal, where ∥offd(A)∥2 ≤ ∥A∥2 + ∥diag(A)∥2 ≤ 2 ∥A∥2. We now +compute +���LT �AL +��� +2 +≤ +����� +n +� +i=1 +Hi ⊗ diag(ei) +����� +2 +2 +��� �A +��� +2 ≤ max +i +∥Hi∥2 +2 +��� �A +��� +2 +where ∥L∥2 += +∥R∥2 = +����� +n +� +i=1 +diag(ei) ⊗ Hi +����� +2 +≤ max +i +∥Hi∥2 +���LT �AL +��� +F +≤ +∥L∥2 +2 +��� �A +��� +F ≤ max +i +∥Hi∥2 +2 +��� �A +��� +F ≤ max +i +∥Hi∥2 +2 +√p ∥A∥F , +48 + +where we use the property of block-diagonal matrix for R = �n +i=1 diag(ei) ⊗ Hi, which is also +known as a direct sum over Hi, i = 1, . . . , n. +Hence for any t > 0, by the Hanson-Wright inequality (Theorem G.3), +P +����vec { Z }T �Avec { Z } +��� > t +� += P +����vec { W }T LT�ALvec { W } +��� > t +� +≤ +2 exp +� +� +�−c min +� +� +� +t2 +C4 +0(maxi ∥Hi∥4 +2) +��� �A +��� +2 +F +, +t +C2 +0(maxi ∥Hi∥2 +2) +��� �A +��� +2 +� +� +� +� +� +� +≤ +2 exp +� +−c min +� +t2 +C4 +0(maxi ∥Hi∥4 +2)p ∥A∥2 +F +, +t +C2 +0(maxi ∥Hi∥2 +2) ∥A∥2 +�� +. +Thus (75) holds. +□ +I.5 +Proof of Lemma I.2 +We prove the lemma with the full generality by allowing each row vector to have its own covariance +Ai = HiHT +i , where Ai ∈ Rp×p, is the covariance for row vector Zj ∈ Rp for j ∈ Ci as shown in (23). +Now we also introduce the positive semidefinite matrix M := HT +i Hi ⪰ 0 ∈ Rm×m. First, we bound +the ℓ2 norm for each anisotropic vector ZT +j = W T +j HT +i ∈ Rp, where Wj ∈ Rm, and j ∈ Ci +∥Zj∥2 +2 += +⟨ HiWj, HiWj ⟩ = W T +j HT +i HiWj =: W T +j MWj +(133) +where +tr(HiHT +i ) += +tr(M) = ∥Hi∥2 +F ≤ (m ∧ p) ∥Hi∥2 +2 , +where Wj is an isotropic sub-gaussian random vectors with independent, mean-zero, coordinates, +and in (133), we use the isotropic property of Wj. Now clearly, +∥M∥F = +��HT +i Hi +�� +F ≤ ∥Hi∥2 ∥Hi∥F +and ∥M∥2 += +∥Hi∥2 +2 ; +(134) +Thus we have for any t > 0, +P +����∥Zj∥2 +2 − E ∥Zj∥2 +2 +��� > t +� += P +����∥HiWj∥2 +2 − E ∥HiWj∥2 +2 +��� > t +� += +P +����W T +j MWj − ∥Hi∥2 +F +��� > t +� +≤ 2 exp +� +−c min +� +t2 +C4 +0 ∥M∥2 +F +, +t +C2 +0 ∥M∥2 +�� +≤ +2 exp +� +−c min +� +t2 +C4 +0 ∥Hi∥2 +2 ∥Hi∥2 +F +, +t +C2 +0 ∥Hi∥2 +2 +�� +, +and hence we can also recover the result in (97) in case Hi = I. +□ +49 + +I.6 +Proof of Theorem 7.2 +First, we choose tdiag = C(maxi ∥Hi∥2 +2)C2 +0(√np ∨ n) and finish the calculations. First, we have +by Lemma I.2, +P +� +max +i +max +j∈Ci +���∥Zj∥2 +2 − E ∥Zj∥2 +2 +��� > tdiag +� +=: P (E0) = +≤ +2n exp +� +−c min +� +(maxi ∥Hi∥4 +2)np +maxi(∥Hi∥2 +2 ∥Hi∥2 +F ) +, (maxi ∥Hi∥2 +2)n +maxi(∥Hi∥2 +2) +�� +≤ +2n exp +� +−c min +� +np +(m ∧ p), n +�� +≤ 2 exp(−c′n), +where for the p × m matrix Hi, we have ∥Hi∥F ≤ √p ∧ m ∥Hi∥2. We use Theorem 7.3 to bound +the off-diagonal part. Hence for all q, h ∈ Sn−1, A(q, h) := offd(q ⊗ h) = (aij), +∥A(q, h)∥2 ≤ ∥A(q, h)∥F ≤ 1 +Let toffd = CoffdC2 +0(maxi ∥Hi∥2 +2)(√pn ∨ n). For a particular realization of q, h ∈ Sn−1 and A(q, h) = +(aij) as defined above, and Theorem 7.3, +P +� +� +������ +n +� +i=1 +n +� +j̸=i +⟨ Zi, Zj ⟩ qihj +������ +> toffd +� +� = P +� +� +n +� +i=1 +n +� +j̸=i +⟨ Zi, Zj ⟩ aij > toffd +� +� +≤ +2 exp +� +−c min +� +C4 +0(maxi ∥Hi∥4 +2)(√pn ∨ n)2 +C4 +0(maxi ∥Hi∥4 +2)p ∥A(q, h)∥2 +F +, C2 +0(maxi ∥Hi∥2 +2)(√pn ∨ n) +C2 +0(maxi ∥Hi∥2 +2) ∥A(q, h)∥2 +�� +≤ +2 exp +� +−cn min(C2 +1, C1) +� +≤ 2 exp(−cn) +for some sufficiently large constants C1 and c > 4 ln 9. Let N be as defined in Lemma G.1. +Taking a union bound over all |N|2 ≤ 92n pairs q, h ∈ N, the ε-net of Sn−1, we conclude that +P +� +� max +q,h∈N +������ +n +� +i=1 +n +� +j̸=i +qihj ⟨ Zi, Zj ⟩ +������ +> toffd +� +� =: P (E8) +≤ +|N|2 · 2 exp +� +−cn min(C2 +1, C1) +� +≤ 2 × 92n exp +� +−cn min(C2 +1, C1) +� +≤ +2 exp (−cn + 2n ln 9) = 2 exp (−c3n) +One can show that (74) holds by a standard approximation argument under Ec +8: we have +��offd(ZZT ) +�� +2 += +sup +q∈Sn−1 +n +� +i=1 +n +� +j̸=i +qiqj ⟨ Zi, Zj ⟩ ≤ +1 +(1 − 2ε) sup +q,h∈N +n +� +i=1 +n +� +j̸=i +qihj ⟨ Zi, Zj ⟩ +≤ +2(max +i +∥Hi∥2 +2)C1C2 +0(√np ∨ n) +(135) +See for example Exercise 4.4.3 [42]. Thus we have on event Ec +8 ∩ Ec +0, +��ZZT − E(ZZT ) +�� +2 +≤ +��diag(ZZT ) − Ediag(ZZT ) +�� +max + +��offd(ZZT ) +�� +2 +≤ +C′C2 +0(max +i +∥Hi∥2 +2)(√np ∨ n) +Moreover, we have P (Ec +8 ∩ Ec +0) ≥ 1 − 2 exp(−cn) upon adjusting the constants. +□ +50 + +References +[1] Abbe, E. (2018). 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Ph.D. thesis, Carnegie Mellon +University, Pittsburgh, PA. +53 + diff --git a/ItAyT4oBgHgl3EQffvgZ/content/tmp_files/load_file.txt b/ItAyT4oBgHgl3EQffvgZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..714dc2b1243d66c0024e77125cfc0b9380d29e82 --- /dev/null +++ b/ItAyT4oBgHgl3EQffvgZ/content/tmp_files/load_file.txt @@ -0,0 +1,1889 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf,len=1888 +page_content='Semidefinite programming on population clustering: a global analysis Shuheng Zhou University of California, Riverside, CA 92521 Abstract In this paper, we consider the problem of partitioning a small data sample of size n drawn from a mixture of 2 sub-gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Our work is motivated by the application of clustering individuals according to their population of origin using markers, when the divergence between the two populations is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' We are interested in the case that individual features are of low average quality γ, and we want to use as few of them as possible to correctly partition the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' We consider semidefinite relaxation of an integer quadratic program which is formulated essentially as finding the maximum cut on a graph where edge weights in the cut represent dissimilarity scores between two nodes based on their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' A small simulation result in Blum, Coja-Oghlan, Frieze and Zhou (2007, 2009) shows that even when the sample size n is small, by increasing p so that np = Ω(1/γ2), one can classify a mixture of two product populations using the spectral method therein with success rate reaching an “oracle” curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' There the “oracle” was computed assuming that distributions were known, where success rate means the ratio between correctly classified individuals and the sample size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' In this work, we show the theoretical underpinning of this observed concentration of measure phenomenon in high dimensions, simultaneously for the semidefinite optimization goal and the spectral method, where the input is based on the gram matrix computed from centered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' We allow a full range of tradeoffs between the sample size and the number of features such that the product of these two is lower bounded by 1/γ2 so long as the number of features p is lower bounded by 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' 1 Introduction We explore a type of classification problem that arises in the context of computational biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' The problem is that we are given a small sample of size n, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=', DNA of n individuals (think of n in the hundreds or thousands), each described by the values of p features or markers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=', SNPs (Single Nucleotide Polymorphisms, think of p as an order of magnitude larger than n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Our goal is to use these features to classify the individuals according to their population of origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Features have slightly different probabilities depending on which population the individual belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Denote by ∆2 the ℓ2 2 distance between two population centers (mean vectors), namely, µ(1), µ(2) ∈ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' We focus on the case where p > n, although it is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Note that ∆ measures the Euclidean distance between µ(1) and µ(2) and thus represents their separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' The objective we consider is to minimize the total data size D = np needed to correctly classify 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content='00344v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content='ST] 1 Jan 2023 the individuals in the sample as a function of the “average quality” γ of the features: γ := ∆2/p, where ∆2 := p � k=1 (µ(1) k − µ(2) k )2 and µ(i) = (µ(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' , µ(i) p ) ∈ Rp, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' (1) Suppose we are given a data matrix X ∈ Rn×p with samples from two populations C1, C2, such that ∀i ∈ Cg, E(Xij) = µ(g) j g = 1, 2, ∀j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' (2) Our goal in the present work is to estimate the group membership vector ¯x ∈ {−1, 1}n such that ¯xi = 1 for i ∈ C1 and ¯xi = −1 for i ∈ C2, (3) where the sizes of clusters |Cj| =: nj, ∀j may not be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Our ultimate goal is to estimate the solution to the discrete optimization problem: maximize xT ¯Ax subject to x ∈ {−1, 1}n (4) where ¯A is a static reference matrix to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' It was previously shown that, in expectation, among all balanced cuts in the complete graph formed among n vertices (sample points), the cut of maximum weight corresponds to the correct partition of the n points according to their distributions in the balanced case (n1 = n2 = n/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Here the weight of a cut is the sum of weights across all edges in the cut, and the edge weight equals the Hamming distance between the bit vectors of the two endpoints [11, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Under suitable conditions, the statement above also holds with high probability (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' In other words, in the context of population clustering, it has been previously shown one can use a random instance of the integer quadratic program: maximize xT Ax subject to x ∈ {−1, 1}n (5) to identify the correct partition of nodes according to their population of origin w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=', so long as the data size D is sufficiently large and the separation metric is at the order of ∆2 = Ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' The analyses focused on the high dimensional setting, where p ≫ n [11, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Here A = (aij) is an n × n symmetric matrix where for 1 ≤ i, j ≤ n, aij = aji denotes the edge weight between nodes i and j, computed from the individuals’ bit vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' This result is structural, rather than algorithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' The integer quadratic program (4) (or (5)) is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' In a groundbreaking paper [21], Goemans and Williamson show that one can use semidefinite program (SDP) as relaxation to solve these approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' In this paper, we propose a semidefinite relaxation framework, inspired by [22], where we design and analyze computational efficient algorithms to partition data into two groups approximately according to their population of origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' More generally, one may consider semidefinite relaxations for the following sub-gaussian mixture model with k centers (implicitly, with rank-k mean matrix embedded), where Xi = µ(ψi) + Zi (6) where Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' , Zn ∈ Rp are independent, sub-gaussian, mean-zero, random vectors and ψi : i → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' , k} assigns node i to a group Cj with the mean µ(j) ∈ Rp for some j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Here we denote by 2 [k] the set of integers {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Here, each row vector of X is a p-dimensional sub-gaussian random vector and we assume rows are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' We will consider a flexible model of parametrization for Zj, j ∈ [n] in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' In particular, we allow each population to have distinct covariance structures, with diagonal matrices as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' The analysis framework for the semidefinite relaxation by Gu´edon and Vershynin [22] was set in the context of community detection in sparse networks, where A represents the adjacency matrix of a random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' In other words, they study the semidefinite relaxation of the integer program (5), where an n×n symmetric random adjacency matrix A (observed) is used to replace the hidden static matrix ¯A in the original problem (4) such that E(A) = ¯A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' The innovative proof strategy of [22] is to apply the Grothendieck’s inequality for the random error A − ¯A rather than the original matrix A as considered in the earlier literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' We call this approach the global analysis, following [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' With proper adjustments, we apply this methodology to our settings and prove the first main Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content='5 regarding the partial recovery of the group memberships based on n sequences of p features following the mean model (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' The important distinction is: here, we replace the random adjacency matrix A arising from stochastic block models as considered in [22] with an instance of symmetric matrix A, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' (9), computed from the centered data matrix which we now elaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Let diag(A) and offd(A) be the diagonal and the off-diagonal part of matrix A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' We propose the following estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' As in many statistical problems, one simple but crucial step is to first obtain the centered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Let 1n = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' , 1] ∈ Rn denote a vector of all 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Let X be a data matrix with row vectors X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' , Xn as defined in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' Denote by Y = X − 1n ⊗ �µn = X − P1X, where P1 = 1 n1n1T n and (7) �µn = 1 n � i Xi is the average over n random vectors in Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' (8) Loosely speaking, this procedure is called “global centering” in the statistical literature, for example, see [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQffvgZ/content/2301.00344v1.pdf'} +page_content=' To estimate the group membership vector ¯x ∈ {−1, 1}n, we use the following adjusted A: A := Y Y T − λ(En − In), where λ = 2 n(n − 1) � i 1; +• exp: exponentiation, y �→ exp(y) for y ∈ R; +• log: natural logarithm, y �→ log(y) for y ∈ R>0; +• entropy: entropy, y �→ +� +−y log(y), +if y > 0, +0, +if y = 0, for y ∈ R≥0; +• sin: sine, y �→ sin(y) for y ∈ R; +• cos: cosine, y �→ cos(y) for y ∈ R; +• abs: absolute value, y �→ |y| for y ∈ R. +In previous versions of SCIP, also high-level structures such as quadratic functions +could be represented as expression types. To avoid ambiguity and reduce complexity, +this has been replaced by a recognition of quadratic expressions that is no longer made +explicit by a change in the expression type. +2.1.2 +Constraint Handler for Nonlinear Constraints +All nonlinear constraints g ≤ g(x) ≤ g of (MINLP) are handled by the constraint +handler for nonlinear constraints in SCIP, while the linear constraints b ≤ Ax ≤ b +are handled by the constraint handlers for linear constraints and its specializations +(e.g., knapsack, set-covering). A constraint handler is responsible for checking whether +solutions satisfy constraints and, if that is not the case, to resolve infeasibility by +enforcing constraints. This applies in particular to solutions of the LP relaxation. The +nonlinear constraint handler currently enforces constraints by the following means: +DOMAINPROP by analyzing the nonlinear constraints with respect to the variable +bounds at the current node of the branch-and-bound tree, infeasibility or a bound +tightening may be deduced, which allow pruning the node or cutting off the given +solution, respectively; this is also known as domain propagation; +SEPARATE a cutting plane that is violated by the given solution may be computed; +BRANCH the current node of the branch-and-bound tree is subdivided, that is, a +variable xi and a branching point ˜xi ∈ [xi, xi] are selected and two child nodes +with xi restricted to [xi, ˜xi] and [˜xi, xi], respectively, are created. +To decide whether a node can be pruned (DOMAINPROP), an overestimate of the +range of g(x) with respect to current variable bounds is computed by means of interval +arithmetics [53]. If a constraint k is found such that gk([x, x]) ∩ [gk, gk] = ∅, then there +exists no point in [x, x] for which this constraint is feasible. A bound tightening may be +computed by applying the same methods in reverse order. That is, interval arithmetic is +used to overestimate g−1([g, g]), the preimage of g(x) on [g, g], and variable bounds are +tightened to [x, x] ∩ g−1([g, g]). This is also known as feasibility-based bound tightening +(FBBT). In the simplest case, callbacks of expression handlers are used to propagate +intervals through expressions. However, in some cases, other methods that take more +structure into account or that use additional information to tighten variable bounds +and constraint sides are used (see, e.g., Sections 2.3.1 and 2.3.2). +To construct a linear relaxation of the nonlinear constraints (SEPARATE option), +5 + +an extended formulation is considered: +min c⊤x, +such that hi(x, wi+1, . . . , w ˆm) ⋚i wi, +i = 1, . . . , ˆm, +b ≤ Ax ≤ b, +x ≤ x ≤ x, +w ≤ w ≤ w, +xI ∈ Z|I|. +(MINLPext) +The functions hi are obtained from the expressions that define functions gi by recursively +annotating subexpressions with auxiliary variables wi+1, . . . , w ˆm for some ˆm ≥ m. +Initially, slack variables w1, . . . , wm are introduced and assigned to the root of all +expressions, i.e., hi := gi, wi := gi, wi := gi, for i = 1, . . . , m. Next, for each function +hi, subexpressions f may be assigned new auxiliary variables wi′, i′ > m, which results +in extending (MINLPext) by additional constraints hi′(x) = wi′ with hi′ := f. Bounds +wi′ and wi′ are initialized to bounds on hi′, if available. Since auxiliary variables in a +subexpression of hi always receive an index larger than max(m, i), the result is referred +to by hi(x, wi+1, . . . , w ˆm) for any i = 1, . . . , ˆm. That is, to simplify notation, wi+1 is +used instead of wmax(i,m)+1. If a subexpression appears in several expressions, it is +assigned at most one auxiliary variable and reindexing may be necessary to have hi +depend on x and wi+1, . . . , w ˆm only. +For the (in)equality sense ⋚i, a valid simplification would be to assume equality +everywhere. For performance reasons, though, it can be beneficial to relax certain +equalities to inequalities if that does not change the feasible space of (MINLPext) when +projected onto x. Therefore, +⋚i := +� +� +� +� +� +=, +if gi > −∞, gi < ∞, +≤, +if gi = −∞, gi < ∞, +≥, +if gi > −∞, gi = ∞, +for i = 1, . . . , m. +For i > m, monotonicity of expressions is taken into account to derive ⋚i. +Whether to annotate a subexpression by an auxiliary variable depends on the +structures that are recognized. In the simplest case, every subexpression that is not +already a variable is annotated with an auxiliary variable. This essentially corresponds +to the Smith Normal Form [65]. For every function hi of (MINLPext), the callbacks +of the corresponding expression handler can be used to compute linear under- and +overestimators, such that a linear relaxation for (MINLPext) is constructed. It can, +however, be beneficial to not add an auxiliary variable for every subexpression, thus +allowing for more complex functions in (MINLPext). This will be the discussed in +Section 2.1.3 below. +Example +Recall Figure 1 and the constraint +log(x)2 + 2 log(x) y + y2 ≤ 4. +By annotating the root of the expression graph with a slack variable w1 and each other +non-variable node with an auxiliary variable, the extended formulation +w2 + 2w3 + w4 ≤ w1, +w2 +5 ≤ w2, +w5 y ≤ w3, +y2 ≤ w4, +log(x) = w5, +w1 ≤ 4. +6 + +is obtained. Bounds on auxiliary variables have been omitted here. The constraints +w2 +5 = w2, w5y = w3, and y2 = w4 were relaxed to inequalities because w2 + 2w3 + w4 is +monotonically increasing in each variable. However, to relax log(x) = w5 to log(x) ≤ w5, +both w2 +5 and w5y would need to be monotonically increasing in w5. This would be the +case if x ≥ 1 and y ≥ 0. +If a constraint hi(x, wi+1, . . . , w ˆm) ≤ wi (the ≥-case is analogous) of (MINLPext) is +violated and hi is nonconvex, then linear underestimators on hi can only be as tight as +the convex envelope of hi. Therefore, it may not be possible find a hyperplane that is +violated by the solution of the LP relaxation. Since the convex envelope of hi depends +on the bounds of variables appearing in hi, these variables are candidates for branching +(BRANCH). More precisely, when an expression handler computes a linear under- or +overestimator for hi(x, wi+1, . . . , w ˆm), it also signals for which variables it used current +variable bounds. Marked original variables are then added to the list of branching +candidates. For an auxiliary variable wi′, i′ > i, the variables in the subexpression that +hi′ represents are considered for branching instead. +The decision on whether to add a cutting plane that separates the solution of the LP +relaxation or to branch is rather complex, but the idea is to branch if either no cutting +plane is found or if the violation of available cutting planes in the relaxation solution is +rather small when compared to the convexification gap of the under/overestimators +that define the cutting planes. In the latter case, it may be beneficial to first reduce +the convexification gap by branching. To select one variable from the list of branching +candidates, the violation of constraints in (MINLPext) and historical information about +the effect of branching on a given variable on the optimal value of the LP relaxation +(“pseudo costs”) are taken into account. The branching point is a convex combination +of the value of the variable in the LP relaxation and the mid-point of the variable’s +interval. +2.1.3 +Nonlinear Handlers +In the previous example, four auxiliary variables were introduced to construct the +extended formulation. This is due to the expression handlers having a rather myopic +view, basically, implementing techniques that can handle only their direct children. +It is clear that, for this example, an extended formulation that only replaces log(x) +by an auxiliary variable w2 could be more efficient to solve. However, this requires +methods to detect the quadratic (or convex) structure and to either compute linear +underestimators for the quadratic (convex) expression w2 +2 + 2w2y + y2 or to separate +cutting planes for the set defined by w2 +2 + 2w2y + y2 ≤ w1. +Such structure detection and handling methods are the task of the new nonlinear +handler plugins that were introduced with SCIP 8. Nonlinear handlers determine the +extended formulation (MINLPext) by deciding when to annotate subexpressions with +auxiliary variables. That is, given a constraint hi(x) ⋚i wi, a nonlinear handler analyses +the expression that defines hi and attempts to detect specific structures. At this point, +it may also request to introduce additional auxiliary variables, thus changing hi(x) +into hi(x, wi+1, . . . , w ˆm). In addition, it informs the constraint handler that it will +now provide separation for hi(x, wi+1, . . . , w ˆm) ≤ wi, or ≥ wi, or both. If none of the +nonlinear handlers declare that they will handle hi(x) ⋚i wi, auxiliary variables are +introduced for each argument of the root of the expression hi and expression handler +callbacks are used to construct cutting planes from linear under-/overestimators. +In addition to separation, nonlinear handlers can also contribute to domain prop- +agation. This is implemented analogously to separation by setting up an additional +extended formulation similarly to (MINLPext), with the main difference that slack and +auxiliary variables are not actually created in SCIP and equalities are currently not +relaxed to inequalities. +7 + +Note that the extended formulations are stored as annotation on the original +expressions. Thus, for each task, the most suitable formulation can be used. For +example, feasibility is checked on the original constraints, domain propagation and +separation use the corresponding extended formulations, but branching is performed, +by default, with respect to original variables only. With SCIP 7 and earlier, only one +extended formulation was constructed explicitly and the connection to the original +formulation was no longer available, leading to issues due to not ensuring that solutions +are also (ε-)feasible for the original constraints. +In addition to the improved numeric reliability, the nonlinear handlers also allow for +a higher flexibility when handling nonlinear structures. For each node in an expression, +more than one nonlinear handler can be attached, each one annotating possibly different +subexpressions with auxiliary variables. +For example, for a nonconvex quadratic +constraint � +i,j ai,jxixj ≤ w, the nonlinear handler for quadratics can declare that it +will provide separation (by intersection cuts, see Section 2.3.5), but that also other +means of separation should be tried. However, since no other nonlinear handler declares +that it will provide separation, auxiliary variables are introduced for each argument +of the sum, that is, an auxiliary variable Xij is assigned to each product xixj. For +the corresponding constraints xixj ≤ Xij (if ai,j ≥ 0), the well-known McCormick +underestimators [49], +Xij ≥ xixj + xjxi − xixj, +Xij ≥ xixj + xjxi − xixj, +(1) +or other means (see Section 2.3.2) will be used to construct a linear relaxation. +2.1.4 +NLP Relaxation +Similar to the central LP relaxation of SCIP, an NLP relaxation is also available. In +contrast to constraint handlers, the NLP relaxation uses a common data structure to +store its constraints. At the moment, constraint handlers for linear constraints and the +constraint handler for nonlinear constraints store a representation of their constraints +in the NLP relaxation, so that in case of a MINLP, the NLP relaxation together with +the integrality conditions on variables provides a unified view of the problem. For +nonlinear constraints, the original (non-extended) form g ≤ g(x) ≤ g is added to the +NLP. To find local optimal solutions for the NLP relaxation, interfaces to the NLP +solvers filterSQP, Ipopt, and Worhp [27, 75, 20] are available. First- and second-order +derivatives for these solvers are computed via CppAD [8]. +The NLP relaxation is mainly used by some primal heuristics (Section 2.8) and +separators (Section 2.4.2) at the moment. +2.2 +Presolving +When presolving nonlinear constraints, expressions are simplified and brought into a +canonical form. For example, recursive sums and products are flattened and fixed or +aggregated variables are replaced by constants or sums of active variables. In addition, +it is ensured that if a subexpression appears several times (in the same or different +constraints), always the same expression object is used. This ensures that in the +extended formulation (MINLPext) at most one auxiliary variable is attached to such +common subexpressions. +2.2.1 +Variable Fixings +Similar to what has been shown by Hansen et al. [38], if a bounded variable xj does +not appear in the objective (cj = 0), but in exactly one constraint gk ≤ gk(x) ≤ gk +where gk(x) is convex in xj for any fixing of other variables and gk = +∞ (or concave +in xj and gk = −∞), then there always exists an optimal solution where xj ∈ {xj, xj}. +8 + +For example, if y ∈ [0, 1] appears only in a constraint xy + yz − y2 ≤ 5, then y can be +changed to a binary variable. +SCIP recognizes such variables for polynomial constraints (under additional assump- +tions [14]) and changes the variable type to binary, if xj = 0 and xj = 1, or adds a +bound disjunction constraint xj ≤ xj ∨ xj ≥ xj. As a consequence, branching on xj +leads to fixing the variable in both children. +2.2.2 +Linearization of Products +The introduction emphasized that with SCIP 8, an explicit extended reformulation of +nonlinear constraints is avoided. An exception that proves this “rule” is the linearization +of products of binary variables in presolving. Doing so has the advantage that more of +SCIP’s techniques for MIP solving can be utilized. +In the simplest case, a product � +i xi is replaced by a new variable z and a constraint +of type “and” that models z = � +i xi is added. The “and”-constraint handler will then +separate a linearization of this product [11]. For a product of only two binary variables, +the linearization is added directly. +For a quadratic function in binary variables with many terms, the number of +variables introduced may be large. Thus, in this case, a linearization that requires fewer +additional variables is used, even though it may lead to a weaker relaxation. +2.2.3 +KKT Strengthening for QPs +A presolving method that aims to tighten the relaxation of a quadratic program (QP) +by adding redundant constraints derived from Karush-Kuhn-Tucker (KKT) conditions +is available. Consider a quadratic program of the form +min +1 +2 x⊤Qx + c⊤x, +such that Ax ≤ b, +(QP) +where Q ∈ Rn×n is symmetric, c ∈ Rn, A ∈ Rm×n, and b ∈ Rm. If (QP) is bounded, +then all optima of (QP) satisfy the following KKT conditions: +Qx + c + A⊤µ = 0, +Ax ≤ b, +µi(Ax − b)i = 0, +i ∈ {1, . . . , m}, +µ ≥ 0, +(KKT) +where µ is the vector of Lagrangian multipliers of the constraints Ax ≤ b. +In a specialized presolver, SCIP recognizes whether (MINLP) is equivalent to (QP) by +checking whether a quadratic objective function has been reformulated into a constraint. +If a (QP) has been found and all variables are bounded, then the equations (KKT) +are added as redundant constraints to the problem, whereby the complementarity +constraints are formulated via special ordered sets of type 1. The redundant constraints +can help to strengthen the linear relaxation and prioritize branching decisions to satisfy +the complementarity constraints, which focuses the search more on the local optima +of (QP). +In addition to (QP), the implementation can also handle mixed-binary quadratic +programs. For all details, see [47, 26]. When this presolver was added to SCIP 4.0, it +has shown to be very beneficial for box-constrained quadratic programs. Due to the +many changes and extensions in SCIP 8, in particular for the handling of quadratic +constraints (Section 2.3), it needs to be reevaluated under which conditions this presolver +should be enabled. Currently, it is disabled by default. +9 + +2.2.4 +Symmetry Detection +Symmetries in a MINLP are automorphisms on Rn that map optimal solutions to optimal +solutions. Such symmetries have an adverse effect on the performance of branch-and- +bound solvers, because symmetric subproblems may be treated repeatedly. Therefore, +SCIP can enforce lexicographically maximal solutions from an orbit of symmetric +solutions via bound tightening and separation of linear inequalities [39, 34, 31, 14]. +Since optimal solutions are naturally not known in advance, the symmetry detection +resorts to find permutations of variables that map the feasible set onto itself and map +each point to one with the same objective function value [48]. These permutations are +given by isomorphisms in an auxiliary symmetry detection graph, which is constructed +from the problem data (e.g., c, A, I, and the expressions that define g(x)) [43, 76]. +2.3 +Quadratics +Since quadratic functions frequently appear in MINLPs (every second instance of +MINLPLib [50] has only linear and quadratic constraints), a number of techniques have +been added to SCIP to handle this structure. Next to the presolving methods that were +discussed in the previous section, three nonlinear handlers and four separators deal with +quadratic structures. When none of the nonlinear handlers are active, then for each +square and bilinear term in a quadratic function, an auxiliary variable is added in the +extended formulation and gradient, secant, and McCormick under- and overestimators +(see (1)) are generated. +2.3.1 +Domain Propagation +If variables appear more than once in a quadratic function, then a term-wise domain +propagation does not necessarily yield the best possible results, due to suffering from +the so-called dependency problem of interval arithmetics. For example, it is easy to +compute the range for x2 + x for given bounds on x, or bounds on x for a given interval +on x2 + x, but standard interval arithmetics would treat the terms x2 and x separately, +which can lead to overestimating the result. +Therefore, a specialized nonlinear handler in SCIP provides a domain propagation +procedure for quadratics that aims to reduce overestimation. For this, the detection +routine of the nonlinear handler writes a quadratic expression as +q(y) = +k +� +i=1 +qi(y) +with +qi(y) = aiy2 +i + ciyi + +� +j∈Pi +bi,jyiyj, +(2) +where yi is either an original variable (x) or another expression, ai, ci ∈ R, bi,j ∈ R\{0}, +j ∈ Pi ⇒ i ̸∈ Pj for all j ∈ Pi, Pi ⊂ {1, . . . , k}, i = 1, . . . , k. For functions qi with +at least two terms (at least two of ai, bi,j, j ∈ Pi, and ci are nonzero), a relaxation +is obtained by replacing each yj by [yj, yj], j ∈ Pi. For this univariate quadratic +interval-term in yi, tight bounds can be computed [24]. +In addition, bounds on variables yj, j ∈ Pi, are computed by considering +� +j∈Pi +bi,jyj ∈ ([q, q] − +� +i′̸=i +qi′(y))/yi − aiyi − ci, +yi ∈ [yi, yi], +(3) +where [q, q] are given bounds on q(y). After relaxing each qi′ to an interval, bounds on +the right-hand side of (3) are computed, which are then used to calculate bounds on +each yj, j ∈ Pi. +10 + +2.3.2 +Bilinear Terms +For a product y1y2, where y1 and y2 are either non-binary variables or other expressions, +the expression handler for products already provides linear under- and overestimators +and domain propagation that is best possible when considering the bounds [y1, y1] × +[y2, y2] only. However, if linear inequalities in y1 and y2 are available, then possibly +tighter linear estimates and variable bounds can be computed. In SCIP, this is done by +a specialized nonlinear handler that implements the algorithm by Locatelli [45]. The +inequalities are found by projection of the LP relaxation onto variables (y1, y2). For +more details, see [54]. An alternative method that uses linear constraints to tighten the +relaxation of quadratic constraints are the RLT cuts described in the following. +2.3.3 +RLT Cuts +The Reformulation-Linearization Technique (RLT) [2, 3] has proven very useful to +tighten relaxations of polynomial programming problems. In SCIP, a separator of cuts +that are computed via RLT for bilinear product relations in (MINLPext) is available. +For simplicity, denote by Xij the auxiliary variable that is associated with a constraint +xixj ⋚ Xij of (MINLPext) (Xji denotes the same variable as Xij). Recall that it is +valid to replace ⋚ by =. Given Xij = xixj, where xi ∈ [xi, xi], xj ∈ [xj, xj], and a +linear constraint a⊤x ≤ b, RLT cuts are derived by first multiplying the constraint by +a nonnegative bound factors (xi − xi), (xi − xi), (xj − xj), or (xj − xj). For instance, +consider multiplication by the factor (xi − xi), which yields a valid nonlinear inequality: +a⊤x (xi − xi) ≤ b (xi − xi). +(4) +This is referred to as the reformulation step. +The linearization step is then performed for all terms xkxi in (4). If a product +relation Xki = xkxi exists, then the product is replaced with Xki. If xk and xi are +contained in the same clique, the product is replaced with an equivalent linear expression. +Otherwise, it is replaced by a linear under- or overestimator such as (1). +In addition, the RLT separator can reveal linearized products between binary and +continuous variables. To do so, it checks whether pairs of linear inequalities that are +defined in the same triple of variables (one of them binary, the other two continuous) +imply a product relation. These implicit products can then be used in the linearization +step of RLT cut generation [16]. +2.3.4 +SDP Cuts +As in the previous section, denote by Xij the auxiliary variable that is associated with +a constraint xixj ⋚ Xij of (MINLPext). A popular convex relaxation of the condition +X = xx⊤ is given by requiring X − xx⊤ to be positive semidefinite. Separation for +the set {(x, X) : X − xx⊤ ⪰ 0} itself is possible, but cuts are typically dense and may +include variables Xij for products that do not exist in the problem. Therefore, only +principal 2 × 2 minors of X − xx⊤, which also need to be positive semidefinite, are +considered. By Schur’s complement, this means that the condition +Aij(x, X) := +� +� +1 +xi +xj +xi +Xii +Xij +xj +Xij +Xjj +� +� ⪰ 0 +(5) +needs to hold for any i, j, i ̸= j. A separator in SCIP detects minors for which Xii, Xjj, +Xij exist in (MINLPext) and enforces Aij(x, X) ⪰ 0. To do so for a solution (ˆx, ˆX) that +violates (5), an eigenvector v ∈ R3 of Aij(ˆx, ˆX) with v⊤Aij(ˆx, ˆX)v < 0 is computed +and the globally valid linear inequality v⊤Aij(x, X)v ≥ 0 is added. +11 + +2.3.5 +Intersection Cuts +Intersection cuts [70, 5] have shown to be an efficient tool to strengthen relaxations +of MIPs. Recently, Mu˜noz and Serrano showed how to compute the tightest possible +intersection cuts for quadratic programs [55]. This method has been implemented in +SCIP [21]. +Assume a nonconvex quadratic constraint of (MINLPext) is q(y) ≤ w with q(y) as +in (2) and w an auxiliary variable. The separation of intersection cuts is implemented +for the set S := {(y, w) ∈ Rk : q(y) ≤ w} that is defined by this constraint. +Let (ˆy, ˆw) be a basic feasible LP solution violating q(y) ≤ w. First, a convex +inequality g(y, w) < 0 is build that is satisfied by (ˆy, ˆw), but by no point of S. This +defines a so-called S-free set C = {(y, w) ∈ Rk+1 : g(y, w) ≤ 0}, that is, a convex set +with (ˆy, ˆw) ∈ int(C) containing no point of S in its interior. The quality of the resulting +cut highly depends on which S-free set is used, but using maximal S-free sets yield the +tightest possible intersection cuts [55]. +By using the conic relaxation K of the LP-feasible region defined by the nonbasic +variables at (ˆy, ˆw), the intersection points between the extreme rays of K and the +boundary of C are computed. The intersection cut is then defined by the hyperplane +going through these points and successfully separates (ˆx, ˆw) and S. See Figure 2 for +an illustration. To obtain even better cuts, there is also a strengthening procedure +implemented that uses the idea of negative edge extension of the cone K [36]. +S +C +K +( ̂y, ̂w) +Figure 2: An intersection cut (red) separating the basic feasible LP solution (ˆy, ˆw) from S (blue). +The cut is computed using the intersection points of an S-free set C (orange) and the rays of a +simplicial cone K ⊇ S (boundary in green) with apex (ˆy, ˆw) ̸∈ S. +In addition to the separation of intersection cuts for a set S given by a constraint +q(y) ≤ w, SCIP can also generate intersection cuts for implied quadratic equations. +Recall the matrix of auxiliary variables X as introduced in Section 2.3.3. The con- +dition X = xx⊤ implies that X needs to have rank 1. Therefore, any 2 × 2 minor +�Xi1j1 +Xi1j2 +Xi2j1 +Xi2j2 +� +of X needs to have determinant zero. That is, for any set of variable +indices i1, i2, j1, j2 with i1 ̸= i2 and j1 ̸= j2, the condition Xi1j1Xi2j2 = Xi1j2Xi2j1 +needs to hold. If all variables in this condition exist in (MINLPext) and a solution +violates this condition, then the previously described procedure to generate intersection +cuts is applied to the set defined by this condition. +Since intersection cuts can be rather dense, it is not clear yet how to decide when it +will be beneficial to generate such cuts. Their separation is therefore currently disabled +by default. For more details, see [21]. +12 + +2 +0 +0 +2 +42.3.6 +Edge-Concave Cuts +Another method to obtain a linear outer-approximation for a quadratic constraint is by +utilizing an edge-concave decomposition of the quadratic function. This has shown to +be particularly useful for randomly generated quadratic instances [51, 52]. A function +is edge-concave over the variables’ domain (e.g., [x, x]) if it is componentwise concave. +Given a quadratic function, the separator for edge-concave cuts solves an auxiliary +MIP to partition the square and bilinear terms into a sum of edge-concave functions +and a remaining function. Since the convex envelope of edge-concave functions is +vertex-polyhedral [67], that is, it is a polyhedral function with vertices corresponding +to the vertices of the box of variable bounds, facets on the convex envelope of each +edge-concave function can be computed by solving an auxiliary linear program (see also +Section 2.4.1). For the function of remaining terms, term-wise linear underestimators +such as (1) are summed up. +Since the current implementation of edge-concave cuts in SCIP has not shown to be +particularly useful for general MINLP, this separator is disabled for now. +2.3.7 +Second-Order Cones +An important connection between MINLP and conic programming is the detection of +constraints that can be represented as a second-order cone (SOC) constraint, since the +latter defines a convex set, while the original constraint may use a nonconvex constraint +function. +A specialized nonlinear handler aims to detect SOC representable structures. In the +detection phase, a constraint hi(x) ≤ wi (the case ≥ is handled similarly) of the extended +formulation (MINLPext) is passed to the nonlinear handler. For this constraint, it is +checked whether it defines a bound on an Euclidian norm ( +��k +j=1(ajy2 +j + bjyj) + c ≤ +wi for some coefficients aj, bj, c ∈ R, aj > 0, where yj is either an original variable or +some subexpression of hi(·)), or is a quadratic constraint that is SOC-representable [46]. +Since the introduction of slack variables wi, i ≤ m, may prevent such a detection, the +equivalent constraint hi(x) ≤ ¯wi is considered instead. +A detected SOC constraint is stored in the form +� +� +� +� +k +� +j=1 +(v⊤ +j y + βj)2 ≤ v⊤ +k+1y + βk+1 +(6) +with vj ∈ Rℓ, j = 1, . . . , k + 1, where y1, . . . , yℓ are variables of (MINLPext). Since +the left-hand side of (6) is convex, a solution ˆy that violates (6) can be separated by +linearization of the left-hand side of (6). +However, if there are many terms on the left-hand side of (6) (k being large), then it +can require many cuts to provide a tight linear relaxation of (6). Thus, a disaggregation +of the cone [72] is used if k ≥ 3: +(v⊤ +j y + βj)2 ≤ zj(v⊤ +k+1y + βk+1), +j = 1, . . . , k, +(7) +k +� +j=1 +zj ≤ v⊤ +k+1y + βk+1, +(8) +where variables z1, . . . , zk are new variables. A solution (ˆy, ˆz) that violates (6) needs +to violate also (7) for some j ∈ {1, . . . , k} or (8). The latter is already linear and can +be added as a cut. If a rotated second-order cone constraint (7) is violated for some j, +then it is transformed into the standard form +� +4(v⊤ +j y + βj)2 + (v⊤ +k+1y + βk+1 − zj)2 ≤ v⊤ +k+1y + βk+1 + zj +and a gradient cut is constructed by linearization of the left-hand side. +13 + +2.4 +Convexity +2.4.1 +Convex and Concave Constraints +For the linear underestimation of functions like x exp(x) or x2 + 2xy + y2, the construc- +tion of an extended formulation (xw, exp(x) = w; w1 + 2w2 + w3, w1 = x2, w2 = xy, +w3 = y2) is not advisable. Instead, hyperplanes that support the epigraph of a convex +function can be used if convexity is recognized. In SCIP, specialized nonlinear handlers +are available to detect for a function hi(x) of (MINLPext) the subexpressions that need +to be replaced by auxiliary variables wi+1, . . . , w ˆm such that the remaining expression +hi(x, wi+1, . . . , w ˆm) is convex or concave. The detection utilizes the often applied +rules for convexity/concavity of function compositions (e.g., f convex and monotone +decreasing, g concave ⇒ f ◦ g convex), but applies them in reverse order. That is, +instead of deciding whether a function is convex/concave based on information on the +convexity/concavity and monotonicity of its arguments, the algorithm formulates condi- +tions on the convexity/concavity of the function arguments given a convexity/concavity +requirement on the function itself. When a condition on an argument cannot be fulfilled, +it is replaced by an auxiliary variable. +Next to “myopic” rules for convexity/concavity that are implemented by the expres- +sion handlers, also rules for product compositions (af(bg(x) + c)g(x) with constants +a, b, c and repeating subexpression g(x)), signomials (c �k +j=1 f pj +j (x) with c, pj ∈ R and +subexpressions fj(x), j = 1, . . . , k), and quadratic forms are available. The latter may +check for definiteness of its Hessian by calculating its eigenvalues. Further, it has +been shown that for a composition of convex functions f ◦ g, it can be beneficial for +the linear relaxation to consider the extended formulation f(w), w ≥ g(x), instead of +the composition f(g(x)) [68]. This is enforced by a small variation of the detection +algorithm. +When a convex constraint hi(x, wi+1, . . . , w ˆm) ≤ wi of (MINLPext) is violated at a +point (ˆx, ˆw), a tangent on the graph of hi at (ˆx, ˆw) is used to compute a separating +hyperplane. The slope of the tangent is given by the gradient of hi at (ˆx, ˆw), which +is calculated via automatic differentiation on the expression graph. If, however, hi is +univariate, that is, hi(x, wi+1, . . . , w ˆm) = f(y) for some variable y, and y is integral, +then taking the hyperplane through the points (⌊ˆy⌋, f(⌊ˆy⌋)) and (⌊ˆy⌋ + 1, f(⌊ˆy⌋ + 1)) +can give a tighter underestimator. +For a concave function hi(x, wi+1, . . . , w ˆm), any hyperplane αx + βw + γ that +underestimates hi(x, wi+1, . . . , w ˆm) in all vertices of the box [x, x] × [wi+1, wi+1] × +· · · × [w ˆm, w ˆm] is a valid linear underestimator, since hi is vertex-polyhedral with +respect to the box. Maximizing αˆx + β ˆw + γ such that αx + βw + γ does not exceed +hi(x, wi+1, . . . , w ˆm) for all vertices gives an underestimator that is as tight as possible +at a given reference point (ˆx, ˆw). For the frequent cases k = 1 and k = 2, routines that +directly compute such an underestimator are available. For k > 2, a linear program +is solved. Since the size of this LP is exponential in k, underestimators for concave +functions in more than 14 variables are currently not computed. +2.4.2 +Tighter Gradient Cuts +The separating hyperplanes generated for convex functions of (MINLPext) as discussed in +the previous section are, in general, not supporting for the feasible region of (MINLPext), +because the point where the functions are linearized is not at the boundary of the +feasible region (which is the reason why it needs to be separated). Therefore, often +several rounds of cut generation and LP solving are required until the relaxation solution +satisfies the convex constraints. Solvers for convex MINLP have handled this problem in +various ways [25, 42], but the basic idea is to build gradient cuts at a suitable boundary +point of the feasible region. +14 + +In SCIP, three procedures for building tighter and/or deeper gradient cuts for convex +relaxations are included. The first two methods compute a point on the boundary of +the set defined by all convex constraints of (MINLP) that is close to the point to be +separated. The first method solves an additional nonlinear program to project the +point to be separated onto the convex set. Since solving an NLP for every point to +be separated can be quite expensive, the second method, going back to an idea by +Veinott [71], does a binary search between an interior point of the convex set and the +point to be separated. The interior point is computed once in the beginning of the +search by solving an auxiliary NLP. For more details, see [47]. +The third method does not aim to separate a given point, but utilizes the feasible +points that are found by primal heuristics of SCIP. When a new solution is found, +gradient cuts are generated at this solution for convex constraints of (MINLPext) and +added to the cutpool. If such a cut is later found to separate the relaxation solution, it +is added to the LP. +All methods are currently disabled as they require more tuning to be efficient in +general. +2.5 +Quotients +Note that the available expression handlers (see Section 2.1.1) do not include a handler +for quotients, since they can equivalently be written using a product and a power +expression. Therefore, the default extended formulation for an expression y1y−1 +2 +is +given by replacing y−1 +2 +by a new auxiliary variable w. The linear outer-approximation is +then obtained by estimating y1w and y−1 +2 +separately. However, tighter linear estimates +are often possible. Therefore, a specialized nonlinear handler checks whether a given +function hi(x) can be cast as +f(y) = ay1 + b +cy2 + d + e +(9) +with a, b, c, d, e ∈ R, a, c ̸= 0, and y1 and y2 being either original variables or subexpres- +sions of hi(x). +Tight linear estimators for (9) are computed by distinguishing a number of cases. +For example, for ay1 + b ≥ 0 and cy2 + d > 0 (if c > 0), a linear underestimator is +obtained by computing a tangent on the graph of the convex underestimator of f +that is given by [78]. A linear overestimator is obtained by computing a facet on the +concave envelope of f, which is easy since −f is vertex-polyhedral. Furthermore, in the +univariate case (y1 = y2), f is either convex or concave on [y1, y1] if −d/c ̸∈ [y2, y2]. +Since in the univariate case the same variable appears twice, also a specialized +domain propagation method that avoids the dependency problem of interval arithmetic +is available. +2.6 +Perspective Strengthening +Perspective reformulations have shown to efficiently tighten relaxations of convex mixed- +integer nonlinear programs with on/off-structures, which are often modeled via big-M +constraints or semi-continuous variables [29]. A variable xj is semi-continuous with +respect to the binary indicator variable xj′, j′ ∈ I, if it is restricted to the domain +[x1 +j, x1 +j] when xj′ = 1 and has a fixed value x0 +j when xj′ = 0. +In SCIP, a strengthening of under- and overestimators for functions that depend on +semi-continuous variables is available. Consider a constraint hi(x, wi+1, . . . , w ˆm) ⋚ wi +of (MINLPext) and write hi as a sum of its nonlinear and linear parts: +hi(x, wi+1, . . . , w ˆm) = hnl +i (xnl, wnl) + hl +i(xl, wl), +15 + +where hnl +i is a nonlinear function, hl +i is a linear function, xnl and wnl are the vectors of +variables x and w, respectively, that appear only in the nonlinear part of hi, and xl and +wl are the vectors of variables x and w, respectively, that appear only in the linear part +of hi. A strengthening of under- or overestimators for hi(x, wi+1, . . . , w ˆm) is attempted +if xnl and wnl are semi-continuous with respect to the same indicator variable xj′. +To determine whether a variable xj is semi-continuous, bounds on xj that are +implied by fixing a binary variable are analyzed. The implied bounds can be obtained +either from linear constraints directly or by probing, and are stored by SCIP in a +globally available data structure. If a pair of implied bounds on xj with the same +binary variable xj′ is found, i.e., +xj ≤ α(u)xj′ + β(u), +xj ≥ α(ℓ)xj′ + β(ℓ), +and β(u) = β(ℓ), then xj is a semi-continuous variable with x0 +j = β(u), x1 +j = α(ℓ) + β(ℓ), +and x1 +j = α(u)+β(u). In addition, an auxiliary variable wi is found to be semi-continuous +if function hi(x, wi+1, . . . , w ˆm) depends only on semi-continuous variables with the same +indicator variable. +Assume that a linear underestimator ℓ(x, wi+1, . . . , w ˆm) of hi(x, wi+1, . . . , w ˆm) has +been computed and split it into parts corresponding to the nonlinear and linear variables +of hi, respectively: +ℓ(x, wi+1, . . . , w ˆm) = ℓnl(xnl, wnl) + ℓl(xl, wl). +The perspective strengthening consists of extending the part of the underestimator that +corresponds to the nonlinear part such that it is tight for xj′ = 0: +ℓnl(xnl, wnl) + +� +hnl +i (x0 +nl, w0 +nl) − ℓnl(x0 +nl, w0 +nl) +� +(1 − xj′) + ℓl(xl, wl). +The linear part remains unchanged, since it shares none of the variables with the +nonlinear part. This extension ensures that the estimator is equal to hi(x, wi+1, . . . , w ˆm) +for xj′ = 0, xnl = x0 +nl, and wnl = w0 +nl, and equal to ℓ(x, wi+1, . . . , w ˆm) for xj′ = 1. If hi +is convex, cuts obtained this way are equivalent to the classic perspective cuts [29]. For +more details on the implementation in SCIP, see [15]. An example is given in Figure 3. +Figure 3: The original set {(x, y, w) : x2 ≤ w, y ∈ {0, 1}, y = 0 → x = 0} (left) and a continuous +relaxation given by {(x, y, w) : x2 ≤ wy, y ∈ [0, 1], w ≥ 0} (right). From the original set, cuts of +the form ˆx2 + 2ˆx(x − ˆx) ≤ w for some reference point ˆx would be generated. With perspective +strengthening, a linearization on the right set is obtained instead, i.e., ˆx2+2ˆx(x− ˆx)+ ˆx2(1−y) ≤ w. +The latter is typically better as it is tight for y = 0 as well. +16 + +w +w≥α,y= +c,y) +)= 0:w +y2.7 +Optimization-Based Bound Tightening +Optimization-Based Bound Tightening (OBBT) is a domain propagation technique +which minimizes and maximizes each variable over the feasible set of the problem or a +relaxation thereof [59]. Whereas FBBT (see Section 2.1.2) propagates the nonlinearities +individually, OBBT considers (a relaxation of) all constraints together, and may hence +compute tighter bounds. However, it is rather expensive compared to FBBT. +In SCIP, OBBT solves two auxiliary LPs for each variable xk that could be subject +to spatial branching: +min / max{xk : Dxx + Dww ≤ d, c⊤x ≤ U, x ∈ [x, x], w ∈ [w, w]} +(10) +where Dxx + Dww ≤ d, Dx ∈ Rℓ×n, Dw ∈ Rℓ× ˆm, d ∈ Rℓ is the linear relaxation of +the feasible region of (MINLPext), and c⊤x ≤ U is an objective cutoff constraint that +excludes solutions with objective value worse than the current incumbent. The optimal +value of (10) may then be used to tighten the lower / upper bound of variable xk. A +variable is subject to spatial branching if cut separation routines use the bounds of the +variable at a node of the branch-and-bound tree. +SCIP, by default, applies OBBT at the root node to tighten bounds globally. It +restricts the computational effort by limiting the amount of LP iterations spent for +solving the auxiliary LPs and interrupting for cheaper domain propagation techniques +to be called between LP solves. +Further, SCIP does not only use the optimal objective values of (10) to tighten the +bounds on xk, but it also applies a computationally cheap approximation of OBBT +during the branch-and-bound search by exploiting the dual solutions from solves of (10) +at the root node. Suppose the maximization LP is solved and feasible dual multipliers +λ1, . . . , λℓ, µ ≥ 0 for Dxx + Dww ≤ d, c⊤x ≤ U, respectively, and the corresponding +reduced cost vectors rx and rw are obtained. Then +xk ≤ +� +j +rx +j xj + +� +j +rw +j wj + λ⊤d + µU +(11) +is a valid inequality, which is called Lagrangian variable bound (LVB), and +� +j:rx +j <0 +rx +j xj + +� +j:rx +j >0 +rx +j xj + +� +j:rw +j <0 +rw +j wj + +� +j:rw +j >0 +rw +j wj + λ⊤d + µU +(12) +is a valid upper bound for xk that equals the OBBT bound if the dual multipliers are +optimal. SCIP learns LVBs at the root node and propagates them during the tree +search whenever the bounds of variables on the right-hand side of (11) become tighter +or an improved primal solution is found. For further details, see [33]. +In addition to OBBT with respect to the LP relaxation, also a variant is available +that optimizes single variables over the potentially tighter convex NLP relaxation that +is given by all linear and convex nonlinear constraints of (MINLP). Also for this +variant, linear Lagrangian variable bounds similar to (11) can be constructed by taking +constraint convexity and KKT conditions into account. Because of the potentially high +computational cost of solving many NLPs, this variant of OBBT is deactivated by +default. For more details, see [47]. +2.8 +Primal Heuristics +The purpose of primal heuristics is to find high quality feasible solutions early in the +search. When given an MINLP, up to 40 primal heuristics are active in SCIP by default. +Many of them aim to find an integer-feasible solution to the LP relaxation. In the +following, primal heuristics that are only active in the presence of nonlinear constraints +are discussed. +17 + +2.8.1 +subNLP +A primal heuristic like subNLP is implemented in virtually any global MINLP solver. +Given a point ˜x that satisfies the integrality requirements (˜xI ∈ Z|I|), the heuristic +starts by fixing all integer variables in (MINLP) to the values given by ˜x. It then calls +the SCIP presolver on this subproblem for possible simplifications. Finally, it triggers a +solution of the remaining NLP, using ˜x as the starting point. If the NLP solver, such +as Ipopt, finds a solution that is feasible (and often also locally optimal) for the NLP +relaxation, then a feasible point for (MINLP) has been found. +The starting point ˜x can be the current solution of the LP relaxation if integer- +feasible, a point found by a primal heuristic that searches for integer-feasible solutions +of the LP relaxation, or a point that is passed on by other primal heuristics for MINLP, +such as those mentioned in the next sections. +How frequently the heuristic should run and how much effort to spend on an NLP +solve is a nontrivial decision. In the current implementation, the heuristic uses a fixed +number for the iteration limit of the NLP solver for its first run. For the following calls, +the limit is set to twice the average number of iterations required in previous runs. If, +however, many of the previous runs hit the iteration limit, then an increased iteration +limit is used. Whether to run the heuristic at a node of the branch-and-bound tree +depends on the number of nodes processed since it ran the last time, the iteration limit +that would be used, and how successful the heuristic has been in finding feasible points +in previous calls. +2.8.2 +Multistart +If (MINLP) is nonconvex after fixing all integer variables, then several local optima +may exist for the NLPs solved by heuristic subNLP. The success of the NLP solver then +strongly depends on the starting point. Therefore, the multistart heuristic aims to +compute several starting points that are passed to the subNLP heuristic. +The algorithm, originally developed in [66], tries to approximate the boundary of +the feasible set of the NLP relaxation by sampling points from [x, x] and pushing +them towards the feasible set by the use of an inexpensive gradient descent method. +Afterwards, points that are relatively close to each other are grouped into clusters. +Ideally, each cluster approximates the boundary of some connected component of the +feasible set. For each cluster, a linear combination of the points is passed as a starting +point to subNLP. For integer variables xi, i ∈ I, the value in the starting point is +rounded to an integral value. +To reduce infeasibility of a point ˆx, the constraint consensus method [66] is used. +The algorithm computes a descent direction for each violated constraint of (MINLP). +For example, if gi(ˆx) > gi for some i ∈ {1, . . . , m}, then the descent direction is given +by − +gi(ˆx) +∥∇gi(ˆx)∥2 ∇gi(ˆx). Point ˆx is then updated by adding the average of the descent +directions for all violated linear and nonlinear constraints. This step is iterated until ˆx +becomes feasible, or a stopping criterion has been fulfilled. +The multistart heuristic currently runs for continuous problems (I = ∅) only by +default, since rounding and fixing integer variables most likely lead to infeasible NLP +subproblems. For more details, see [47]. +2.8.3 +NLP Diving +As an alternative to finding a good fixing for all integer variables of (MINLP), the NLP +diving heuristic starts by solving the NLP relaxation at the current branch-and-bound +node with an NLP solver. It then iteratively fixes integer variables with fractional value +and resolves both the LP and NLP relaxations, thereby simulating a depth-first-search +in a branch-and-bound tree. By default, variables for which the sum of the distances +18 + +from the solutions of the LP and NLP relaxations to a common integer value is minimal +are rounded to the nearest integer value. Further, binary variables and nonlinear +variables are preferred. If the resulting NLP is found to be (locally) infeasible, one-level +backtracking is applied, that is, the last fixing is undone, and the opposite fixing is +tried. If this is infeasible, too, the heuristic aborts. +2.8.4 +MPEC +While the NLP diving heuristic either completely omits or enforces integrality restrictions +in the NLP relaxation, the MPEC heuristic adds a relaxation of the integrality restriction +to the NLP and tightens this relaxation iteratively. The heuristic is only applicable to +mixed-binary nonlinear programs at the moment. +The basic idea of the heuristic, originally developed in [61], is to reformulate +(MINLP) as a mathematical program with equilibrium constraints (MPEC) and to +solve this MPEC to local optimality. +The MPEC is obtained from (MINLP) by +rewriting the condition xi ∈ {0, 1}, i ∈ I, as complementarity constraint xi ⊥ 1 − xi. +This reformulation is again reformulated to an NLP by writing it as xi (1 − xi) = 0. +However, since these reformulated complementarity constraints will not, in general, +satisfy constraint qualifications, solving this NLP reformulation with a generic NLP +solver will often fail. +Therefore, in order to increase the chances of solving the NLP reformulation, +the heuristic solves regularized versions of the NLP by relaxing xi(1 − xi) = 0 to +xi(1 − xi) ≤ θ, for different, ever smaller θ > 0. The solution of one NLP is thereby +used as the starting point for the next solve. If the NLP solution is close to satisfying +xI ∈ {0, 1}|I|, it is passed as starting point to the subNLP heuristic. If an NLP is +(locally) infeasible, the heuristic does two more attempts where the values for binary +variables that are already close to 0 or 1 are flipped to 1 or 0, respectively. For more +details, see [34]. +2.8.5 +Undercover +While the previous heuristics focused mainly on enforcing the integrality condition on an +NLP, heuristic undercover [10] starts from a completely different angle. The heuristic +is based on the observation that it sometimes suffices to fix only a comparatively small +number of variables of (MINLP) to yield a subproblem with all constraints being linear. +For example, for a bilinear term, only one of the variables needs to be fixed. The +variables to fix are chosen by solving a set covering problem, which aims at minimizing +the number of variables to fix. The values for the fixed variables are taken from the +solution of the LP or NLP relaxation or a known feasible solution of the MINLP. +The resulting sub-MIP is less complex to solve, and does not need to be solved to +proven optimality. The solutions of the sub-MIP are immediately feasible for (MINLP). +However, the best one is also passed as starting point to heuristic subnlp to try for +further improvement. For more details, see [10]. +3 +Benchmark +The following aims to present a fair comparison of SCIP with several other state-of-the- +art solvers for general MINLP. Doing so is not trivial at all. First, a set of instances +needs to be selected that is suitable as a benchmark set. Second, solver parameters +have to be set such that all solvers solve the same instances with the same working +limits and the same requirements on feasibility and optimality – this goal could not be +reached completely. Third, the solver’s results have to be checked for correctness, or, +when this is not possible, plausibility. +19 + +GAMS was used for the experiments, as it provides various facilities to help on +solver comparisons and comes with current versions of SCIP and the commercial solvers +BARON [40], Lindo API [44], and Octeract included. ANTIGONE has not been +included in the comparison, as its development seems to have stopped years ago. +All computations were run on a Linux cluster with Intel Xeon E5-2670 v2 CPUs +(20 cores). The GAMS version is 41.2.0, which includes SCIP 8.0.2, BARON 22.9.30, +Lindo API 14.0.5099.162, and Octeract 4.5.1. A GAMS license with all solvers enabled +was used, so that SCIP uses CPLEX 22.1.0.0 as LP solver and Ipopt with HSL MA27 +as NLP solver, BARON can choose between all LP/MIP/NLP solvers that it interfaces +with, and Octeract uses CPLEX 22.1.0.0 as LP/MIP/QP/QCP solver. +3.1 +Test Set +To construct a test set suitable for benchmarking, the MINLPLib [50] collection of 1595 +MINLP instances was used as source. First, all instances that could not be handled +by some of the considered solvers were excluded, e.g., instances with trigonometric +functions, as they are not supported by BARON. All solvers were then run in serial +mode (that is, with parallelization features disabled) on the remaining 1505 instances +and using the parameter settings described below. The results of these runs were then +used to select a set of 200 instances that could be solved by at least one solver, that +were not all trivial, had a varying degree of integrality and nonlinearity, and such that +having many instances with a similar name is avoided. The latter was done to avoid +overrepresentation of optimization problems for which many instances were added to +MINLPLib. +Since small changes to an instance can lead to large variations in the solver’s +performance, the benchmark’s reliability is improved by considering for each instance +four additional variants where the order of variables and equations has been permuted. +The permuted instances were generated with GAMS/Convert. Thus, a test set of 1000 +instances is obtained. +The following approach was used to select the 200 instances before permutation: +Let I be the set of 1505 instances, di be the fraction of integer variables in instance +i ∈ I, and ei be the fraction of nonzeros in the Jacobian and objective function gradient +that correspond to nonlinear terms. Next, assign to each instance an identifier fi ∈ F +such that instances that seem to come from the same model are assigned the same +identifier. This goal is approximated by mapping i to the name of the instance until +the first digit, underscore, or dash, except for the block layout design instances fo*, m*, +no*, o*, which were all assigned to the same identifier. |F| = 230 different identifiers +were found this way. +Further, let ti be the largest time in seconds that any solver who did not produce +wrong results on instance i spend on instance i. Finally, let S be the number of instances +that could be solved by at least one solver. +To ensure that instances with a varying amount of integer variables and nonlinearity +are included, the interval [0, 1] was split once at breakpoints 0.05, 0.25, 0.5, 0.9 and once +at 0.1, 0.25, 0.5. Let D and E be the resulting partitions of [0, 1]. For every interval +from D and E, the aim is to have roughly the same number of instances with di and +ei in the respective intervals. For the choice of breakpoints that define D and E, the +distribution of di and ei, i ∈ I, have been taken into account. For example, MINLPLib +contains many purely continuous and purely discrete instances, but not many instances +that are mostly linear or completely nonlinear. +To avoid including too many instances originating from the same model, including +more than two instances for each identifier in F is discouraged. Further, instances that +seem trivial, i.e., which are solved by all solvers in no more than five seconds, or could +not be solved by any solver are excluded. Introducing penalty terms, the following +20 + +optimization problem for instance selection is obtained: +min +� +d∈D +λ2 +d + +� +e∈E +λ2 +e + 10 +� +f∈F +λ2 +f +such that +� +i∈I:di∈d +zi = +� N +|D| +� ++ λd +∀d ∈ D, +� +i∈I:ei∈e +zi = +� N +|E| +� ++ λe +∀e ∈ E, +� +i∈I:fi=f +zi ≤ 2 + λf +∀f ∈ F, +zi = 0 +∀i ∈ I : ti ≤ 5, +zi = 0 +∀i ∈ I : i ̸∈ S, +z ∈ {0, 1}|I|, λ ∈ Z|D|+|E|+|F | +This problem was solved for N varying between 180 and 220. For N = 208, this yield a +selection of 200 instances with an acceptable penalty value of 106. See Section A for a +list of all selected instances. Table 1 shows the number of instances for each element of +D × E. For five identifiers from F, three instead of two instances were selected, i.e., +λf = 1 for five f ∈ F. +E ↓ | D → +[0, 0.05) +[0.05, 0.25) +[0.25, 0.5) +[0.5, 0.9) +[0.9, 1] +[0, 1] +[0, 0.1) +3 +7 +19 +15 +6 +50 +[0.1, 0.25) +8 +22 +9 +7 +4 +50 +[0.25, 0.5) +8 +8 +6 +10 +18 +50 +[0.5, 1] +25 +2 +5 +7 +11 +50 +[0, 1] +44 +39 +39 +39 +39 +200 +Table 1: Number of instances selected with “discreteness” di and “nonlinearity” ei in intervals +from D and E. +3.2 +Parameter Settings +3.2.1 +Missing Variable Bounds +To compute a lower bound on the optimal value of a minimization problem, all solvers +considered here construct a convex relaxation of the given problem. For nonconvex +constraints, this often relies on the computation of valid convex underestimators or +concave overestimators. As these typically depend on variables’ bounds (recall the +McCormick underestimators (1)), missing or very large bounds on variables in nonconvex +terms can mean that an instance will be very hard or impossible to solve. +Even when the user forgot to specify some variable bounds, the solver may still +be able to derive bounds via domain propagation. Further, once a feasible solution +ˆx has been found, additional bounds may be derived from the inequality c⊤x ≤ c⊤ˆx. +However, as there are always cases where bounds are still missing after presolve, solvers +invented different ways to deal with this obstacle. +If SCIP cannot construct an under- or overestimator because of missing variable +bounds, it continues by branching on an unbounded variable. This way, there will +eventually be a node in the branch-and-bound tree where all variables are bounded. +Nodes that still contain unbounded variable domains may be pruned due to a derived +lower bound on the objective function exceeding the incumbents objective function +21 + +value. But it may also be the case that pruning will not be possible and SCIP does not +terminate. However, variable bounds after branching cannot grow indefinitely in SCIP, +but are limited by ±1020 by default. That is, SCIP does not search for solutions with +variable values beyond this value. +The other solvers considered here add variable bounds based on a heuristic decision. +If BARON is still missing bounds on variables in nonconvex terms after presolve, it sets +the bound to a value that depends on the type of nonlinearity involved. Typically, this +value is around ±1010. BARON also prints a warning to the log and no longer claim +to have solved a problem to global optimality, i.e., it does not return a lower bound. +Lindo API adjusts the bounds for all variables that are involved in convexification to +be within [−1010, 1010]. At termination, it returns the lower bound for the restricted +problem. Octeract proceeds similarly and introduces a bound of ±107 for every missing +bound and returns the lower bound for the restricted problem at termination. +Evidently, just passing an instance with unbounded variables to a solver with default +settings may mean that each solver solves a different subproblem of the actual problem +and often also reports a lower bound that corresponds to the solved subproblem only. +Fortunately, for every solver considered here, parameters are available to adjust the +treatment of unbounded variables. A first impulse could be to tell all solvers to set +missing bounds to infinity, but this is not possible as each solver treats values beyond a +certain finite value as “infinity” (BARON: 1050, Octeract: 10308, SCIP: 1020). Changing +this value is either not possible or not advisable. +We therefore decided to aim for ±1012 as replacement for a missing variable bound. +For BARON and SCIP, the GAMS interface can replace any missing bound by ±1012 +before the instance is passed to the solver. BARON will hence also return a lower +bound for this restricted problem. For Lindo API, a solver parameter can be changed so +that bounds for all variables subject to convexification are bounded by ±1012 (instead +of ±1010). Finally, also for Octeract, all missing bounds are set to ±1012 (instead of +±107) by changing of a solver parameter. Note, that this still does not ensure that all +solvers solve the same instance, since Lindo API would still change initial finite bounds +beyond 1012 and may also not set any bounds for variables that are not involved in +convexification. +Next to missing bounds on problem variables, also singularities in functions (e.g., 1/x, +log(x)) can prevent finite under- or overestimators from being available. Unfortunately, +there are no parameters available to ensure a uniform treatment of this case in all +solvers. SCIP ensures that a variable x in xp, p < 0, or log(x) is bounded away from +zero by 10−9, and terminates with a lower bound for this modified problem. BARON +applies the same method as the one for missing bounds on problem variables to choose +a suitable bound on x. No lower bound is returned at termination then. The methods +in Lindo API and Octeract are not known to us. +3.2.2 +Solution Quality +To ensure that all solvers return solutions of the same quality, constraints of (MINLP) +are required to be satisfied with an absolute tolerance of 10−6. This applies to linear +and nonlinear equations, variable bounds, and integrality. +In addition, a tolerance on the proof of optimality is set. For this purpose, typically, +solvers are allowed to stop when the absolute or relative gap between lower and upper +bounds on the optimal value are sufficiently small. Since the test set is diverse and has +optimal values of varying magnitude, setting only a relative gap limit and no absolute +gap limit would be preferable. Unfortunately, Octeract does not permit different values +for these limits. As a compromise, BARON, Lindo API, and SCIP are run with 10−4 +as relative gap limit and 10−6 as absolute gap limit, while for Octeract, 10−6 is used for +both the absolute and relative gap limit. Below, the impact of using a tighter optimality +tolerance for Octeract is analyzed in a separate comparison. +22 + +3.2.3 +Working Limits +As working limits, a time limit of two hours is used and the jobs on the cluster are +restricted to 50 GB of RAM. Further, the amount parallelization (multiple threads or +processes) that a solver is allowed to use is limited in varying degrees. To simplify the +presentation, the term “threads” is used also for Octeract, even though it uses multiple +processes instead of threads to parallelize its solving process. +3.2.4 +Summary +To summarize, the following parameters are used: +GAMS (applied to all solvers): optcr=1e-4, optca=1e-6, reslim=7200, workspace= +50000, threads ∈ {1, 4, 8, 16} +BARON: InfBnd=1e12, AbsConFeasTol=1e-6, AbsIntFeasTol=1e-6 +Lindo API: GOP BNDLIM=1e12, SOLVER FEASTOL=1e-6 +Octeract: INFINITY=1e12, INTEGRALITY VIOLATION TOLERANCE=1e-6 +SCIP: gams/infbound=1e12, constraints/nonlinear/linearizeheursol=o (this +undoes a change in the algorithmic settings of SCIP that is part of the GAMS/SCIP +interface) +3.3 +Correctness Checks +The GAMS/Examiner 2.0 tool is used to evaluate the violation of constraints, bounds, +and integrality in the solutions reported by the solver. Examiner generates for each +solver a file that contains for each instance the solving time, returned lower and upper +bound, and solution infeasibility. +A run of a solver on an instance is marked as failed if the solver terminated +abnormally, the solution is not feasible with respect to the feasibility tolerance, or the +lower or upper bound contradicts with the bounds on the optimal value that are specified +on the MINLPLib page. Note, that the primal and dual bounds on the MINLPLib page +were calculated without enforcing the ±1012 limit on unbounded variables. However, in +order for an instance to be accepted into the test set, one of the solvers considered here +must have solved the instance and found an optimal value that fits within the lower +and upper bounds given at MINLPLib. It is therefore acceptable to use these bounds +for checking. +A run that has not failed is marked as solved if the relative or absolute limits on the +gap between lower and upper bound are satisfied. If a solver stopped without closing +the gap before the time limit, then the solver time is changed to the time limit. The +only exception here is BARON, which stops on two instances before the time limit +without reporting a lower bound due to singularities in functions (see Section 3.2.1). To +be consistent with the treatment of other solvers, these two instances were accounted +as solved by BARON with the original solver time. +3.4 +Results +3.4.1 +Serial Mode +For the main comparison, all parallelization features in the solvers were disabled, that +is, GAMS was run with option threads set to 1. In addition to the solver itself, results +for the virtual best and virtual worst solver are reported, which are obtained by picking +for each instance the fastest or the slowest solver, respectively. +Table 2 shows for each solver the number of instances that could be solved, the +number of times the time limit was reached, and the number of runs that were marked +as failed. Further, the shifted geometric mean of the running time of the solver is +23 + +provided. The shift has been set to 1 second. Here, instances that failed are accounted +with the time limit. The performance profile [23] in Figure 4 shows the number of +instances a solver solved within a time that is at most a factor of the fastest solvers +time. Section B.1 provides detailed results. +solved +timeout +fail +time +BARON +790 +183 +27 +75.4 +Lindo API +538 +323 +139 +489.1 +Octeract +671 +279 +50 +184.1 +SCIP +776 +183 +41 +85.2 +virt. worst +368 +405 +227 +1505.2 +virt. best +967 +33 +0 +19.7 +Table 2: Aggregated performance data for all solvers on test set of 1000 instances with parallelization +disabled. +100 +101 +102 +103 +104 +0 +200 +400 +600 +800 +1,000 +time factor to best (τ) +# instances solved +BARON +Lindo API +Octeract +SCIP +Figure 4: Performance profile comparing all solvers with parallelization disabled. +The results show a small lead of BARON before SCIP with respect to both number +of instances solved and average time. Since the number of timeouts is almost equal, +one could argue that it is the higher stability of BARON that moves it onto the first +place here. In fact, the 41 fails of SCIP are due to returning a wrong optimal value 16 +times, returning an infeasible solution 23 times, and aborts due to numerical troubles +for two instances. For BARON, fails are due to returning a wrong optimal value 26 +times and an infeasible solution only once. While SCIP 8.0 has made a large step +forward in ensuring that nonlinear constraints are satisfied in the non-presolved problem, +violations in linear constraints or variable bounds still occur for a few instances. These +are typically due to variables being aggregated during presolve. +Even though Octeract and Lindo API solved considerably fewer instances than +BARON and SCIP, which also results in an increased mean time, it is noteworthy that +each of the two is also the fastest solver on 270 and 66 instances, respectively. Octeract +also produced correct results for 95% of the test set, while for Lindo API a relatively +high number of wrong optimal values, infeasible solutions, or aborts is observed. +The large differences between the real and virtual solvers show that none of the +solvers dominates all others or is dominated. +Next, the effect of changing the gap limit for Octeract has been investigated. Recall +from Section 3.2.2 that relative and absolute gap limits of 10−6 and 10−4, respectively, +were used for all solvers except for Octeract. Since Octeract does not allow choosing +24 + +these limits separately, it had been run with the tighter relative gap limit of 10−6. To +check whether this lead to a considerable disadvantage for this solver, the solver was +rerun on the 200 non-permuted instances with both relative and absolute gap limit set +to 10−4. The table in Section B.2 shows that the change in the convergence tolerance +had essentially no effect on the solver’s performance. In both cases, the same 134 +instances could be solved. The mean time changed from 178.6 for a limit of 10−6 to +179.0 for a limit of 10−4. +3.4.2 +Parallel Mode +In the next comparison, each solver is allowed to use multiple threads or processes. +Since SCIP’s use of multiple threads is limited to presolving MIPs, checking quadratic +functions for convexity, and the linear algebra in Ipopt, FiberSCIP [64] is used to +run SCIP in parallel mode. FiberSCIP is a shared-memory instantiation of the UG +framework [62] for the parallelization of branch-and-bound based solvers. The framework +parallelizes the search of the branch-and-bound tree by collecting and distributing open +problems between independent instances of SCIP. In addition, the first seconds of the +solving process are used for a “racing ramp-up” phase. Here, multiple SCIP instances +with differing parameter sets are run concurrently, and the one with the best lower +bound is used for the remaining solve. The UG version was 1.0.0 beta3. +For the runs in serial mode, reaching the memory limit of 50 GB was not observed +for any solver. But since parallelization often increases memory requirements, a memory +limit of 100 GB has been used for the runs in parallel mode. Since this meant a +reduction in available computing resources, only the 200 non-permuted instances are +used for comparisons. +Table 3 shows, for an increasing number of threads, the number of instances that +could be solved by each solver and the mean time spent. In addition, Figure 5 provides +a performance profile that compares SCIP and FiberSCIP only. Section B.3 gives +detailed results. +1 thread +4 threads +8 threads +16 threads +solved +time +solved +time +solved +time +solved +time +BARON +161 +64.3 +160 +58.2 +160 +57.1 +158 +58.6 +Lindo API +114 +423.6 +114 +379.2 +106 +459.5 +107 +456.4 +Octeract +134 +178.6 +133 +146.9 +138 +118.1 +135 +123.2 +(Fiber)SCIP +161 +76.9 +145 +94.3 +147 +77.8 +152 +74.8 +Table 3: Aggregated performance data for all solvers on test set of 200 instances when run with +parallelization allowed. +Apparently, enabling parallelization seldom has a considerable advantage on this +test set. For Octeract, where parallelization was part of its original design, a small +increase in the number of instances that could be solved and a reduction in time by 34% +when using up to 8 parallel processes is observed. As far as we know, BARON’s use +of multiple threads is currently limited to enabling this feature in the solver for a +MIP relaxation. As a consequence, only moderate improvement of running time by up +to 11% are seen. For Lindo API, an improvement due to parallelization seems to be +impeded by a further increase in fails when using multiple threads (1 thread: 24, 4: +28, 8: 35, 16: 43). Finally, for SCIP/FiberSCIP the additional overhead due to the +parallelization being build on top of the solver instead of being tightly integrated is not +compensated by the use of multiple threads. However, in contrast to other solvers, a +monotonous improvement in both number of instances solved and mean solving time +when increasing from 4 to 16 threads is observed. Further, the virtual solvers in the +25 + +100 +101 +102 +103 +0 +50 +100 +150 +200 +time factor to best (τ) +# instances solved +1 thread +4 threads +8 threads +16 threads +virt. best +virt. worst +Figure 5: Performance profile comparing SCIP and FiberSCIP. +performance profile show that FiberSCIP can solve instances that SCIP on one thread +couldn’t solve. +Finally, note that a benefit due to parallelization can usually only be expected for +rather challenging instances because of the additional overhead in duplicating and +synchronizing data and processes. However, the test set deliberately included only +instances that could already be solved by some solver in serial mode, and only instances +that were trivial for all solvers, though they may be solved quickly by some, were +excluded. As a small experiment, for each solver only those instances that required at +least 10 or 100 seconds to solve in serial mode were considered. Unfortunately, this +essentially repeated the trends shown in Table 3, so details are omitted here. A more +thorough analysis of the parallelization capabilities of MINLP solvers using a set of +challenging instances only would be necessary, but exceeds the scope of this paper. +4 +Conclusion +The development of the MINLP solver in SCIP has come a long way. In a recent version- +to-version comparison [57, slides 49-51], a steady improvement in the performance of +SCIP on MINLP over the last ten years has been measured, resulting in SCIP 8 solving +twice as many instances as SCIP 3 and a speed-up of factor three. Partially, this +improvement has been achieved by improving and adding features particular for MINLP. +However, due to the generality of SCIP as a CIP solver, also many developments that +targeted MIP solving were immediately available for MINLP solving. +With version 8.0, the MINLP solving capabilities of SCIP have been largely reworked +and extended, which resulted in a considerable improvement in both robustness and +performance [14, 57]. As a result, SCIP’s performance is currently on par with the +state-of-the-art commercial solver BARON. +In contrast to the commercial solvers considered here, SCIP offers a variety of +possibilities for a user, developer, or researcher to interact with the solving process. In +particular, the newly added “nonlinear handler” plugin type sets SCIP apart from most +other MINLP solvers, as it allows focusing on experimenting with new algorithms to +handle certain structures in nonlinear functions without modifications to the solver’s +code. +The rather large number of features that are disabled by default shows that tuning +and improving the existing code base has become increasingly necessary. Future work +will of course also include the addition of new features, e.g., improved separation for +signomial functions [77], the use of alternative relaxations for polynomial functions [17], +26 + +or monoidal strengthening of intersection cuts for quadratic constraints [22]. +The increasing number of cores in present-day CPUs means that to fully utilize +an ordinary desktop computer, a solver needs to be parallelized. +While the UG +framework provides such a possibility for SCIP in both shared and distributed memory +environments, the experiments with FiberSCIP on up to 16 threads show that more +tuning is necessary to ensure that the additional overhead can be compensated by the use +of additional computing resources. Since the development of UG was initially motivated +and has focused primarily on the use of large-scale parallel computing environments [63], +an investigation on using UG with SCIP to solve challenging MINLPs in distributed +memory environments with many CPU cores could be interesting as well. +Acknowledgments +We are very much in all SCIP developers’ debt – the extensions to support nonlinear +constraints and solve MINLPs would not have been possible without the framework’s +existence and the powerful MIP solver that we could build upon. While the authors of +this paper are the main developers of the new MINLP features in SCIP 8, many have +contributed to the MINLP capabilities in previous releases of SCIP, namely Martin +Ballerstein, Timo Berthold, Tobias Fischer, Thorsten Gellermann, Ambros Gleixner, +Renke Kuhlmann, Dennis Michaels, Marc Pfetsch, and Stefan Weltge. 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Technical report, arXiv, 2022. +doi:10.48550/ARXIV.2212.02857. +[78] J. M. Zamora and I. E. Grossmann. Continuous global optimization of structured +process systems models. Computers and Chemical Engineering, 22(12):1749–1770, +1998. doi:10.1016/S0098-1354(98)00244-0. +32 + +A +Test Set +The following table provides details on the test set of 200 instances that was constructed +by the selection process described in Section 3.1. For each instance, the number of +variables (n), the number of discrete variables (|I|), the number of constraints (m + ˜m), +the number of nonzeros in the Jacobian and objective function gradient (nz), and the +number of nonzeros that correspond to nonlinear terms (nlnz) is given. +instance +n +|I| +m+ ˜m +nz +nlnz +alan +8 +4 +7 +23 +3 +autocorr bern20-05 +20 +20 +0 +20 +20 +autocorr bern35-04 +35 +35 +0 +35 +35 +ball mk2 10 +10 +10 +1 +20 +10 +ball mk2 30 +30 +30 +1 +60 +30 +ball mk3 10 +10 +10 +1 +20 +10 +batch0812 nc +76 +36 +205 +472 +232 +batchs101006m +278 +129 +1019 +2865 +49 +batchs121208m +406 +203 +1511 +4255 +59 +bayes2 20 +86 +0 +77 +615 +440 +bayes2 30 +86 +0 +77 +618 +440 +blend029 +102 +36 +213 +542 +64 +blend146 +222 +87 +624 +1721 +256 +camshape100 +199 +0 +200 +696 +299 +cardqp inlp +50 +50 +1 +100 +50 +cardqp iqp +50 +50 +1 +100 +50 +carton7 +328 +256 +687 +3979 +678 +carton9 +360 +288 +893 +4917 +758 +casctanks +500 +40 +517 +1605 +514 +cecil 13 +840 +180 +898 +2811 +360 +celar6-sub0 +640 +640 +16 +1280 +640 +chakra +62 +0 +41 +142 +41 +chem +11 +0 +4 +36 +11 +chenery +43 +0 +38 +132 +56 +chimera k64maxcut-01 +1101 +1101 +0 +1101 +1101 +chimera mis-01 +2032 +2032 +0 +2032 +2032 +chp shorttermplan1a +1008 +144 +2068 +6118 +576 +chp shorttermplan2a +1584 +240 +3896 +10160 +1152 +chp shorttermplan2b +1392 +192 +2552 +7672 +1440 +clay0204m +52 +32 +90 +284 +64 +clay0205m +80 +50 +135 +430 +80 +color lab3 3x0 +316 +316 +80 +632 +237 +crossdock 15x7 +210 +210 +44 +630 +210 +crossdock 15x8 +240 +240 +46 +720 +240 +crudeoil lee1 07 +749 +56 +1776 +8124 +896 +crudeoil pooling ct2 +403 +108 +732 +2523 +140 +csched1 +76 +63 +22 +173 +8 +csched1a +28 +15 +22 +77 +7 +cvxnonsep psig20 +20 +10 +0 +20 +20 +cvxnonsep psig30 +30 +15 +0 +30 +30 +du-opt +20 +13 +9 +46 +20 +du-opt5 +20 +13 +9 +46 +20 +edgecross10-040 +90 +90 +480 +1530 +90 +edgecross10-080 +90 +74 +480 +1528 +88 +eg all s +7 +7 +27 +219 +196 +eigena2 +2500 +0 +1275 +127500 +127500 +elec50 +150 +0 +50 +300 +300 +elf +54 +24 +38 +177 +30 +eniplac +141 +24 +189 +510 +48 +enpro56pb +127 +73 +191 +650 +24 +ex1244 +95 +23 +129 +468 +52 +ex1252a +24 +9 +34 +93 +36 +faclay20h +190 +190 +2280 +7030 +190 +faclay80 +3160 +3160 +164320 +496120 +3160 +feedtray +97 +7 +91 +450 +282 +fin2bb +588 +175 +618 +9413 +42 +flay04m +42 +24 +42 +154 +4 +flay05m +62 +40 +65 +242 +5 +flay06m +86 +60 +93 +350 +6 +fo7 ar25 1 +112 +42 +269 +1054 +14 +fo7 ar3 1 +112 +42 +269 +1054 +14 +forest +236 +73 +309 +1013 +178 +gabriel01 +215 +72 +467 +1789 +512 +gabriel02 +261 +71 +597 +2608 +1024 +gasnet +90 +10 +69 +266 +130 +gasprod sarawak16 +1526 +38 +2252 +6453 +1088 +gastrans582 cold13 95 +2186 +250 +3732 +8538 +2139 +33 + +instance +n +|I| +m+ ˜m +nz +nlnz +gastrans582 mild11 +2186 +250 +3732 +8538 +2139 +gear +4 +4 +0 +4 +4 +gear2 +28 +24 +4 +32 +4 +gear4 +6 +4 +1 +8 +4 +genpooling lee1 +49 +9 +82 +369 +128 +genpooling lee2 +53 +9 +92 +453 +192 +ghg 1veh +29 +12 +37 +130 +91 +gilbert +1000 +0 +1 +2000 +2000 +graphpart 2g-0066-0066 +108 +108 +36 +216 +108 +graphpart clique-60 +180 +180 +60 +360 +180 +gsg 0001 +77 +0 +111 +368 +44 +hadamard 5 +25 +25 +0 +25 +25 +heatexch spec1 +56 +12 +64 +224 +32 +heatexch spec2 +76 +16 +90 +300 +42 +hhfair +29 +0 +25 +80 +21 +himmel16 +18 +0 +21 +96 +84 +house +8 +0 +8 +25 +9 +hs62 +3 +0 +1 +6 +6 +hvb11 +9817 +9537 +10251 +36005 +64 +hybriddynamic var +81 +10 +100 +286 +61 +hybriddynamic varcc +151 +0 +110 +388 +101 +hydroenergy1 +288 +96 +428 +1212 +120 +ibs2 +3010 +1500 +1821 +13510 +3000 +johnall +194 +190 +192 +957 +573 +kall circles c6b +17 +0 +53 +148 +86 +kall congruentcircles c72 +17 +0 +59 +160 +86 +kissing2 +772 +0 +10000 +154400 +154400 +kport20 +101 +40 +27 +189 +116 +kriging peaks-red020 +2 +0 +0 +2 +2 +kriging peaks-red100 +2 +0 +0 +2 +2 +lop97icx +986 +899 +87 +1890 +704 +mathopt5 7 +1 +0 +0 +1 +1 +mathopt5 8 +1 +0 +0 +1 +1 +maxcsp-geo50-20-d4-75-36 +1000 +1000 +50 +2000 +1000 +meanvar-orl400 05 e 7 +2000 +400 +2003 +7200 +1600 +meanvar-orl400 05 e 8 +1600 +400 +1603 +6400 +800 +mhw4d +5 +0 +3 +13 +10 +milinfract +1000 +500 +501 +502000 +1000 +minlphi +64 +0 +79 +206 +36 +multiplants mtg1a +193 +93 +256 +1972 +95 +multiplants mtg2 +229 +112 +306 +2689 +126 +nd netgen-3000-1-1-b-b-ns 7 +15000 +3000 +12155 +48000 +9000 +netmod kar1 +456 +136 +666 +1848 +4 +netmod kar2 +456 +136 +666 +1848 +4 +nous1 +50 +2 +43 +196 +122 +nous2 +50 +2 +43 +196 +122 +nvs02 +8 +5 +3 +19 +16 +nvs06 +2 +2 +0 +2 +2 +oil2 +936 +2 +926 +2214 +440 +optmass +30010 +0 +25005 +80020 +10006 +ortez +87 +18 +74 +268 +54 +p ball 10b 5p 3d m +95 +50 +129 +518 +150 +p ball 15b 5p 2d m +105 +75 +139 +523 +150 +parabol5 2 3 +40400 +0 +40200 +240004 +601 +parallel +205 +25 +115 +751 +155 +pedigree ex485 +485 +426 +296 +1925 +485 +pedigree ex485 2 +485 +426 +296 +1925 +485 +pointpack06 +12 +0 +20 +86 +60 +pointpack08 +16 +0 +35 +155 +112 +pooling epa1 +214 +30 +340 +1154 +257 +pooling epa2 +331 +45 +524 +1913 +554 +portfol buyin +17 +8 +19 +58 +16 +portfol card +17 +8 +20 +66 +16 +powerflow0014r +118 +0 +197 +652 +461 +powerflow0057r +440 +0 +725 +2462 +1795 +prob07 +14 +0 +35 +109 +63 +process +10 +0 +7 +27 +11 +procurement1mot +784 +60 +749 +2444 +12 +procurement2mot +796 +60 +761 +2480 +12 +product +1553 +107 +1925 +5555 +264 +product2 +2842 +128 +3125 +8249 +1056 +prolog +20 +0 +22 +128 +14 +qp3 +100 +0 +52 +2747 +100 +qspp 0 10 0 1 10 1 +180 +180 +100 +540 +180 +qspp 0 11 0 1 10 1 +220 +220 +121 +660 +220 +radar-2000-10-a-6 lat 7 +10000 +2000 +8001 +28000 +6000 +radar-3000-10-a-8 lat 7 +15000 +3000 +12001 +42000 +9000 +ravempb +112 +54 +186 +610 +28 +34 + +instance +n +|I| +m+ ˜m +nz +nlnz +risk2bpb +463 +14 +580 +2288 +3 +routingdelay bigm +1123 +396 +2977 +7739 +1827 +rsyn0815m +205 +79 +347 +909 +11 +rsyn0815m03m +705 +282 +1647 +4120 +33 +sfacloc2 2 95 +186 +39 +239 +595 +76 +sfacloc2 3 90 +291 +75 +496 +1282 +135 +sjup2 +1696 +8 +17085 +151716 +88800 +slay06m +102 +60 +135 +462 +12 +slay07m +140 +84 +189 +644 +14 +smallinvDAXr1b010-011 +30 +30 +3 +120 +30 +smallinvDAXr1b020-022 +30 +30 +3 +120 +30 +sonet17v4 +136 +136 +2057 +6527 +272 +sonet18v6 +153 +153 +2466 +7802 +306 +sonetgr17 +152 +152 +152 +694 +302 +spectra2 +69 +30 +72 +408 +240 +sporttournament24 +276 +276 +0 +276 +276 +sporttournament30 +435 +435 +0 +435 +435 +sssd12-05persp +95 +75 +52 +305 +45 +sssd18-06persp +150 +126 +66 +474 +54 +st testgr1 +10 +10 +5 +51 +10 +st testgr3 +20 +20 +20 +181 +20 +steenbrf +468 +0 +108 +972 +108 +stockcycle +480 +432 +97 +1008 +48 +supplychainp1 022020 +2940 +460 +5300 +15040 +40 +supplychainp1 030510 +445 +70 +835 +2330 +15 +supplychainr1 022020 +1440 +460 +1840 +7000 +40 +supplychainr1 030510 +230 +70 +280 +1005 +15 +syn15m04m +340 +120 +806 +1986 +44 +syn30m02m +320 +120 +604 +1502 +40 +synheat +56 +12 +64 +224 +28 +tanksize +46 +9 +73 +290 +63 +telecomsp pacbell +3570 +3528 +2940 +121302 +74088 +tln5 +35 +35 +30 +155 +50 +tln7 +63 +63 +42 +287 +98 +tls2 +37 +33 +24 +209 +8 +tls4 +105 +89 +64 +613 +32 +topopt-mbb 60x40 50 +33600 +2400 +14363 +259956 +33600 +toroidal2g20 5555 +400 +400 +0 +400 +400 +toroidal3g7 6666 +343 +343 +0 +343 +343 +transswitch0009r +69 +9 +103 +346 +255 +tricp +169 +0 +190 +1493 +1140 +tspn08 +44 +28 +18 +136 +60 +tspn15 +135 +105 +34 +502 +165 +unitcommit1 +960 +720 +5329 +12404 +240 +unitcommit2 +960 +720 +5329 +12404 +480 +wager +156 +84 +142 +532 +240 +waste +2484 +400 +1991 +9242 +2736 +wastepaper3 +52 +27 +30 +177 +108 +wastepaper4 +76 +44 +38 +274 +176 +wastepaper6 +136 +90 +54 +528 +360 +water4 +195 +126 +137 +756 +46 +waternd1 +74 +20 +83 +301 +114 +waterno2 02 +332 +18 +410 +1088 +202 +waterno2 03 +498 +27 +616 +1635 +303 +waterund01 +40 +0 +38 +152 +78 +B +Detailed Computational Results +The following tables show the outcome from running each solver on instances from +the test set. If an instance has been solved to optimality, the time spend is reported. +Note that due to differences in formulas for the relative gap in the various solvers, an +instance may be accounted as solved even though the solver stopped at the time limit. +If a run has been flagged as failed, the reason for this decision is given: “abort” if the +solver did not return with a result, “nonopt” if the reported upper or lower bound were +not consistent with those given by MINLPLib, and “infeas” if the reported solution +is not feasible with respect to the feasibility tolerance. Otherwise, the relative gap at +termination is reported, which is ∞ if no feasible solution or lower bound has been +computed. An exception here is BARON, where an instance is considered as solved if +the solver only decided to not return a lower bound due to singularities in functions +(see Section 3.2.1). This is the case for instances mhw4d and multiplants mtg2 and +35 + +their permutations. For each instance, a time or gap that is at most 10% worse than +the one from the best solver on this instance is printed in bold font. +B.1 +Serial Mode +The following table shows the outcome from running each solver on the test set of 200 +instances and their permutations in serial mode. +instance +perm +BARON +Lindo API +Octeract +SCIP +alan +- +0.33 +0.89% +0.05 +0.19 +1 +0.30 +0.89% +0.05 +0.14 +2 +0.19 +0.89% +0.13 +0.11 +3 +0.19 +0.89% +0.05 +0.11 +4 +0.18 +0.89% +0.05 +0.13 +autocorr bern20-05 +- +7.29 +100.43 +4.36 +16.34 +1 +5.18 +90.50 +3.73 +13.26 +2 +6.19 +86.68 +4.58 +23.15 +3 +6.01 +70.76 +3.30 +15.95 +4 +5.35 +63.71 +4.33 +19.75 +autocorr bern35-04 +- +29.91 +13.5% +12.7% +107.46 +1 +45.39 +infeas +3524.97 +93.55 +2 +55.21 +infeas +2425.34 +104.74 +3 +79.60 +infeas +6572.33 +88.83 +4 +39.39 +infeas +4446.69 +77.29 +ball mk2 10 +- +0.07 +3.98 +0.01 +0.05 +1 +0.07 +3.96 +0.01 +0.10 +2 +0.07 +3.98 +0.01 +0.01 +3 +0.07 +3.66 +0.01 +0.01 +4 +0.07 +3.66 +0.01 +0.01 +ball mk2 30 +- +0.25 +100.0% +0.02 +0.06 +1 +0.08 +100.0% +0.02 +0.08 +2 +0.09 +100.0% +0.02 +0.03 +3 +0.08 +100.0% +0.02 +0.03 +4 +0.13 +100.0% +0.04 +0.03 +ball mk3 10 +- +10.58 +3.58 +0.04 +0.00 +1 +10.30 +3.85 +0.04 +0.00 +2 +10.47 +3.88 +0.04 +0.00 +3 +10.15 +3.68 +0.04 +0.00 +4 +10.20 +3.56 +0.04 +0.00 +batch0812 nc +- +6.57 +25.46 +8.37 +1.45 +1 +8.95 +22.13 +9.06 +1.18 +2 +5.85 +20.34 +8.84 +1.01 +3 +6.41 +20.35 +8.65 +1.45 +4 +70.42 +23.51 +9.21 +1.05 +batchs101006m +- +6.91 +134.04 +4.84 +7.63 +1 +18.23 +59.63 +5.28 +6.55 +2 +10.39 +51.85 +5.20 +6.44 +3 +10.03 +81.83 +5.13 +6.31 +4 +8.77 +57.22 +4.80 +6.19 +batchs121208m +- +47.33 +198.76 +79.1% +6.99 +1 +27.10 +221.28 +abort +14.64 +2 +30.58 +180.28 +abort +15.32 +3 +33.04 +184.44 +abort +14.49 +4 +26.62 +152.58 +60.6% +12.04 +bayes2 20 +- +67.77 +0.033% +0.033% +0.033% +1 +80.24 +0.033% +0.033% +0.033% +2 +69.41 +0.033% +0.033% +0.033% +3 +58.54 +0.033% +6051.51 +0.033% +4 +372.69 +0.033% +0.033% +0.033% +bayes2 30 +- +21.02 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+38.73 +waterund01 +- +0.35% +infeas +0.18% +1525.91 +1 +0.34% +0.84% +0.18% +1988.24 +2 +0.34% +infeas +0.18% +3003.09 +3 +0.35% +0.98% +0.18% +0.025% +4 +0.35% +infeas +0.18% +16.83 +48 + +B.2 +Octeract Gap Limit +The following table shows the outcome from running Octeract in serial mode with both +gap limits set to either 10−6 or 10−4 on the test set of 200 non-permuted instances. +instance +10−6 +10−4 +alan +0.05 +0.05 +autocorr bern20-05 +4.36 +4.39 +autocorr bern35-04 +12.7% +12.7% +ball mk2 10 +0.01 +0.01 +ball mk2 30 +0.02 +0.02 +ball mk3 10 +0.04 +0.04 +batch0812 nc +8.37 +8.63 +batchs101006m +4.84 +4.92 +batchs121208m +79.1% +79.1% +bayes2 20 +0.033% +0.033% +bayes2 30 +7200.00 +7200.00 +blend029 +29.33 +30.14 +blend146 +8.8% +8.8% +camshape100 +19.40 +19.42 +cardqp inlp +792.06 +788.89 +cardqp iqp +785.02 +793.77 +carton7 +6.03 +6.34 +carton9 +344.94 +346.42 +casctanks +11.8% +12.4% +cecil 13 +188.40 +205.63 +celar6-sub0 +∞ +∞ +chakra +7200.00 +7200.00 +chem +0.18 +0.18 +chenery +200.71 +200.09 +chimera k64maxcut-01 +168.60 +169.03 +chimera mis-01 +1.14 +1.45 +chp shorttermplan1a +64.74 +65.18 +chp shorttermplan2a +17.59 +17.69 +chp shorttermplan2b +0.67% +0.69% +clay0204m +1.47 +1.45 +clay0205m +15.10 +14.97 +color lab3 3x0 +∞ +∞ +crossdock 15x7 +∞ +∞ +crossdock 15x8 +6306.65 +6336.71 +crudeoil lee1 07 +8.56 +7.58 +crudeoil pooling ct2 +nonopt +nonopt +csched1 +8.1% +8.1% +csched1a +17.77 +17.43 +cvxnonsep psig20 +35.0% +35.0% +cvxnonsep psig30 +45.6% +45.6% +du-opt +122.38 +122.77 +du-opt5 +101.32 +102.03 +edgecross10-040 +4.16 +4.11 +edgecross10-080 +6% +5.6% +eg all s +102% +102% +eigena2 +416.89 +523.38 +elec50 +66.4% +66.4% +elf +2.54 +2.54 +eniplac +1.54 +1.30 +enpro56pb +1.98 +2.31 +ex1244 +81.99 +82.60 +ex1252a +245.41 +248.28 +faclay20h +785.28 +781.38 +faclay80 +∞ +∞ +feedtray +82.1% +82.1% +fin2bb +100.0% +100.0% +flay04m +0.87 +0.90 +flay05m +infeas +infeas +flay06m +infeas +infeas +fo7 ar25 1 +8.95 +8.99 +fo7 ar3 1 +18.33 +18.57 +forest +nonopt +nonopt +gabriel01 +2% +4.3% +gabriel02 +7078.04 +6811.48 +gasnet +96.7% +96.7% +gasprod sarawak16 +0.74% +0.68% +gastrans582 cold13 95 +∞ +∞ +gastrans582 mild11 +∞ +∞ +gear +0.11 +0.07 +gear2 +0.14 +0.13 +gear4 +17.56 +17.33 +genpooling lee1 +117.04 +117.72 +49 + +instance +10−6 +10−4 +genpooling lee2 +186.63 +186.60 +ghg 1veh +12.42 +12.36 +gilbert +0.9% +0.9% +graphpart 2g-0066-0066 +0.75 +0.43 +graphpart clique-60 +2890.51 +2884.75 +gsg 0001 +8.58 +8.56 +hadamard 5 +20.45 +20.23 +heatexch spec1 +17.7% +17.7% +heatexch spec2 +5% +5% +hhfair +∞ +∞ +himmel16 +12.95 +13.27 +house +108.41 +108.71 +hs62 +0.023% +0.023% +hvb11 +7.3% +7.3% +hybriddynamic var +0.32% +0.32% +hybriddynamic varcc +158.14 +158.12 +hydroenergy1 +0.65% +0.65% +ibs2 +5.1% +5.1% +johnall +44.23 +43.95 +kall circles c6b +463.62 +466.28 +kall congruentcircles c72 +49.42 +49.78 +kissing2 +100.0% +100.0% +kport20 +3596.06 +3593.09 +kriging peaks-red020 +85.61 +85.98 +kriging peaks-red100 +629.86 +630.04 +lop97icx +5.90 +5.90 +mathopt5 7 +0.08 +0.08 +mathopt5 8 +0.06 +0.07 +maxcsp-geo50-20-d4-75-36 +7.71 +7.78 +meanvar-orl400 05 e 7 +∞ +∞ +meanvar-orl400 05 e 8 +6.00 +5.98 +mhw4d +0.43 +0.62 +milinfract +75.5% +75.5% +minlphi +100% +100% +multiplants mtg1a +5.1% +5.1% +multiplants mtg2 +2% +2% +nd netgen-3000-1-1-b-b-ns 7 +3.67 +3.59 +netmod kar1 +41.89 +41.79 +netmod kar2 +42.01 +42.04 +nous1 +52.93 +52.90 +nous2 +8.29 +8.04 +nvs02 +0.22 +0.15 +nvs06 +0.06 +0.06 +oil2 +nonopt +nonopt +optmass +463.14 +463.57 +ortez +9.48 +9.48 +p ball 10b 5p 3d m +49.66 +49.32 +p ball 15b 5p 2d m +infeas +infeas +parabol5 2 3 +4095.70 +4110.54 +parallel +102.24 +102.42 +pedigree ex485 +276.45 +277.27 +pedigree ex485 2 +3.66 +3.94 +pointpack06 +4.76 +4.77 +pointpack08 +208.18 +209.02 +pooling epa1 +21.07 +21.10 +pooling epa2 +1694.37 +2039.75 +portfol buyin +3.56 +3.37 +portfol card +5.29 +5.26 +powerflow0014r +100.0% +100.0% +powerflow0057r +∞ +∞ +prob07 +16.19 +15.83 +process +0.93 +0.92 +procurement1mot +79.5% +79.5% +procurement2mot +2.71 +2.69 +product +nonopt +nonopt +product2 +2.53 +2.84 +prolog +100.0% +100.0% +qp3 +4.6% +4.6% +qspp 0 10 0 1 10 1 +26.39 +26.51 +qspp 0 11 0 1 10 1 +124.07 +124.05 +radar-2000-10-a-6 lat 7 +33.34 +32.97 +radar-3000-10-a-8 lat 7 +209.09 +208.61 +ravempb +1.26 +0.96 +risk2bpb +10.70 +10.68 +routingdelay bigm +18.4% +18.4% +rsyn0815m +1.02 +1.05 +rsyn0815m03m +44.8% +44.8% +sfacloc2 2 95 +1.11 +0.79 +50 + +instance +10−6 +10−4 +sfacloc2 3 90 +67.86 +67.78 +sjup2 +∞ +∞ +slay06m +0.69 +0.48 +slay07m +0.61 +0.91 +smallinvDAXr1b010-011 +8.69 +8.99 +smallinvDAXr1b020-022 +205.80 +208.43 +sonet17v4 +160.55 +160.19 +sonet18v6 +184.16 +184.08 +sonetgr17 +65.24 +65.03 +spectra2 +105% +105% +sporttournament24 +21.48 +21.47 +sporttournament30 +1946.51 +1959.43 +sssd12-05persp +780.87 +780.44 +sssd18-06persp +19.9% +20.2% +st testgr1 +0.07 +0.07 +st testgr3 +0.07 +0.07 +steenbrf +98.0% +98.0% +stockcycle +39.57 +39.61 +supplychainp1 022020 +8.5% +8.8% +supplychainp1 030510 +3.08 +3.10 +supplychainr1 022020 +2966.92 +2966.61 +supplychainr1 030510 +0.74 +0.74 +syn15m04m +1.04 +1.36 +syn30m02m +1.83 +1.56 +synheat +17.1% +17.1% +tanksize +54.61 +54.45 +telecomsp pacbell +1125.10 +1111.55 +tln5 +2.08 +2.10 +tln7 +274.33 +274.65 +tls2 +0.18 +0.22 +tls4 +36.82 +37.19 +topopt-mbb 60x40 50 +∞ +∞ +toroidal2g20 5555 +3.23 +2.98 +toroidal3g7 6666 +67.16 +67.08 +transswitch0009r +5.6% +5.5% +tricp +100.0% +100.0% +tspn08 +2.3% +2.3% +tspn15 +81.4% +81.4% +unitcommit1 +1082.69 +1092.08 +unitcommit2 +11.20 +11.23 +wager +252.73 +252.96 +waste +99.9% +99.9% +wastepaper3 +22.76 +22.60 +wastepaper4 +2099.99 +2110.22 +wastepaper6 +0.16% +0.21% +water4 +1648.53 +2062.46 +waternd1 +1.34 +1.08 +waterno2 02 +4.18 +4.11 +waterno2 03 +440.65 +440.06 +waterund01 +0.18% +0.18% +B.3 +Parallel Mode +The following table shows the outcome from running BARON in serial and parallel +mode. +instance +1 thread +4 threads +8 threads +16 threads +alan +0.33 +0.38 +0.37 +0.36 +autocorr bern20-05 +7.29 +1.91 +1.33 +1.31 +autocorr bern35-04 +29.91 +8.44 +5.71 +4.66 +ball mk2 10 +0.07 +0.07 +0.07 +0.20 +ball mk2 30 +0.25 +0.08 +0.08 +0.08 +ball mk3 10 +10.58 +11.19 +11.35 +11.71 +batch0812 nc +6.57 +5.33 +6.47 +5.32 +batchs101006m +6.91 +13.82 +9.41 +11.02 +batchs121208m +47.33 +26.40 +24.95 +32.16 +bayes2 20 +67.77 +68.42 +70.96 +68.64 +bayes2 30 +21.02 +22.46 +21.94 +23.08 +blend029 +1.03 +1.08 +1.25 +0.95 +blend146 +3648.64 +2175.95 +2854.41 +3654.96 +camshape100 +9.2% +9.2% +9.2% +9.2% +cardqp inlp +12.40 +10.82 +12.25 +18.77 +cardqp iqp +12.92 +10.80 +12.49 +18.70 +carton7 +987.64 +1964.21 +1441.91 +832.10 +51 + +instance +1 thread +4 threads +8 threads +16 threads +carton9 +45.0% +36.0% +46.9% +37.9% +casctanks +256.46 +251.20 +248.18 +251.93 +cecil 13 +280.62 +238.03 +201.60 +195.64 +celar6-sub0 +100.0% +100.0% +100.0% +100.0% +chakra +0.14 +0.15 +0.14 +0.15 +chem +0.04 +0.04 +0.04 +0.04 +chenery +1.40 +1.42 +1.23 +1.16 +chimera k64maxcut-01 +560.62 +183.06 +119.55 +171.40 +chimera mis-01 +13.69 +12.39 +11.84 +12.54 +chp shorttermplan1a +2296.84 +2329.95 +0.061% +1276.12 +chp shorttermplan2a +937.38 +1002.49 +999.57 +956.66 +chp shorttermplan2b +0.34% +0.31% +0.3% +0.31% +clay0204m +0.47 +0.37 +0.24 +1.28 +clay0205m +4.92 +3.78 +2.63 +4.94 +color lab3 3x0 +2654.00 +606.50 +383.87 +346.70 +crossdock 15x7 +385.56 +87.79 +49.36 +38.69 +crossdock 15x8 +788.81 +322.11 +128.24 +102.94 +crudeoil lee1 07 +7.50 +7.47 +27.59 +6.98 +crudeoil pooling ct2 +4.95 +41.75 +21.47 +4.28 +csched1 +13.03 +12.42 +29.59 +31.28 +csched1a +0.70 +0.76 +0.70 +0.78 +cvxnonsep psig20 +0.99 +0.92 +0.91 +0.95 +cvxnonsep psig30 +4.25 +4.06 +4.13 +4.21 +du-opt +2.37 +2.18 +1.95 +2.26 +du-opt5 +3.61 +3.34 +3.32 +3.54 +edgecross10-040 +7.25 +2.73 +2.13 +2.54 +edgecross10-080 +51.58 +15.21 +10.25 +9.23 +eg all s +infeas +infeas +infeas +90.9% +eigena2 +1146.69 +1150.94 +1136.30 +1132.85 +elec50 +66.5% +66.5% +66.5% +66.5% +elf +5.91 +4.52 +4.24 +5.65 +eniplac +4.54 +4.04 +5.48 +9.66 +enpro56pb +1.84 +3.38 +2.01 +2.92 +ex1244 +4.86 +4.56 +4.49 +4.23 +ex1252a +9.12 +9.19 +8.97 +10.85 +faclay20h +349.36 +91.42 +98.32 +47.33 +faclay80 +120% +120% +120% +120% +feedtray +80.5% +80.5% +80.5% +80.5% +fin2bb +114.16 +83.45 +97.44 +91.84 +flay04m +10.56 +9.18 +8.80 +8.94 +flay05m +394.85 +370.14 +353.95 +411.75 +flay06m +7% +7.2% +6.4% +6.6% +fo7 ar25 1 +17.00 +4.55 +5.17 +3.37 +fo7 ar3 1 +25.69 +52.69 +30.43 +21.73 +forest +747.74 +1311.21 +877.73 +1470.56 +gabriel01 +1760.52 +1619.79 +1272.26 +1131.82 +gabriel02 +11.1% +18.1% +9% +15.6% +gasnet +22.38 +57.1% +50.7% +50.7% +gasprod sarawak16 +4578.18 +5171.96 +2524.08 +2214.82 +gastrans582 cold13 95 +1442.44 +732.29 +954.39 +1435.95 +gastrans582 mild11 +687.72 +1206.29 +1283.65 +2130.36 +gear +0.12 +0.11 +0.11 +0.28 +gear2 +0.64 +0.62 +0.69 +0.67 +gear4 +1.63 +1.53 +1.90 +2.39 +genpooling lee1 +9.37 +5.21 +5.37 +5.46 +genpooling lee2 +85.93 +84.33 +83.67 +85.79 +ghg 1veh +2.77 +2.96 +3.11 +2.98 +gilbert +2.35 +2.60 +2.56 +2.38 +graphpart 2g-0066-0066 +0.65 +0.76 +0.46 +0.37 +graphpart clique-60 +34.6% +100.0% +100.0% +100.0% +gsg 0001 +13.68 +16.00 +16.69 +11.94 +hadamard 5 +54.92 +17.35 +8.54 +9.77 +heatexch spec1 +1.39 +1.64 +1.76 +1.32 +heatexch spec2 +6.60 +7.91 +8.50 +7.88 +hhfair +0.43 +0.39 +0.42 +0.35 +himmel16 +41.46 +36.26 +36.42 +36.46 +house +0.37 +0.38 +0.38 +0.57 +hs62 +1.07 +1.08 +0.94 +0.88 +hvb11 +175.10 +174.88 +179.60 +177.88 +hybriddynamic var +0.75 +0.75 +0.98 +1.09 +hybriddynamic varcc +0.91 +0.75 +0.77 +0.78 +hydroenergy1 +1026.99 +1057.36 +1054.54 +1057.15 +ibs2 +nonopt +nonopt +nonopt +nonopt +johnall +2.85 +2.51 +2.50 +2.23 +kall circles c6b +310.13 +313.79 +307.19 +306.85 +kall congruentcircles c72 +26.45 +27.78 +24.66 +22.91 +kissing2 +184.78 +207.89 +184.05 +207.42 +kport20 +6.1% +3.3% +6.8% +5.7% +52 + +instance +1 thread +4 threads +8 threads +16 threads +kriging peaks-red020 +10.33 +9.90 +9.00 +9.43 +kriging peaks-red100 +196.08 +198.30 +195.92 +193.18 +lop97icx +1709.08 +450.05 +315.56 +317.81 +mathopt5 7 +0.29 +0.13 +0.12 +0.12 +mathopt5 8 +0.26 +0.43 +0.40 +0.26 +maxcsp-geo50-20-d4-75-36 +16.94 +18.01 +15.92 +17.40 +meanvar-orl400 05 e 7 +95.4% +94.7% +94.3% +94.8% +meanvar-orl400 05 e 8 +600.24 +541.81 +508.87 +455.48 +mhw4d +0.83 +0.70 +0.67 +0.80 +milinfract +55.06 +54.78 +54.71 +55.20 +minlphi +∞ +∞ +∞ +∞ +multiplants mtg1a +2245.95 +2338.78 +4026.31 +1138.55 +multiplants mtg2 +2115.70 +2189.56 +2175.33 +2125.52 +nd netgen-3000-1-1-b-b-ns 7 +∞ +∞ +∞ +∞ +netmod kar1 +4.53 +5.70 +5.71 +6.81 +netmod kar2 +5.26 +5.46 +5.75 +6.88 +nous1 +51.20 +47.60 +44.39 +53.14 +nous2 +0.40 +0.35 +0.34 +0.36 +nvs02 +0.04 +0.05 +0.05 +0.05 +nvs06 +0.08 +0.08 +0.10 +0.10 +oil2 +3.30 +3.02 +3.00 +2.67 +optmass +∞ +∞ +∞ +∞ +ortez +nonopt +nonopt +nonopt +nonopt +p ball 10b 5p 3d m +32.22 +23.87 +20.81 +25.42 +p ball 15b 5p 2d m +90.52 +63.59 +63.02 +77.58 +parabol5 2 3 +0.051% +0.051% +0.051% +0.051% +parallel +19.19 +21.16 +20.85 +18.15 +pedigree ex485 +68.37 +39.15 +79.06 +414.20 +pedigree ex485 2 +17.30 +14.20 +16.23 +14.65 +pointpack06 +5.59 +6.60 +4.94 +5.96 +pointpack08 +159.31 +153.67 +154.66 +151.67 +pooling epa1 +12.84 +11.97 +14.83 +14.15 +pooling epa2 +2.4% +0.44% +5.5% +3% +portfol buyin +0.34 +0.33 +0.34 +0.32 +portfol card +0.31 +0.30 +0.29 +0.30 +powerflow0014r +50.50 +51.54 +50.22 +53.19 +powerflow0057r +∞ +∞ +∞ +∞ +prob07 +75.62 +81.05 +76.59 +71.80 +process +0.51 +0.52 +0.42 +0.42 +procurement1mot +59.57 +59.24 +63.12 +61.09 +procurement2mot +4.13 +4.24 +3.78 +3.65 +product +48.09 +24.31 +49.29 +nonopt +product2 +2.90 +225.61 +76.64 +289.08 +prolog +0.31 +0.32 +0.33 +0.36 +qp3 +13.7% +14.0% +14.2% +14.4% +qspp 0 10 0 1 10 1 +24.2% +23.8% +23.7% +23.8% +qspp 0 11 0 1 10 1 +30.5% +30.1% +30.1% +30.1% +radar-2000-10-a-6 lat 7 +75.1% +91.0% +91.0% +91.0% +radar-3000-10-a-8 lat 7 +100.0% +100.0% +100.0% +100.0% +ravempb +0.82 +0.80 +1.05 +0.94 +risk2bpb +0.70 +0.43 +0.46 +0.49 +routingdelay bigm +26.08 +23.23 +23.91 +24.85 +rsyn0815m +0.29 +0.26 +0.26 +0.26 +rsyn0815m03m +31.23 +38.94 +34.49 +39.53 +sfacloc2 2 95 +0.31 +0.28 +0.41 +0.27 +sfacloc2 3 90 +29.12 +24.28 +14.83 +16.30 +sjup2 +∞ +∞ +∞ +∞ +slay06m +0.46 +0.41 +0.48 +0.59 +slay07m +0.62 +0.64 +0.58 +0.58 +smallinvDAXr1b010-011 +1.62 +1.52 +1.44 +1.52 +smallinvDAXr1b020-022 +1.95 +2.26 +1.63 +1.75 +sonet17v4 +339.62 +62.99 +33.05 +33.46 +sonet18v6 +429.30 +97.11 +62.50 +60.56 +sonetgr17 +346.77 +74.19 +44.37 +29.83 +spectra2 +1.37 +1.29 +1.47 +1.38 +sporttournament24 +4982.31 +1521.61 +722.06 +763.20 +sporttournament30 +11.8% +67.7% +67.7% +67.7% +sssd12-05persp +27.8% +28.0% +27.8% +27.6% +sssd18-06persp +41.4% +40.5% +41.4% +41.3% +st testgr1 +0.24 +0.14 +0.16 +0.22 +st testgr3 +0.24 +0.28 +0.35 +0.50 +steenbrf +14.39 +15.32 +15.07 +17.55 +stockcycle +7.82 +6.48 +6.94 +7.08 +supplychainp1 022020 +nonopt +nonopt +nonopt +nonopt +supplychainp1 030510 +nonopt +nonopt +nonopt +nonopt +supplychainr1 022020 +6.13 +5.67 +5.47 +5.25 +supplychainr1 030510 +0.28 +0.26 +0.25 +0.37 +syn15m04m +0.54 +0.45 +0.47 +0.49 +53 + +instance +1 thread +4 threads +8 threads +16 threads +syn30m02m +2.51 +2.20 +2.38 +2.21 +synheat +35.92 +28.74 +26.21 +25.37 +tanksize +4.81 +4.17 +4.75 +4.07 +telecomsp pacbell +3393.96 +1396.50 +1189.08 +1401.72 +tln5 +5.26 +1.33 +5.03 +5.53 +tln7 +2935.35 +1739.45 +3085.42 +2.7% +tls2 +0.13 +0.12 +0.22 +0.12 +tls4 +35.15 +16.87 +14.67 +20.79 +topopt-mbb 60x40 50 +∞ +∞ +∞ +∞ +toroidal2g20 5555 +3747.17 +114.31 +81.54 +190.59 +toroidal3g7 6666 +6.2% +74.7% +74.7% +74.7% +transswitch0009r +8% +8% +8% +8% +tricp +0.82 +1.25 +1.11 +1.14 +tspn08 +10.81 +9.28 +11.25 +9.49 +tspn15 +884.44 +1003.34 +412.58 +410.34 +unitcommit1 +0.27% +0.27% +0.27% +0.27% +unitcommit2 +7200.00 +725.90 +6169.57 +732.94 +wager +15.30 +27.54 +25.87 +27.99 +waste +22.55 +24.58 +23.21 +29.11 +wastepaper3 +9.19 +14.02 +14.63 +12.67 +wastepaper4 +871.50 +417.33 +763.71 +556.85 +wastepaper6 +0.022% +0.022% +7200.00 +0.022% +water4 +44.4% +43.7% +47.6% +49.2% +waternd1 +5.50 +6.83 +6.98 +6.07 +waterno2 02 +5.20 +4.18 +4.33 +5.03 +waterno2 03 +260.33 +262.06 +272.21 +272.70 +waterund01 +0.35% +0.35% +0.35% +0.35% +The following table shows the outcome from running Lindo API in serial and parallel +mode. +instance +1 thread +4 threads +8 threads +16 threads +alan +0.89% +0.89% +0.89% +0.89% +autocorr bern20-05 +100.43 +33.83 +23.88 +21.91 +autocorr bern35-04 +13.5% +2655.83 +1788.33 +2461.81 +ball mk2 10 +3.98 +2.17 +1.28 +1.43 +ball mk2 30 +100.0% +100.0% +100.0% +100.0% +ball mk3 10 +3.58 +1.79 +2.05 +2.17 +batch0812 nc +25.46 +18.73 +17.51 +17.68 +batchs101006m +134.04 +abort +abort +abort +batchs121208m +198.76 +abort +abort +abort +bayes2 20 +0.033% +0.033% +0.033% +infeas +bayes2 30 +7200.00 +abort +7200.00 +infeas +blend029 +abort +205.94 +166.19 +96.01 +blend146 +abort +5.8% +4.2% +6.8% +camshape100 +9.2% +8.9% +8.7% +8.4% +cardqp inlp +134.08 +nonopt +nonopt +nonopt +cardqp iqp +133.22 +nonopt +nonopt +nonopt +carton7 +110.53 +110.35 +110.48 +110.50 +carton9 +287.45 +287.68 +287.13 +287.83 +casctanks +3.4% +3.4% +3.4% +3.4% +cecil 13 +0.32% +0.16% +0.11% +5951.36 +celar6-sub0 +100% +100% +100% +100% +chakra +1.28 +0.34 +infeas +infeas +chem +1172.22 +471.44 +314.57 +329.14 +chenery +0.43% +0.43% +0.43% +abort +chimera k64maxcut-01 +16.4% +16.4% +16.1% +16.7% +chimera mis-01 +nonopt +nonopt +nonopt +nonopt +chp shorttermplan1a +∞ +∞ +∞ +∞ +chp shorttermplan2a +∞ +∞ +∞ +∞ +chp shorttermplan2b +16.6% +6045.30 +5089.35 +3782.12 +clay0204m +4.75 +2.51 +2.56 +2.44 +clay0205m +51.80 +18.76 +11.84 +10.23 +color lab3 3x0 +65.6% +65.6% +65.6% +65.6% +crossdock 15x7 +124% +124% +124% +124% +crossdock 15x8 +124% +124% +124% +124% +crudeoil lee1 07 +63.26 +63.10 +63.51 +63.54 +crudeoil pooling ct2 +1.8% +3.1% +1.6% +1.5% +csched1 +66.81 +65.05 +46.49 +36.59 +csched1a +4.85 +2.82 +2.44 +2.36 +cvxnonsep psig20 +0.48 +0.25 +0.25 +0.25 +cvxnonsep psig30 +3.26 +3.00 +3.05 +2.95 +du-opt +5.35 +5.94 +5.92 +6.74 +du-opt5 +4.70 +4.77 +4.63 +5.17 +edgecross10-040 +0.29 +0.12 +0.12 +0.12 +edgecross10-080 +nonopt +nonopt +nonopt +nonopt +54 + +instance +1 thread +4 threads +8 threads +16 threads +eg all s +abort +abort +abort +abort +eigena2 +∞ +∞ +∞ +∞ +elec50 +798.72 +648.65 +757.22 +656.49 +elf +nonopt +2.56 +infeas +3.02 +eniplac +3% +3952.97 +3085.99 +2683.08 +enpro56pb +1454.20 +564.02 +92.37 +42.35 +ex1244 +9.69 +6.99 +6.71 +6.71 +ex1252a +infeas +infeas +infeas +infeas +faclay20h +nonopt +nonopt +nonopt +nonopt +faclay80 +5096.67 +5092.40 +5092.51 +5120.74 +feedtray +13.79 +13.86 +13.58 +13.71 +fin2bb +1294.03 +3136.42 +0.59% +3467.31 +flay04m +647.41 +231.73 +166.42 +123.11 +flay05m +0.54% +0.038% +5375.73 +3837.92 +flay06m +14.3% +9.5% +8.2% +6.8% +fo7 ar25 1 +1282.08 +662.73 +661.65 +661.84 +fo7 ar3 1 +1847.41 +796.80 +752.45 +954.82 +forest +1.18 +1.18 +1.23 +1.53 +gabriel01 +5.4% +4.5% +4.2% +3.8% +gabriel02 +44.0% +43.0% +42.2% +42.1% +gasnet +64.9% +64.1% +63.9% +63.8% +gasprod sarawak16 +infeas +0.4% +0.4% +1.2% +gastrans582 cold13 95 +∞ +∞ +∞ +∞ +gastrans582 mild11 +∞ +∞ +∞ +∞ +gear +0.05 +0.06 +0.06 +0.06 +gear2 +0.23 +0.23 +0.24 +0.24 +gear4 +infeas +infeas +infeas +infeas +genpooling lee1 +0.41% +680.62 +136.93 +infeas +genpooling lee2 +3724.02 +169.90 +111.71 +86.41 +ghg 1veh +106.61 +60.51 +46.93 +45.15 +gilbert +14.25 +28.89 +25.67 +34.60 +graphpart 2g-0066-0066 +8.7% +8.7% +8.7% +8.7% +graphpart clique-60 +83.3% +83.3% +83.3% +83.3% +gsg 0001 +351.30 +578.61 +338.79 +308.84 +hadamard 5 +129.27 +73.47 +59.47 +35.89 +heatexch spec1 +0.34% +0.14% +0.16% +0.25% +heatexch spec2 +0.041% +0.052% +0.039% +0.038% +hhfair +100.0% +100.0% +100.0% +100.0% +himmel16 +20.42 +8.10 +5.33 +3.63 +house +0.89 +infeas +infeas +infeas +hs62 +4.04 +3.33 +2.13 +1.81 +hvb11 +40.7% +27.4% +20.5% +nonopt +hybriddynamic var +4.11 +1.54 +1.26 +0.78 +hybriddynamic varcc +12.00 +4.55 +3.17 +2.71 +hydroenergy1 +0.93% +0.74% +0.69% +0.59% +ibs2 +abort +abort +abort +abort +johnall +22.36 +22.56 +22.26 +22.48 +kall circles c6b +62.28 +120.62 +90.79 +80.68 +kall congruentcircles c72 +4.74 +7.27 +5.22 +5.33 +kissing2 +867.71 +866.79 +868.19 +869.92 +kport20 +13.1% +13.1% +12.4% +11.5% +kriging peaks-red020 +21.79 +7.24 +4.12 +3.65 +kriging peaks-red100 +140.14 +59.63 +41.23 +32.35 +lop97icx +39.83 +13.81 +nonopt +nonopt +mathopt5 7 +12.35 +10.93 +10.78 +11.87 +mathopt5 8 +9.72 +8.27 +8.20 +9.06 +maxcsp-geo50-20-d4-75-36 +101% +101% +101% +101% +meanvar-orl400 05 e 7 +23.95 +16.40 +14.93 +nonopt +meanvar-orl400 05 e 8 +19.32 +19.40 +abort +19.05 +mhw4d +0.78 +0.63 +0.41 +0.65 +milinfract +infeas +68.0% +67.8% +68.1% +minlphi +1.86 +1.14 +infeas +infeas +multiplants mtg1a +15.6% +11.9% +5.6% +7% +multiplants mtg2 +0.12% +1303.99 +1336.22 +1444.62 +nd netgen-3000-1-1-b-b-ns 7 +680.80 +infeas +760.69 +infeas +netmod kar1 +5948.45 +5.7% +6.2% +4040.34 +netmod kar2 +5917.59 +6.5% +7.6% +infeas +nous1 +2.9% +0.79% +6836.04 +7023.44 +nous2 +2.16 +1.27 +1.08 +1.36 +nvs02 +69.68 +70.03 +69.93 +70.27 +nvs06 +6.64 +6.36 +6.68 +6.68 +oil2 +43.56 +43.27 +43.63 +43.29 +optmass +11.3% +12.8% +12.8% +12.8% +ortez +2.42 +2.38 +2.28 +2.38 +p ball 10b 5p 3d m +26.41 +11.49 +7.87 +9.79 +p ball 15b 5p 2d m +101.62 +32.66 +13.84 +19.15 +parabol5 2 3 +15.7% +15.6% +15.5% +15.5% +parallel +38.93 +153.77 +89.39 +66.77 +55 + +instance +1 thread +4 threads +8 threads +16 threads +pedigree ex485 +5.8% +5.7% +5.1% +2.4% +pedigree ex485 2 +192.97 +128.78 +83.00 +81.80 +pointpack06 +27.80 +11.29 +6.94 +10.78 +pointpack08 +3611.87 +1638.66 +2129.26 +1356.41 +pooling epa1 +infeas +infeas +infeas +abort +pooling epa2 +0.041% +4885.54 +0.081% +6389.62 +portfol buyin +2.34 +2.27 +2.28 +2.29 +portfol card +2.15 +2.19 +1.96 +2.20 +powerflow0014r +100.0% +82.1% +71.3% +60.6% +powerflow0057r +∞ +abort +abort +abort +prob07 +68.41 +155.23 +105.41 +100.26 +process +1.38 +1.88 +1.10 +0.96 +procurement1mot +86.4% +88.5% +89.2% +89.3% +procurement2mot +13.59 +abort +abort +abort +product +465.36 +448.67 +267.76 +infeas +product2 +145.38 +145.40 +145.40 +145.42 +prolog +100.0% +0.40 +infeas +1.50 +qp3 +0.01 +0.01 +0.01 +0.01 +qspp 0 10 0 1 10 1 +632.32 +624.71 +631.86 +623.84 +qspp 0 11 0 1 10 1 +3676.78 +3669.93 +3677.76 +3673.84 +radar-2000-10-a-6 lat 7 +33.14 +33.30 +33.04 +33.00 +radar-3000-10-a-8 lat 7 +558.69 +495.51 +498.02 +538.74 +ravempb +3438.61 +305.70 +26.47 +infeas +risk2bpb +1.46 +1.20 +1.20 +1.19 +routingdelay bigm +nonopt +nonopt +nonopt +nonopt +rsyn0815m +92.08 +18.96 +15.88 +14.40 +rsyn0815m03m +nonopt +nonopt +infeas +nonopt +sfacloc2 2 95 +5.25 +5.49 +5.55 +5.50 +sfacloc2 3 90 +588.76 +496.17 +400.13 +0.24% +sjup2 +3456.91 +2496.08 +2490.04 +2495.30 +slay06m +5.66 +4.64 +4.32 +5.06 +slay07m +6.96 +6.39 +6.64 +6.65 +smallinvDAXr1b010-011 +3.51 +4.58 +4.67 +5.10 +smallinvDAXr1b020-022 +3.46 +4.79 +nonopt +nonopt +sonet17v4 +nonopt +nonopt +nonopt +nonopt +sonet18v6 +nonopt +nonopt +nonopt +nonopt +sonetgr17 +nonopt +1464.38 +nonopt +nonopt +spectra2 +infeas +infeas +infeas +infeas +sporttournament24 +6.7% +5.4% +4.7% +4.7% +sporttournament30 +12.5% +10.4% +11.2% +10.8% +sssd12-05persp +378.13 +abort +abort +abort +sssd18-06persp +7200.00 +7200.00 +abort +abort +st testgr1 +5.51 +2.49 +2.78 +nonopt +st testgr3 +3.67 +3.20 +3.79 +4.19 +steenbrf +0.75 +0.73 +0.77 +1.52 +stockcycle +3.18 +3.97 +4.50 +3.75 +supplychainp1 022020 +20.8% +13.9% +11.9% +11.8% +supplychainp1 030510 +63.91 +66.23 +63.40 +48.63 +supplychainr1 022020 +12.5% +8.8% +7% +2.9% +supplychainr1 030510 +7.19 +8.73 +8.30 +8.18 +syn15m04m +42.37 +19.93 +13.59 +14.97 +syn30m02m +239.18 +123.93 +79.51 +85.36 +synheat +0.36% +0.19% +0.28% +0.16% +tanksize +13.85 +7.22 +5.12 +4.44 +telecomsp pacbell +2.8% +3.5% +2.4% +5.1% +tln5 +30.1% +26.2% +7.8% +1682.28 +tln7 +61.4% +60.0% +9.9% +28.3% +tls2 +5.43 +3.96 +4.13 +3.86 +tls4 +833.24 +319.16 +276.69 +302.31 +topopt-mbb 60x40 50 +195.38 +194.93 +194.58 +194.68 +toroidal2g20 5555 +nonopt +nonopt +nonopt +nonopt +toroidal3g7 6666 +nonopt +nonopt +nonopt +nonopt +transswitch0009r +16.0% +abort +abort +abort +tricp +infeas +infeas +infeas +infeas +tspn08 +10.9% +10.8% +10.8% +10.7% +tspn15 +12.8% +12.6% +12.4% +12.4% +unitcommit1 +81.17 +68.02 +67.44 +78.21 +unitcommit2 +18.0% +18.0% +17.8% +17.7% +wager +∞ +32.4% +∞ +∞ +waste +46.3% +46.1% +46.1% +46.1% +wastepaper3 +33.90 +26.89 +0.83% +23.07 +wastepaper4 +2320.60 +1388.31 +1206.13 +1041.52 +wastepaper6 +abort +7200.00 +7200.00 +7200.00 +water4 +46.1% +39.1% +42.4% +34.1% +waternd1 +1130.72 +853.76 +525.28 +560.05 +waterno2 02 +26.45 +12.97 +9.72 +10.04 +waterno2 03 +1886.92 +737.09 +532.41 +465.62 +waterund01 +infeas +infeas +infeas +infeas +56 + +The following table shows the outcome from running Octeract in serial and parallel +mode. +instance +1 thread +4 threads +8 threads +16 threads +alan +0.05 +0.06 +0.08 +0.09 +autocorr bern20-05 +4.36 +4.91 +4.88 +7.71 +autocorr bern35-04 +12.7% +4064.07 +1891.23 +2241.67 +ball mk2 10 +0.01 +0.01 +0.02 +0.02 +ball mk2 30 +0.02 +0.02 +0.02 +0.03 +ball mk3 10 +0.04 +0.04 +0.04 +0.04 +batch0812 nc +8.37 +4.85 +3.34 +4.12 +batchs101006m +4.84 +5.55 +5.40 +7.16 +batchs121208m +79.1% +79.1% +79.1% +abort +bayes2 20 +0.033% +0.033% +3777.02 +6089.69 +bayes2 30 +7200.00 +7200.00 +7200.00 +7200.00 +blend029 +29.33 +13.45 +10.27 +11.02 +blend146 +8.8% +3.4% +6.8% +2.6% +camshape100 +19.40 +9.17 +7.12 +9.91 +cardqp inlp +792.06 +258.89 +147.93 +124.77 +cardqp iqp +785.02 +259.33 +148.69 +124.80 +carton7 +6.03 +1.00 +0.89 +0.76 +carton9 +344.94 +7.49 +3.62 +3.55 +casctanks +11.8% +12.1% +11.6% +12.5% +cecil 13 +188.40 +99.55 +134.75 +166.38 +celar6-sub0 +∞ +1637.26 +1641.73 +1562.09 +chakra +7200.00 +0.021% +0.022% +7200.00 +chem +0.18 +0.19 +0.19 +0.28 +chenery +200.71 +77.26 +35.42 +25.05 +chimera k64maxcut-01 +168.60 +55.75 +40.84 +39.93 +chimera mis-01 +1.14 +1.50 +1.38 +1.94 +chp shorttermplan1a +64.74 +0.028% +0.023% +0.042% +chp shorttermplan2a +17.59 +16.76 +16.30 +21.88 +chp shorttermplan2b +0.67% +0.67% +0.71% +0.7% +clay0204m +1.47 +2.10 +2.28 +3.10 +clay0205m +15.10 +13.45 +13.77 +29.70 +color lab3 3x0 +∞ +∞ +5460.67 +∞ +crossdock 15x7 +∞ +6047.29 +2087.00 +1851.98 +crossdock 15x8 +6306.65 +4889.74 +2772.27 +2837.68 +crudeoil lee1 07 +8.56 +8.03 +4.93 +5.83 +crudeoil pooling ct2 +nonopt +nonopt +nonopt +nonopt +csched1 +8.1% +4306.97 +2116.58 +1592.40 +csched1a +17.77 +11.89 +5.44 +5.07 +cvxnonsep psig20 +35.0% +28.1% +24.8% +25.2% +cvxnonsep psig30 +45.6% +36.6% +35.3% +34.4% +du-opt +122.38 +18.42 +15.18 +8.15 +du-opt5 +101.32 +48.22 +17.33 +14.29 +edgecross10-040 +4.16 +0.79 +0.44 +0.63 +edgecross10-080 +6% +3.8% +5205.96 +3.8% +eg all s +102% +76.0% +41.6% +198% +eigena2 +416.89 +461.71 +602.17 +∞ +elec50 +66.4% +66.4% +66.4% +66.3% +elf +2.54 +infeas +infeas +infeas +eniplac +1.54 +1.06 +1.01 +1.61 +enpro56pb +1.98 +2.48 +2.25 +3.52 +ex1244 +81.99 +35.00 +0.056% +0.082% +ex1252a +245.41 +94.34 +42.83 +31.72 +faclay20h +785.28 +413.15 +463.78 +418.22 +faclay80 +∞ +∞ +∞ +∞ +feedtray +82.1% +82.1% +82.2% +82.1% +fin2bb +100.0% +100.0% +100.0% +100.0% +flay04m +0.87 +0.34 +0.27 +0.28 +flay05m +infeas +infeas +infeas +infeas +flay06m +infeas +infeas +566.98 +413.02 +fo7 ar25 1 +8.95 +3.12 +2.85 +1.49 +fo7 ar3 1 +18.33 +3.23 +2.13 +2.18 +forest +nonopt +nonopt +nonopt +nonopt +gabriel01 +2% +0.28% +3548.92 +2168.66 +gabriel02 +7078.04 +2457.11 +1088.23 +588.74 +gasnet +96.7% +96.4% +96.2% +96.1% +gasprod sarawak16 +0.74% +0.68% +0.5% +0.4% +gastrans582 cold13 95 +∞ +∞ +∞ +∞ +gastrans582 mild11 +∞ +∞ +∞ +∞ +gear +0.11 +0.09 +0.09 +0.14 +gear2 +0.14 +0.13 +0.13 +0.26 +gear4 +17.56 +7.25 +3.29 +2.45 +genpooling lee1 +117.04 +40.77 +20.27 +15.25 +genpooling lee2 +186.63 +77.47 +36.61 +23.73 +ghg 1veh +12.42 +infeas +3.58 +4.05 +gilbert +0.9% +9.7% +9.7% +22.5% +57 + +instance +1 thread +4 threads +8 threads +16 threads +graphpart 2g-0066-0066 +0.75 +0.19 +0.21 +0.22 +graphpart clique-60 +2890.51 +1502.07 +287.82 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+0.18% +0.13% +0.11% +0.097% +The following table shows the outcome from running SCIP (1 thread) and FiberSCIP +(4, 8, 16 threads). +instance +1 thread +4 threads +8 threads +16 threads +alan +0.19 +1.05 +1.06 +1.13 +autocorr bern20-05 +16.34 +33.08 +17.10 +18.19 +autocorr bern35-04 +107.46 +158.09 +101.10 +64.17 +ball mk2 10 +0.05 +1.05 +1.06 +1.12 +ball mk2 30 +0.06 +1.06 +1.06 +1.14 +ball mk3 10 +0.00 +0.00 +0.00 +0.00 +batch0812 nc +1.45 +1.06 +1.08 +1.13 +batchs101006m +7.63 +abort +abort +abort +batchs121208m +6.99 +abort +7.15 +abort +bayes2 20 +0.033% +0.97% +0.033% +0.033% +bayes2 30 +7200.00 +7200.00 +7200.00 +7200.00 +blend029 +3.67 +3.06 +3.09 +2.12 +blend146 +471.12 +280.13 +119.13 +176.20 +camshape100 +5.4% +10.5% +10.4% +9.7% +cardqp inlp +2751.70 +2408.46 +1083.38 +1285.54 +cardqp iqp +2769.87 +2406.76 +1190.43 +758.44 +carton7 +13.54 +10.89 +6.94 +8.69 +carton9 +36.30 +34.26 +23.15 +20.50 +casctanks +199% +114% +114% +114% +cecil 13 +605.03 +0.36% +72.23 +71.31 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+unitcommit2 +439.70 +3.81 +5.85 +6.92 +wager +3.46 +2.14 +1.16 +2.22 +waste +38.85 +57.1% +55.4% +55.0% +wastepaper3 +4.22 +6.07 +5.09 +5.14 +wastepaper4 +167.16 +abort +abort +40.14 +wastepaper6 +0.023% +0.04% +0.034% +7200.00 +water4 +nonopt +nonopt +nonopt +3473.45 +waternd1 +7.21 +10.06 +4.08 +5.13 +waterno2 02 +2.89 +2.10 +2.11 +1.19 +waterno2 03 +nonopt +33.13 +nonopt +39.3% +waterund01 +1525.91 +1.5% +1.5% +abort +62 + diff --git a/M9AyT4oBgHgl3EQfs_mP/content/tmp_files/load_file.txt b/M9AyT4oBgHgl3EQfs_mP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e6f133885ebf738599b96c864f1cadce2771dbb --- /dev/null +++ b/M9AyT4oBgHgl3EQfs_mP/content/tmp_files/load_file.txt @@ -0,0 +1,9690 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf,len=9689 +page_content='Global Optimization of Mixed-Integer Nonlinear Programs with SCIP 8 Ksenia Bestuzheva∗, Antonia Chmiela†, Benjamin M¨uller‡, Felipe Serrano§, Stefan Vigerske¶, Fabian Wegscheider‖ January 3, 2023 Abstract For over ten years, the constraint integer programming framework SCIP has been extended by capabilities for the solution of convex and nonconvex mixed-integer nonlinear programs (MINLPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' With the recently published version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0, these capabilities have been largely reworked and extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This paper discusses the motivations for recent changes and provides an overview of features that are particular to MINLP solving in SCIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, difficulties in benchmarking global MINLP solvers are discussed and a comparison with several state-of-the- art global MINLP solvers is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 1 Introduction Mixed-integer nonlinear programming (MINLP) concerns with the optimization of an objective function such that a finite set of linear or nonlinear constraints and integrality conditions is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The generality of this problem class means that many real-world applications can be modeled as MINLPs [28, 37, 58, 69], but also that software that can handle this class efficiently becomes extremely complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Solvers for MINLP [19] are often built on top of or by combining solvers for mixed-integer linear programming (MIP) and solvers that find locally optimal solutions for nonlinear programs (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In fact, one of the first commercial MINLP solvers, SCICONIC [7], extends a MIP solver by piecewise linear approximations of low dimensional nonlinear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The first general purpose solver was DICOPT [41], which decomposes the solution of an MINLP into a sequence of MIP and NLP solves [25], thereby building on established software for these two program classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' DICOPT can solve MINLPs where nonlinear constraints are convex to optimality, but works only as a heuristic on nonconvex MINLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The first general purpose solvers to solve also nonconvex MINLPs to optimality were αBB, BARON, and GLOP [4, 60, 65], all based on convexification techniques for nonconvex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Also the solver SCIP (Solving Constraint Integer Programs), for which this paper provides an overview, belongs to the latter category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' ∗Zuse Institute Berlin, Department AIS2T, bestuzheva@zib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='de, ORCID: 0000-0002-7018-7099 †Zuse Institute Berlin, Department AIS2T, chmiela@zib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='de, ORCID: 0000-0002-4809-2958 ‡Zuse Institute Berlin, Department AIS2T, benjamin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='mueller@zib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='de, ORCID: 0000-0002-4463-2873 §Zuse Institute Berlin, Department AIS2T, serrano@zib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='de, ORCID: 0000-0002-7892-3951 ¶GAMS Software GmbH, c/o Zuse Institute Berlin, Department AIS2T, svigerske@gams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='com ‖Zuse Institute Berlin, Department AIS2T 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='00587v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='OC] 2 Jan 2023 In the following, MINLPs of the form min c⊤x, such that g ≤ g(x) ≤ g, b ≤ Ax ≤ b, x ≤ x ≤ x, xI ∈ Z|I|, (MINLP) are considered, where x, x ∈ R n, R := R ∪ {±∞}, x ≤ x, I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , n}, c ∈ Rn, g, g ∈ R m, g ≤ g, g : Rn → R m is specified explicitly in algebraic form, b, b ∈ R ˜m, b ≤ b, and A ∈ R ˜m×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The restriction to a linear objective function is a technical detail of SCIP and without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The software SCIP has been designed as a branch-cut-and-price framework to solve different types of optimization problems, most generally constraint integer programs (CIPs), and most importantly MIPs and MINLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Roughly speaking, CIPs are finite- dimensional optimization problems with arbitrary constraints and a linear objective function that satisfy the following property: if all integer variables are fixed, the remaining subproblem must form a linear or nonlinear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The problem class of CIP was motivated by the modeling flexibility of constraint programming and the algorithmic requirements of integrating it with efficient solution techniques available for MIP [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In order to solve CIPs, SCIP constructs relaxations – typically linear programs (LPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If the relaxation solution is not feasible for the current subproblem, the plugins that handle the violated constraints need to take measures to eventually render the relaxation solution infeasible for the updated relaxation, for example by branching or separation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A plethora of additional plugin types, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', for presolving, finding feasible solutions, or tightening variable bounds, allow accelerating the solution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' After 20 years of development of the framework itself and included plugins, SCIP includes mature solvers for MIP, MINLP, as well as several other problem classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since November 2022, SCIP is freely available under an open-source license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SCIP solves problems like (MINLP) to global optimality via a spatial branch-and- bound algorithm that mixes branch-and-infer and branch-and-cut [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Important parts of the solution algorithm are presolving, domain propagation (that is, tightening of variable bounds), linear relaxation, and branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A distinguishing feature of SCIP is that its capabilities to handle nonlinear constraints are not limited to MINLPs, but can be used for any CIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For example, problems can be handled where linear and nonlinear constraints are mixed with typical constraints from constraint programming, as long as appropriate constraint handlers have been included in SCIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since most constraint handlers in SCIP construct a linear relaxation of their constraints, also the handling of nonlinear constraints focuses on linear relaxations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The emphasis on handling CIPs with nonlinear constraints rather than MINLP only is also a reason that the use of nonlinear relaxations or reformulations of complete MINLPs into other problem types, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', mixed-integer conic programs, has not been explored much so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The development of SCIP initially focused on solving CIPs where fixing all integer variables resulted in a linear program [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, it was soon realized that this requirement was not actually enforced by the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' As long as constraint handlers were able to resolve infeasibilities by separation, branching, or other means, the problem could be handled by SCIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' First experiments to handle nonlinear constraints in continuous variables were conducted for bilinear mixing constraints in mine production planning [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The positive results of these experiments motivated the decision to include support for more general nonlinear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' With version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 (2009), initial support for quadratic constraints (convex or nonconvex) and solving quadratically constrained 2 programs (QCPs) to local optimality by Ipopt [75] was added [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 (2010), a primal heuristic that solves sub-MIPs was added [10] and other large-neighborhood- search heuristics were extended to create sub-MINLPs [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, second-order cone constraints in three variables could be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' More general nonlinear constraints, specified in algebraic form, were first supported by SCIP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 (2011) [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Next to the specialized treatment for quadratic constraints, also handlers for signpower constraints (x|x|p = z for some p ≥ 1) [32] and 1-convex bivariate constraints (f(x, y) = z for f being convex or concave whenever x or y has been fixed) [6] were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' With the basic handling of nonlinear constraints in place [74], the next releases were dedicated to adding features that improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SCIP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 (2012) brought optimization-based bound tightening (OBBT) [33] and an NLP diving heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SCIP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 (2015) added a reformulation of general quadratic constraints into second- order cone constraints and separation for edge-concave decompositions of quadratic constraints [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' With SCIP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 (2017), higher-dimensional second-order cone con- straints were disaggregated, KKT conditions for quadratic programs were utilized, multiple starting points were tried for NLP solves, solutions of the LP relaxation were projected onto a convex NLP relaxation, and also OBBT could be performed on the NLP instead of the LP relaxation [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Improved convexification of bilinear constraints by use of additional linear constraints [54], a new primal heuristic that solves a sequence of NLP reformulations, and interfaces to the NLP solvers filterSQP and Worhp [27, 20, 56] were added for SCIP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 (2017) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The following two major releases brought a branch-and-price based solver for ring-packing [35] (SCIP 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0, 2018) and support for convex nonlinear subproblems in Benders Decomposition (SCIP 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 [31], 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' That versions 6 and 7 added comparatively few features for MINLP was due to an ongoing complete overhaul on the way how nonlinear constraints were handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The primary motivation for this change, which was released with SCIP 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 (2022) [14], was to increase the reliability of the solver and to alleviate numerical issues that arose from problem reformulations and led to SCIP returning solutions that are feasible in the reformulated problem, but infeasible in the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' More precisely, previous SCIP versions built an extended formulation of (MINLP) explicitly, with the consequence that the original constraints were no longer included in the presolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Even though the formulations were theoretically equivalent, it was possible that ε-feasible solutions for the reformulated problem were not ε-feasible in the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SCIP 8 remedies this by building an implicit extended formulation as an annotation to the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A second motivation for the major changes in SCIP 8 was to reduce the ambiguity of expression and nonlinear structure types by implementing different plugin types for low-level structure types that define expressions, and high-level structure types that add functionality for particular, sometimes overlapping structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Finally, new features for improving the solver’s performance on MINLPs were introduced with SCIP 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' These include intersection, SDP (semi-definite programming), and RLT (reformulation linearization technique) cuts for quadratic expressions [21, 16], perspective strengthening [15], and symmetry detection [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SCIP can read MINLPs from files in the following formats: LP, MPS, NL (AMPL), OSiL, PIP, and ZIMPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition, problems can be passed to SCIP via interfaces to a variety of programming languages and modeling packages, including AMPL, C, GAMS, Java, Julia, Python, and MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The following section provides an overview of the MINLP solving capabilities of SCIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Afterwards, the performance of SCIP is compared with that of other state-of-the-art global solvers for MINLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 3 2 MINLP capabilities of SCIP In the following, an overview of the facilities available in SCIP that are specific to the handling of MINLPs is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' First, available nonlinear functions are listed and the integration of nonlinear constraints into the branch-and-cut solver of SCIP is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Next, the concept of a nonlinear handler is introduced, which is a new plug-in type that has been added with SCIP 8 and facilitates the integration of extensions that handle specific nonlinear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The remainder of this section gives an overview of features available in SCIP that increase the efficiency of MINLP solving, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', cut generators to tighten the linear relaxation, presolve reductions to simplify the problem, and primal heuristics to find feasible solutions early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To be concise, the presentation has been limited to high-level descriptions that spare technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Unless specified otherwise, more details are often found in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Framework 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Expressions Algebraic expressions are well-formed combinations of constants, variables, and various algebraic operations such as addition, multiplication, and exponentiation, that are used to describe mathematical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' They are often represented by a directed acyclic graph with nodes representing variables, constants, and operations and arcs indicating the flow of computation, see Figure 1 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' � 2 log x � 2 2 y Figure 1: Expression graph for algebraic expression log(x)2 + 2 log(x)y + y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Also in SCIP, expressions are stored as directed acyclic graphs, while all semantics of expression operands are defined by expression handler plugins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' These handler implement callbacks that are used by methods in the SCIP core to manage expressions (create, modify, copy, free, parse, print), to evaluate and compute derivatives at a point, to evaluate over intervals, to simplify, to identify common subexpressions, to check curvature and integrality, and to iterate over it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Some additional expression handler callbacks are used by the constraint handler for nonlinear constraints (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2) exclusively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Expression handlers for the following operators are included in SCIP 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0: val: scalar constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' var: a SCIP variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' sum: an affine-linear function, y �→ a0 + �k j=1 ajyj for y ∈ Rk with constant coefficients a ∈ Rk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' prod: a product, y �→ c �k j=1 yj for y ∈ Rk with constant factor c ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 4 pow: a power with a constant exponent, y �→ yp for y ∈ R and exponent p ∈ R (if p ̸∈ Z, then y ≥ 0 is required);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' signpower: a signed power, y �→ sign(y)|y|p for y ∈ R and constant exponent p ∈ R, p > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' exp: exponentiation, y �→ exp(y) for y ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' log: natural logarithm, y �→ log(y) for y ∈ R>0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' entropy: entropy, y �→ � −y log(y), if y > 0, 0, if y = 0, for y ∈ R≥0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' sin: sine, y �→ sin(y) for y ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' cos: cosine, y �→ cos(y) for y ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' abs: absolute value, y �→ |y| for y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In previous versions of SCIP, also high-level structures such as quadratic functions could be represented as expression types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To avoid ambiguity and reduce complexity, this has been replaced by a recognition of quadratic expressions that is no longer made explicit by a change in the expression type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 Constraint Handler for Nonlinear Constraints All nonlinear constraints g ≤ g(x) ≤ g of (MINLP) are handled by the constraint handler for nonlinear constraints in SCIP, while the linear constraints b ≤ Ax ≤ b are handled by the constraint handlers for linear constraints and its specializations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', knapsack, set-covering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A constraint handler is responsible for checking whether solutions satisfy constraints and, if that is not the case, to resolve infeasibility by enforcing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This applies in particular to solutions of the LP relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The nonlinear constraint handler currently enforces constraints by the following means: DOMAINPROP by analyzing the nonlinear constraints with respect to the variable bounds at the current node of the branch-and-bound tree, infeasibility or a bound tightening may be deduced, which allow pruning the node or cutting off the given solution, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' this is also known as domain propagation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SEPARATE a cutting plane that is violated by the given solution may be computed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' BRANCH the current node of the branch-and-bound tree is subdivided, that is, a variable xi and a branching point ˜xi ∈ [xi, xi] are selected and two child nodes with xi restricted to [xi, ˜xi] and [˜xi, xi], respectively, are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To decide whether a node can be pruned (DOMAINPROP), an overestimate of the range of g(x) with respect to current variable bounds is computed by means of interval arithmetics [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If a constraint k is found such that gk([x, x]) ∩ [gk, gk] = ∅, then there exists no point in [x, x] for which this constraint is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A bound tightening may be computed by applying the same methods in reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' That is, interval arithmetic is used to overestimate g−1([g, g]), the preimage of g(x) on [g, g], and variable bounds are tightened to [x, x] ∩ g−1([g, g]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This is also known as feasibility-based bound tightening (FBBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In the simplest case, callbacks of expression handlers are used to propagate intervals through expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, in some cases, other methods that take more structure into account or that use additional information to tighten variable bounds and constraint sides are used (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To construct a linear relaxation of the nonlinear constraints (SEPARATE option), 5 an extended formulation is considered: min c⊤x, such that hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) ⋚i wi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , ˆm, b ≤ Ax ≤ b, x ≤ x ≤ x, w ≤ w ≤ w, xI ∈ Z|I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' (MINLPext) The functions hi are obtained from the expressions that define functions gi by recursively annotating subexpressions with auxiliary variables wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm for some ˆm ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Initially, slack variables w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , wm are introduced and assigned to the root of all expressions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', hi := gi, wi := gi, wi := gi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Next, for each function hi, subexpressions f may be assigned new auxiliary variables wi′, i′ > m, which results in extending (MINLPext) by additional constraints hi′(x) = wi′ with hi′ := f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Bounds wi′ and wi′ are initialized to bounds on hi′, if available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since auxiliary variables in a subexpression of hi always receive an index larger than max(m, i), the result is referred to by hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) for any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , ˆm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' That is, to simplify notation, wi+1 is used instead of wmax(i,m)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If a subexpression appears in several expressions, it is assigned at most one auxiliary variable and reindexing may be necessary to have hi depend on x and wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For the (in)equality sense ⋚i, a valid simplification would be to assume equality everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For performance reasons, though, it can be beneficial to relax certain equalities to inequalities if that does not change the feasible space of (MINLPext) when projected onto x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, ⋚i := � � � � � =, if gi > −∞, gi < ∞, ≤, if gi = −∞, gi < ∞, ≥, if gi > −∞, gi = ∞, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For i > m, monotonicity of expressions is taken into account to derive ⋚i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Whether to annotate a subexpression by an auxiliary variable depends on the structures that are recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In the simplest case, every subexpression that is not already a variable is annotated with an auxiliary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This essentially corresponds to the Smith Normal Form [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For every function hi of (MINLPext), the callbacks of the corresponding expression handler can be used to compute linear under- and overestimators, such that a linear relaxation for (MINLPext) is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' It can, however, be beneficial to not add an auxiliary variable for every subexpression, thus allowing for more complex functions in (MINLPext).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This will be the discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Example Recall Figure 1 and the constraint log(x)2 + 2 log(x) y + y2 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' By annotating the root of the expression graph with a slack variable w1 and each other non-variable node with an auxiliary variable, the extended formulation w2 + 2w3 + w4 ≤ w1, w2 5 ≤ w2, w5 y ≤ w3, y2 ≤ w4, log(x) = w5, w1 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 6 is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Bounds on auxiliary variables have been omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The constraints w2 5 = w2, w5y = w3, and y2 = w4 were relaxed to inequalities because w2 + 2w3 + w4 is monotonically increasing in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, to relax log(x) = w5 to log(x) ≤ w5, both w2 5 and w5y would need to be monotonically increasing in w5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This would be the case if x ≥ 1 and y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If a constraint hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) ≤ wi (the ≥-case is analogous) of (MINLPext) is violated and hi is nonconvex, then linear underestimators on hi can only be as tight as the convex envelope of hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, it may not be possible find a hyperplane that is violated by the solution of the LP relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since the convex envelope of hi depends on the bounds of variables appearing in hi, these variables are candidates for branching (BRANCH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' More precisely, when an expression handler computes a linear under- or overestimator for hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm), it also signals for which variables it used current variable bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Marked original variables are then added to the list of branching candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For an auxiliary variable wi′, i′ > i, the variables in the subexpression that hi′ represents are considered for branching instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The decision on whether to add a cutting plane that separates the solution of the LP relaxation or to branch is rather complex, but the idea is to branch if either no cutting plane is found or if the violation of available cutting planes in the relaxation solution is rather small when compared to the convexification gap of the under/overestimators that define the cutting planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In the latter case, it may be beneficial to first reduce the convexification gap by branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To select one variable from the list of branching candidates, the violation of constraints in (MINLPext) and historical information about the effect of branching on a given variable on the optimal value of the LP relaxation (“pseudo costs”) are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The branching point is a convex combination of the value of the variable in the LP relaxation and the mid-point of the variable’s interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 Nonlinear Handlers In the previous example, four auxiliary variables were introduced to construct the extended formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This is due to the expression handlers having a rather myopic view, basically, implementing techniques that can handle only their direct children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' It is clear that, for this example, an extended formulation that only replaces log(x) by an auxiliary variable w2 could be more efficient to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, this requires methods to detect the quadratic (or convex) structure and to either compute linear underestimators for the quadratic (convex) expression w2 2 + 2w2y + y2 or to separate cutting planes for the set defined by w2 2 + 2w2y + y2 ≤ w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Such structure detection and handling methods are the task of the new nonlinear handler plugins that were introduced with SCIP 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Nonlinear handlers determine the extended formulation (MINLPext) by deciding when to annotate subexpressions with auxiliary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' That is, given a constraint hi(x) ⋚i wi, a nonlinear handler analyses the expression that defines hi and attempts to detect specific structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' At this point, it may also request to introduce additional auxiliary variables, thus changing hi(x) into hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition, it informs the constraint handler that it will now provide separation for hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) ≤ wi, or ≥ wi, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If none of the nonlinear handlers declare that they will handle hi(x) ⋚i wi, auxiliary variables are introduced for each argument of the root of the expression hi and expression handler callbacks are used to construct cutting planes from linear under-/overestimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition to separation, nonlinear handlers can also contribute to domain prop- agation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This is implemented analogously to separation by setting up an additional extended formulation similarly to (MINLPext), with the main difference that slack and auxiliary variables are not actually created in SCIP and equalities are currently not relaxed to inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 7 Note that the extended formulations are stored as annotation on the original expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Thus, for each task, the most suitable formulation can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For example, feasibility is checked on the original constraints, domain propagation and separation use the corresponding extended formulations, but branching is performed, by default, with respect to original variables only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' With SCIP 7 and earlier, only one extended formulation was constructed explicitly and the connection to the original formulation was no longer available, leading to issues due to not ensuring that solutions are also (ε-)feasible for the original constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition to the improved numeric reliability, the nonlinear handlers also allow for a higher flexibility when handling nonlinear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For each node in an expression, more than one nonlinear handler can be attached, each one annotating possibly different subexpressions with auxiliary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For example, for a nonconvex quadratic constraint � i,j ai,jxixj ≤ w, the nonlinear handler for quadratics can declare that it will provide separation (by intersection cuts, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5), but that also other means of separation should be tried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, since no other nonlinear handler declares that it will provide separation, auxiliary variables are introduced for each argument of the sum, that is, an auxiliary variable Xij is assigned to each product xixj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For the corresponding constraints xixj ≤ Xij (if ai,j ≥ 0), the well-known McCormick underestimators [49], Xij ≥ xixj + xjxi − xixj, Xij ≥ xixj + xjxi − xixj, (1) or other means (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2) will be used to construct a linear relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4 NLP Relaxation Similar to the central LP relaxation of SCIP, an NLP relaxation is also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In contrast to constraint handlers, the NLP relaxation uses a common data structure to store its constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' At the moment, constraint handlers for linear constraints and the constraint handler for nonlinear constraints store a representation of their constraints in the NLP relaxation, so that in case of a MINLP, the NLP relaxation together with the integrality conditions on variables provides a unified view of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For nonlinear constraints, the original (non-extended) form g ≤ g(x) ≤ g is added to the NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To find local optimal solutions for the NLP relaxation, interfaces to the NLP solvers filterSQP, Ipopt, and Worhp [27, 75, 20] are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' First- and second-order derivatives for these solvers are computed via CppAD [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The NLP relaxation is mainly used by some primal heuristics (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='8) and separators (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2) at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 Presolving When presolving nonlinear constraints, expressions are simplified and brought into a canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For example, recursive sums and products are flattened and fixed or aggregated variables are replaced by constants or sums of active variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition, it is ensured that if a subexpression appears several times (in the same or different constraints), always the same expression object is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This ensures that in the extended formulation (MINLPext) at most one auxiliary variable is attached to such common subexpressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Variable Fixings Similar to what has been shown by Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' [38], if a bounded variable xj does not appear in the objective (cj = 0), but in exactly one constraint gk ≤ gk(x) ≤ gk where gk(x) is convex in xj for any fixing of other variables and gk = +∞ (or concave in xj and gk = −∞), then there always exists an optimal solution where xj ∈ {xj, xj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 8 For example, if y ∈ [0, 1] appears only in a constraint xy + yz − y2 ≤ 5, then y can be changed to a binary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SCIP recognizes such variables for polynomial constraints (under additional assump- tions [14]) and changes the variable type to binary, if xj = 0 and xj = 1, or adds a bound disjunction constraint xj ≤ xj ∨ xj ≥ xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' As a consequence, branching on xj leads to fixing the variable in both children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 Linearization of Products The introduction emphasized that with SCIP 8, an explicit extended reformulation of nonlinear constraints is avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' An exception that proves this “rule” is the linearization of products of binary variables in presolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Doing so has the advantage that more of SCIP’s techniques for MIP solving can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In the simplest case, a product � i xi is replaced by a new variable z and a constraint of type “and” that models z = � i xi is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The “and”-constraint handler will then separate a linearization of this product [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For a product of only two binary variables, the linearization is added directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For a quadratic function in binary variables with many terms, the number of variables introduced may be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Thus, in this case, a linearization that requires fewer additional variables is used, even though it may lead to a weaker relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 KKT Strengthening for QPs A presolving method that aims to tighten the relaxation of a quadratic program (QP) by adding redundant constraints derived from Karush-Kuhn-Tucker (KKT) conditions is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Consider a quadratic program of the form min 1 2 x⊤Qx + c⊤x, such that Ax ≤ b, (QP) where Q ∈ Rn×n is symmetric, c ∈ Rn, A ∈ Rm×n, and b ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If (QP) is bounded, then all optima of (QP) satisfy the following KKT conditions: Qx + c + A⊤µ = 0, Ax ≤ b, µi(Ax − b)i = 0, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , m}, µ ≥ 0, (KKT) where µ is the vector of Lagrangian multipliers of the constraints Ax ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In a specialized presolver, SCIP recognizes whether (MINLP) is equivalent to (QP) by checking whether a quadratic objective function has been reformulated into a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If a (QP) has been found and all variables are bounded, then the equations (KKT) are added as redundant constraints to the problem, whereby the complementarity constraints are formulated via special ordered sets of type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The redundant constraints can help to strengthen the linear relaxation and prioritize branching decisions to satisfy the complementarity constraints, which focuses the search more on the local optima of (QP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition to (QP), the implementation can also handle mixed-binary quadratic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For all details, see [47, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' When this presolver was added to SCIP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0, it has shown to be very beneficial for box-constrained quadratic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Due to the many changes and extensions in SCIP 8, in particular for the handling of quadratic constraints (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3), it needs to be reevaluated under which conditions this presolver should be enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Currently, it is disabled by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4 Symmetry Detection Symmetries in a MINLP are automorphisms on Rn that map optimal solutions to optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Such symmetries have an adverse effect on the performance of branch-and- bound solvers, because symmetric subproblems may be treated repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, SCIP can enforce lexicographically maximal solutions from an orbit of symmetric solutions via bound tightening and separation of linear inequalities [39, 34, 31, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since optimal solutions are naturally not known in advance, the symmetry detection resorts to find permutations of variables that map the feasible set onto itself and map each point to one with the same objective function value [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' These permutations are given by isomorphisms in an auxiliary symmetry detection graph, which is constructed from the problem data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', c, A, I, and the expressions that define g(x)) [43, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 Quadratics Since quadratic functions frequently appear in MINLPs (every second instance of MINLPLib [50] has only linear and quadratic constraints), a number of techniques have been added to SCIP to handle this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Next to the presolving methods that were discussed in the previous section, three nonlinear handlers and four separators deal with quadratic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' When none of the nonlinear handlers are active, then for each square and bilinear term in a quadratic function, an auxiliary variable is added in the extended formulation and gradient, secant, and McCormick under- and overestimators (see (1)) are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Domain Propagation If variables appear more than once in a quadratic function, then a term-wise domain propagation does not necessarily yield the best possible results, due to suffering from the so-called dependency problem of interval arithmetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For example, it is easy to compute the range for x2 + x for given bounds on x, or bounds on x for a given interval on x2 + x, but standard interval arithmetics would treat the terms x2 and x separately, which can lead to overestimating the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, a specialized nonlinear handler in SCIP provides a domain propagation procedure for quadratics that aims to reduce overestimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For this, the detection routine of the nonlinear handler writes a quadratic expression as q(y) = k � i=1 qi(y) with qi(y) = aiy2 i + ciyi + � j∈Pi bi,jyiyj, (2) where yi is either an original variable (x) or another expression, ai, ci ∈ R, bi,j ∈ R\\{0}, j ∈ Pi ⇒ i ̸∈ Pj for all j ∈ Pi, Pi ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , k}, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For functions qi with at least two terms (at least two of ai, bi,j, j ∈ Pi, and ci are nonzero), a relaxation is obtained by replacing each yj by [yj, yj], j ∈ Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For this univariate quadratic interval-term in yi, tight bounds can be computed [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition, bounds on variables yj, j ∈ Pi, are computed by considering � j∈Pi bi,jyj ∈ ([q, q] − � i′̸=i qi′(y))/yi − aiyi − ci, yi ∈ [yi, yi], (3) where [q, q] are given bounds on q(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' After relaxing each qi′ to an interval, bounds on the right-hand side of (3) are computed, which are then used to calculate bounds on each yj, j ∈ Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 Bilinear Terms For a product y1y2, where y1 and y2 are either non-binary variables or other expressions, the expression handler for products already provides linear under- and overestimators and domain propagation that is best possible when considering the bounds [y1, y1] × [y2, y2] only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, if linear inequalities in y1 and y2 are available, then possibly tighter linear estimates and variable bounds can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In SCIP, this is done by a specialized nonlinear handler that implements the algorithm by Locatelli [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The inequalities are found by projection of the LP relaxation onto variables (y1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For more details, see [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' An alternative method that uses linear constraints to tighten the relaxation of quadratic constraints are the RLT cuts described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 RLT Cuts The Reformulation-Linearization Technique (RLT) [2, 3] has proven very useful to tighten relaxations of polynomial programming problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In SCIP, a separator of cuts that are computed via RLT for bilinear product relations in (MINLPext) is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For simplicity, denote by Xij the auxiliary variable that is associated with a constraint xixj ⋚ Xij of (MINLPext) (Xji denotes the same variable as Xij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Recall that it is valid to replace ⋚ by =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Given Xij = xixj, where xi ∈ [xi, xi], xj ∈ [xj, xj], and a linear constraint a⊤x ≤ b, RLT cuts are derived by first multiplying the constraint by a nonnegative bound factors (xi − xi), (xi − xi), (xj − xj), or (xj − xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For instance, consider multiplication by the factor (xi − xi), which yields a valid nonlinear inequality: a⊤x (xi − xi) ≤ b (xi − xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' (4) This is referred to as the reformulation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The linearization step is then performed for all terms xkxi in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If a product relation Xki = xkxi exists, then the product is replaced with Xki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If xk and xi are contained in the same clique, the product is replaced with an equivalent linear expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Otherwise, it is replaced by a linear under- or overestimator such as (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition, the RLT separator can reveal linearized products between binary and continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To do so, it checks whether pairs of linear inequalities that are defined in the same triple of variables (one of them binary, the other two continuous) imply a product relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' These implicit products can then be used in the linearization step of RLT cut generation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4 SDP Cuts As in the previous section, denote by Xij the auxiliary variable that is associated with a constraint xixj ⋚ Xij of (MINLPext).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A popular convex relaxation of the condition X = xx⊤ is given by requiring X − xx⊤ to be positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Separation for the set {(x, X) : X − xx⊤ ⪰ 0} itself is possible, but cuts are typically dense and may include variables Xij for products that do not exist in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, only principal 2 × 2 minors of X − xx⊤, which also need to be positive semidefinite, are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' By Schur’s complement, this means that the condition Aij(x, X) := � � 1 xi xj xi Xii Xij xj Xij Xjj � � ⪰ 0 (5) needs to hold for any i, j, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A separator in SCIP detects minors for which Xii, Xjj, Xij exist in (MINLPext) and enforces Aij(x, X) ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To do so for a solution (ˆx, ˆX) that violates (5), an eigenvector v ∈ R3 of Aij(ˆx, ˆX) with v⊤Aij(ˆx, ˆX)v < 0 is computed and the globally valid linear inequality v⊤Aij(x, X)v ≥ 0 is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5 Intersection Cuts Intersection cuts [70, 5] have shown to be an efficient tool to strengthen relaxations of MIPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Recently, Mu˜noz and Serrano showed how to compute the tightest possible intersection cuts for quadratic programs [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This method has been implemented in SCIP [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Assume a nonconvex quadratic constraint of (MINLPext) is q(y) ≤ w with q(y) as in (2) and w an auxiliary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The separation of intersection cuts is implemented for the set S := {(y, w) ∈ Rk : q(y) ≤ w} that is defined by this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Let (ˆy, ˆw) be a basic feasible LP solution violating q(y) ≤ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' First, a convex inequality g(y, w) < 0 is build that is satisfied by (ˆy, ˆw), but by no point of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This defines a so-called S-free set C = {(y, w) ∈ Rk+1 : g(y, w) ≤ 0}, that is, a convex set with (ˆy, ˆw) ∈ int(C) containing no point of S in its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The quality of the resulting cut highly depends on which S-free set is used, but using maximal S-free sets yield the tightest possible intersection cuts [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' By using the conic relaxation K of the LP-feasible region defined by the nonbasic variables at (ˆy, ˆw), the intersection points between the extreme rays of K and the boundary of C are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The intersection cut is then defined by the hyperplane going through these points and successfully separates (ˆx, ˆw) and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' See Figure 2 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To obtain even better cuts, there is also a strengthening procedure implemented that uses the idea of negative edge extension of the cone K [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' S C K ( ̂y, ̂w) Figure 2: An intersection cut (red) separating the basic feasible LP solution (ˆy, ˆw) from S (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The cut is computed using the intersection points of an S-free set C (orange) and the rays of a simplicial cone K ⊇ S (boundary in green) with apex (ˆy, ˆw) ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition to the separation of intersection cuts for a set S given by a constraint q(y) ≤ w, SCIP can also generate intersection cuts for implied quadratic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Recall the matrix of auxiliary variables X as introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The con- dition X = xx⊤ implies that X needs to have rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, any 2 × 2 minor �Xi1j1 Xi1j2 Xi2j1 Xi2j2 � of X needs to have determinant zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' That is, for any set of variable indices i1, i2, j1, j2 with i1 ̸= i2 and j1 ̸= j2, the condition Xi1j1Xi2j2 = Xi1j2Xi2j1 needs to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If all variables in this condition exist in (MINLPext) and a solution violates this condition, then the previously described procedure to generate intersection cuts is applied to the set defined by this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since intersection cuts can be rather dense, it is not clear yet how to decide when it will be beneficial to generate such cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Their separation is therefore currently disabled by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For more details, see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 12 2 0 0 2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='6 Edge-Concave Cuts Another method to obtain a linear outer-approximation for a quadratic constraint is by utilizing an edge-concave decomposition of the quadratic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This has shown to be particularly useful for randomly generated quadratic instances [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A function is edge-concave over the variables’ domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', [x, x]) if it is componentwise concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Given a quadratic function, the separator for edge-concave cuts solves an auxiliary MIP to partition the square and bilinear terms into a sum of edge-concave functions and a remaining function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since the convex envelope of edge-concave functions is vertex-polyhedral [67], that is, it is a polyhedral function with vertices corresponding to the vertices of the box of variable bounds, facets on the convex envelope of each edge-concave function can be computed by solving an auxiliary linear program (see also Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For the function of remaining terms, term-wise linear underestimators such as (1) are summed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since the current implementation of edge-concave cuts in SCIP has not shown to be particularly useful for general MINLP, this separator is disabled for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='7 Second-Order Cones An important connection between MINLP and conic programming is the detection of constraints that can be represented as a second-order cone (SOC) constraint, since the latter defines a convex set, while the original constraint may use a nonconvex constraint function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A specialized nonlinear handler aims to detect SOC representable structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In the detection phase, a constraint hi(x) ≤ wi (the case ≥ is handled similarly) of the extended formulation (MINLPext) is passed to the nonlinear handler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For this constraint, it is checked whether it defines a bound on an Euclidian norm ( ��k j=1(ajy2 j + bjyj) + c ≤ wi for some coefficients aj, bj, c ∈ R, aj > 0, where yj is either an original variable or some subexpression of hi(·)), or is a quadratic constraint that is SOC-representable [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since the introduction of slack variables wi, i ≤ m, may prevent such a detection, the equivalent constraint hi(x) ≤ ¯wi is considered instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A detected SOC constraint is stored in the form � � � � k � j=1 (v⊤ j y + βj)2 ≤ v⊤ k+1y + βk+1 (6) with vj ∈ Rℓ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , k + 1, where y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , yℓ are variables of (MINLPext).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since the left-hand side of (6) is convex, a solution ˆy that violates (6) can be separated by linearization of the left-hand side of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, if there are many terms on the left-hand side of (6) (k being large), then it can require many cuts to provide a tight linear relaxation of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Thus, a disaggregation of the cone [72] is used if k ≥ 3: (v⊤ j y + βj)2 ≤ zj(v⊤ k+1y + βk+1), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , k, (7) k � j=1 zj ≤ v⊤ k+1y + βk+1, (8) where variables z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , zk are new variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A solution (ˆy, ˆz) that violates (6) needs to violate also (7) for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , k} or (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The latter is already linear and can be added as a cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If a rotated second-order cone constraint (7) is violated for some j, then it is transformed into the standard form � 4(v⊤ j y + βj)2 + (v⊤ k+1y + βk+1 − zj)2 ≤ v⊤ k+1y + βk+1 + zj and a gradient cut is constructed by linearization of the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4 Convexity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Convex and Concave Constraints For the linear underestimation of functions like x exp(x) or x2 + 2xy + y2, the construc- tion of an extended formulation (xw, exp(x) = w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' w1 + 2w2 + w3, w1 = x2, w2 = xy, w3 = y2) is not advisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Instead, hyperplanes that support the epigraph of a convex function can be used if convexity is recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In SCIP, specialized nonlinear handlers are available to detect for a function hi(x) of (MINLPext) the subexpressions that need to be replaced by auxiliary variables wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm such that the remaining expression hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) is convex or concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The detection utilizes the often applied rules for convexity/concavity of function compositions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', f convex and monotone decreasing, g concave ⇒ f ◦ g convex), but applies them in reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' That is, instead of deciding whether a function is convex/concave based on information on the convexity/concavity and monotonicity of its arguments, the algorithm formulates condi- tions on the convexity/concavity of the function arguments given a convexity/concavity requirement on the function itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' When a condition on an argument cannot be fulfilled, it is replaced by an auxiliary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Next to “myopic” rules for convexity/concavity that are implemented by the expres- sion handlers, also rules for product compositions (af(bg(x) + c)g(x) with constants a, b, c and repeating subexpression g(x)), signomials (c �k j=1 f pj j (x) with c, pj ∈ R and subexpressions fj(x), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , k), and quadratic forms are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The latter may check for definiteness of its Hessian by calculating its eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, it has been shown that for a composition of convex functions f ◦ g, it can be beneficial for the linear relaxation to consider the extended formulation f(w), w ≥ g(x), instead of the composition f(g(x)) [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This is enforced by a small variation of the detection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' When a convex constraint hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) ≤ wi of (MINLPext) is violated at a point (ˆx, ˆw), a tangent on the graph of hi at (ˆx, ˆw) is used to compute a separating hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The slope of the tangent is given by the gradient of hi at (ˆx, ˆw), which is calculated via automatic differentiation on the expression graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If, however, hi is univariate, that is, hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) = f(y) for some variable y, and y is integral, then taking the hyperplane through the points (⌊ˆy⌋, f(⌊ˆy⌋)) and (⌊ˆy⌋ + 1, f(⌊ˆy⌋ + 1)) can give a tighter underestimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For a concave function hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm), any hyperplane αx + βw + γ that underestimates hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) in all vertices of the box [x, x] × [wi+1, wi+1] × · · × [w ˆm, w ˆm] is a valid linear underestimator, since hi is vertex-polyhedral with respect to the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Maximizing αˆx + β ˆw + γ such that αx + βw + γ does not exceed hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) for all vertices gives an underestimator that is as tight as possible at a given reference point (ˆx, ˆw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For the frequent cases k = 1 and k = 2, routines that directly compute such an underestimator are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For k > 2, a linear program is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since the size of this LP is exponential in k, underestimators for concave functions in more than 14 variables are currently not computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 Tighter Gradient Cuts The separating hyperplanes generated for convex functions of (MINLPext) as discussed in the previous section are, in general, not supporting for the feasible region of (MINLPext), because the point where the functions are linearized is not at the boundary of the feasible region (which is the reason why it needs to be separated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, often several rounds of cut generation and LP solving are required until the relaxation solution satisfies the convex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Solvers for convex MINLP have handled this problem in various ways [25, 42], but the basic idea is to build gradient cuts at a suitable boundary point of the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 14 In SCIP, three procedures for building tighter and/or deeper gradient cuts for convex relaxations are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The first two methods compute a point on the boundary of the set defined by all convex constraints of (MINLP) that is close to the point to be separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The first method solves an additional nonlinear program to project the point to be separated onto the convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since solving an NLP for every point to be separated can be quite expensive, the second method, going back to an idea by Veinott [71], does a binary search between an interior point of the convex set and the point to be separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The interior point is computed once in the beginning of the search by solving an auxiliary NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For more details, see [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The third method does not aim to separate a given point, but utilizes the feasible points that are found by primal heuristics of SCIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' When a new solution is found, gradient cuts are generated at this solution for convex constraints of (MINLPext) and added to the cutpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If such a cut is later found to separate the relaxation solution, it is added to the LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' All methods are currently disabled as they require more tuning to be efficient in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5 Quotients Note that the available expression handlers (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1) do not include a handler for quotients, since they can equivalently be written using a product and a power expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, the default extended formulation for an expression y1y−1 2 is given by replacing y−1 2 by a new auxiliary variable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The linear outer-approximation is then obtained by estimating y1w and y−1 2 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, tighter linear estimates are often possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, a specialized nonlinear handler checks whether a given function hi(x) can be cast as f(y) = ay1 + b cy2 + d + e (9) with a, b, c, d, e ∈ R, a, c ̸= 0, and y1 and y2 being either original variables or subexpres- sions of hi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Tight linear estimators for (9) are computed by distinguishing a number of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For example, for ay1 + b ≥ 0 and cy2 + d > 0 (if c > 0), a linear underestimator is obtained by computing a tangent on the graph of the convex underestimator of f that is given by [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A linear overestimator is obtained by computing a facet on the concave envelope of f, which is easy since −f is vertex-polyhedral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Furthermore, in the univariate case (y1 = y2), f is either convex or concave on [y1, y1] if −d/c ̸∈ [y2, y2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since in the univariate case the same variable appears twice, also a specialized domain propagation method that avoids the dependency problem of interval arithmetic is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='6 Perspective Strengthening Perspective reformulations have shown to efficiently tighten relaxations of convex mixed- integer nonlinear programs with on/off-structures, which are often modeled via big-M constraints or semi-continuous variables [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A variable xj is semi-continuous with respect to the binary indicator variable xj′, j′ ∈ I, if it is restricted to the domain [x1 j, x1 j] when xj′ = 1 and has a fixed value x0 j when xj′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In SCIP, a strengthening of under- and overestimators for functions that depend on semi-continuous variables is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Consider a constraint hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) ⋚ wi of (MINLPext) and write hi as a sum of its nonlinear and linear parts: hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) = hnl i (xnl, wnl) + hl i(xl, wl), 15 where hnl i is a nonlinear function, hl i is a linear function, xnl and wnl are the vectors of variables x and w, respectively, that appear only in the nonlinear part of hi, and xl and wl are the vectors of variables x and w, respectively, that appear only in the linear part of hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A strengthening of under- or overestimators for hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) is attempted if xnl and wnl are semi-continuous with respect to the same indicator variable xj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To determine whether a variable xj is semi-continuous, bounds on xj that are implied by fixing a binary variable are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The implied bounds can be obtained either from linear constraints directly or by probing, and are stored by SCIP in a globally available data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If a pair of implied bounds on xj with the same binary variable xj′ is found, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', xj ≤ α(u)xj′ + β(u), xj ≥ α(ℓ)xj′ + β(ℓ), and β(u) = β(ℓ), then xj is a semi-continuous variable with x0 j = β(u), x1 j = α(ℓ) + β(ℓ), and x1 j = α(u)+β(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition, an auxiliary variable wi is found to be semi-continuous if function hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) depends only on semi-continuous variables with the same indicator variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Assume that a linear underestimator ℓ(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) of hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) has been computed and split it into parts corresponding to the nonlinear and linear variables of hi, respectively: ℓ(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) = ℓnl(xnl, wnl) + ℓl(xl, wl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The perspective strengthening consists of extending the part of the underestimator that corresponds to the nonlinear part such that it is tight for xj′ = 0: ℓnl(xnl, wnl) + � hnl i (x0 nl, w0 nl) − ℓnl(x0 nl, w0 nl) � (1 − xj′) + ℓl(xl, wl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The linear part remains unchanged, since it shares none of the variables with the nonlinear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This extension ensures that the estimator is equal to hi(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) for xj′ = 0, xnl = x0 nl, and wnl = w0 nl, and equal to ℓ(x, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , w ˆm) for xj′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If hi is convex, cuts obtained this way are equivalent to the classic perspective cuts [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For more details on the implementation in SCIP, see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' An example is given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Figure 3: The original set {(x, y, w) : x2 ≤ w, y ∈ {0, 1}, y = 0 → x = 0} (left) and a continuous relaxation given by {(x, y, w) : x2 ≤ wy, y ∈ [0, 1], w ≥ 0} (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' From the original set, cuts of the form ˆx2 + 2ˆx(x − ˆx) ≤ w for some reference point ˆx would be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' With perspective strengthening, a linearization on the right set is obtained instead, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', ˆx2+2ˆx(x− ˆx)+ ˆx2(1−y) ≤ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The latter is typically better as it is tight for y = 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 16 w w≥α,y= c,y) )= 0:w y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='7 Optimization-Based Bound Tightening Optimization-Based Bound Tightening (OBBT) is a domain propagation technique which minimizes and maximizes each variable over the feasible set of the problem or a relaxation thereof [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Whereas FBBT (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2) propagates the nonlinearities individually, OBBT considers (a relaxation of) all constraints together, and may hence compute tighter bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, it is rather expensive compared to FBBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In SCIP, OBBT solves two auxiliary LPs for each variable xk that could be subject to spatial branching: min / max{xk : Dxx + Dww ≤ d, c⊤x ≤ U, x ∈ [x, x], w ∈ [w, w]} (10) where Dxx + Dww ≤ d, Dx ∈ Rℓ×n, Dw ∈ Rℓ× ˆm, d ∈ Rℓ is the linear relaxation of the feasible region of (MINLPext), and c⊤x ≤ U is an objective cutoff constraint that excludes solutions with objective value worse than the current incumbent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The optimal value of (10) may then be used to tighten the lower / upper bound of variable xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A variable is subject to spatial branching if cut separation routines use the bounds of the variable at a node of the branch-and-bound tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SCIP, by default, applies OBBT at the root node to tighten bounds globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' It restricts the computational effort by limiting the amount of LP iterations spent for solving the auxiliary LPs and interrupting for cheaper domain propagation techniques to be called between LP solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, SCIP does not only use the optimal objective values of (10) to tighten the bounds on xk, but it also applies a computationally cheap approximation of OBBT during the branch-and-bound search by exploiting the dual solutions from solves of (10) at the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Suppose the maximization LP is solved and feasible dual multipliers λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , λℓ, µ ≥ 0 for Dxx + Dww ≤ d, c⊤x ≤ U, respectively, and the corresponding reduced cost vectors rx and rw are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Then xk ≤ � j rx j xj + � j rw j wj + λ⊤d + µU (11) is a valid inequality, which is called Lagrangian variable bound (LVB), and � j:rx j <0 rx j xj + � j:rx j >0 rx j xj + � j:rw j <0 rw j wj + � j:rw j >0 rw j wj + λ⊤d + µU (12) is a valid upper bound for xk that equals the OBBT bound if the dual multipliers are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SCIP learns LVBs at the root node and propagates them during the tree search whenever the bounds of variables on the right-hand side of (11) become tighter or an improved primal solution is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For further details, see [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition to OBBT with respect to the LP relaxation, also a variant is available that optimizes single variables over the potentially tighter convex NLP relaxation that is given by all linear and convex nonlinear constraints of (MINLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Also for this variant, linear Lagrangian variable bounds similar to (11) can be constructed by taking constraint convexity and KKT conditions into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Because of the potentially high computational cost of solving many NLPs, this variant of OBBT is deactivated by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For more details, see [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='8 Primal Heuristics The purpose of primal heuristics is to find high quality feasible solutions early in the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' When given an MINLP, up to 40 primal heuristics are active in SCIP by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Many of them aim to find an integer-feasible solution to the LP relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In the following, primal heuristics that are only active in the presence of nonlinear constraints are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 subNLP A primal heuristic like subNLP is implemented in virtually any global MINLP solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Given a point ˜x that satisfies the integrality requirements (˜xI ∈ Z|I|), the heuristic starts by fixing all integer variables in (MINLP) to the values given by ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' It then calls the SCIP presolver on this subproblem for possible simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Finally, it triggers a solution of the remaining NLP, using ˜x as the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If the NLP solver, such as Ipopt, finds a solution that is feasible (and often also locally optimal) for the NLP relaxation, then a feasible point for (MINLP) has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The starting point ˜x can be the current solution of the LP relaxation if integer- feasible, a point found by a primal heuristic that searches for integer-feasible solutions of the LP relaxation, or a point that is passed on by other primal heuristics for MINLP, such as those mentioned in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' How frequently the heuristic should run and how much effort to spend on an NLP solve is a nontrivial decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In the current implementation, the heuristic uses a fixed number for the iteration limit of the NLP solver for its first run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For the following calls, the limit is set to twice the average number of iterations required in previous runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If, however, many of the previous runs hit the iteration limit, then an increased iteration limit is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Whether to run the heuristic at a node of the branch-and-bound tree depends on the number of nodes processed since it ran the last time, the iteration limit that would be used, and how successful the heuristic has been in finding feasible points in previous calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 Multistart If (MINLP) is nonconvex after fixing all integer variables, then several local optima may exist for the NLPs solved by heuristic subNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The success of the NLP solver then strongly depends on the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, the multistart heuristic aims to compute several starting points that are passed to the subNLP heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The algorithm, originally developed in [66], tries to approximate the boundary of the feasible set of the NLP relaxation by sampling points from [x, x] and pushing them towards the feasible set by the use of an inexpensive gradient descent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Afterwards, points that are relatively close to each other are grouped into clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Ideally, each cluster approximates the boundary of some connected component of the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For each cluster, a linear combination of the points is passed as a starting point to subNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For integer variables xi, i ∈ I, the value in the starting point is rounded to an integral value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To reduce infeasibility of a point ˆx, the constraint consensus method [66] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The algorithm computes a descent direction for each violated constraint of (MINLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For example, if gi(ˆx) > gi for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' , m}, then the descent direction is given by − gi(ˆx) ∥∇gi(ˆx)∥2 ∇gi(ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Point ˆx is then updated by adding the average of the descent directions for all violated linear and nonlinear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This step is iterated until ˆx becomes feasible, or a stopping criterion has been fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The multistart heuristic currently runs for continuous problems (I = ∅) only by default, since rounding and fixing integer variables most likely lead to infeasible NLP subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For more details, see [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 NLP Diving As an alternative to finding a good fixing for all integer variables of (MINLP), the NLP diving heuristic starts by solving the NLP relaxation at the current branch-and-bound node with an NLP solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' It then iteratively fixes integer variables with fractional value and resolves both the LP and NLP relaxations, thereby simulating a depth-first-search in a branch-and-bound tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' By default, variables for which the sum of the distances 18 from the solutions of the LP and NLP relaxations to a common integer value is minimal are rounded to the nearest integer value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, binary variables and nonlinear variables are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If the resulting NLP is found to be (locally) infeasible, one-level backtracking is applied, that is, the last fixing is undone, and the opposite fixing is tried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If this is infeasible, too, the heuristic aborts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4 MPEC While the NLP diving heuristic either completely omits or enforces integrality restrictions in the NLP relaxation, the MPEC heuristic adds a relaxation of the integrality restriction to the NLP and tightens this relaxation iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The heuristic is only applicable to mixed-binary nonlinear programs at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The basic idea of the heuristic, originally developed in [61], is to reformulate (MINLP) as a mathematical program with equilibrium constraints (MPEC) and to solve this MPEC to local optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The MPEC is obtained from (MINLP) by rewriting the condition xi ∈ {0, 1}, i ∈ I, as complementarity constraint xi ⊥ 1 − xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This reformulation is again reformulated to an NLP by writing it as xi (1 − xi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, since these reformulated complementarity constraints will not, in general, satisfy constraint qualifications, solving this NLP reformulation with a generic NLP solver will often fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Therefore, in order to increase the chances of solving the NLP reformulation, the heuristic solves regularized versions of the NLP by relaxing xi(1 − xi) = 0 to xi(1 − xi) ≤ θ, for different, ever smaller θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The solution of one NLP is thereby used as the starting point for the next solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If the NLP solution is close to satisfying xI ∈ {0, 1}|I|, it is passed as starting point to the subNLP heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If an NLP is (locally) infeasible, the heuristic does two more attempts where the values for binary variables that are already close to 0 or 1 are flipped to 1 or 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For more details, see [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5 Undercover While the previous heuristics focused mainly on enforcing the integrality condition on an NLP, heuristic undercover [10] starts from a completely different angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The heuristic is based on the observation that it sometimes suffices to fix only a comparatively small number of variables of (MINLP) to yield a subproblem with all constraints being linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For example, for a bilinear term, only one of the variables needs to be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The variables to fix are chosen by solving a set covering problem, which aims at minimizing the number of variables to fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The values for the fixed variables are taken from the solution of the LP or NLP relaxation or a known feasible solution of the MINLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The resulting sub-MIP is less complex to solve, and does not need to be solved to proven optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The solutions of the sub-MIP are immediately feasible for (MINLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, the best one is also passed as starting point to heuristic subnlp to try for further improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For more details, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 3 Benchmark The following aims to present a fair comparison of SCIP with several other state-of-the- art solvers for general MINLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Doing so is not trivial at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' First, a set of instances needs to be selected that is suitable as a benchmark set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Second, solver parameters have to be set such that all solvers solve the same instances with the same working limits and the same requirements on feasibility and optimality – this goal could not be reached completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Third, the solver’s results have to be checked for correctness, or, when this is not possible, plausibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 19 GAMS was used for the experiments, as it provides various facilities to help on solver comparisons and comes with current versions of SCIP and the commercial solvers BARON [40], Lindo API [44], and Octeract included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' ANTIGONE has not been included in the comparison, as its development seems to have stopped years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' All computations were run on a Linux cluster with Intel Xeon E5-2670 v2 CPUs (20 cores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The GAMS version is 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0, which includes SCIP 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2, BARON 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='30, Lindo API 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='162, and Octeract 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A GAMS license with all solvers enabled was used, so that SCIP uses CPLEX 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 as LP solver and Ipopt with HSL MA27 as NLP solver, BARON can choose between all LP/MIP/NLP solvers that it interfaces with, and Octeract uses CPLEX 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 as LP/MIP/QP/QCP solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Test Set To construct a test set suitable for benchmarking, the MINLPLib [50] collection of 1595 MINLP instances was used as source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' First, all instances that could not be handled by some of the considered solvers were excluded, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', instances with trigonometric functions, as they are not supported by BARON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' All solvers were then run in serial mode (that is, with parallelization features disabled) on the remaining 1505 instances and using the parameter settings described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The results of these runs were then used to select a set of 200 instances that could be solved by at least one solver, that were not all trivial, had a varying degree of integrality and nonlinearity, and such that having many instances with a similar name is avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The latter was done to avoid overrepresentation of optimization problems for which many instances were added to MINLPLib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since small changes to an instance can lead to large variations in the solver’s performance, the benchmark’s reliability is improved by considering for each instance four additional variants where the order of variables and equations has been permuted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The permuted instances were generated with GAMS/Convert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Thus, a test set of 1000 instances is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The following approach was used to select the 200 instances before permutation: Let I be the set of 1505 instances, di be the fraction of integer variables in instance i ∈ I, and ei be the fraction of nonzeros in the Jacobian and objective function gradient that correspond to nonlinear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Next, assign to each instance an identifier fi ∈ F such that instances that seem to come from the same model are assigned the same identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This goal is approximated by mapping i to the name of the instance until the first digit, underscore, or dash, except for the block layout design instances fo*, m*, no*, o*, which were all assigned to the same identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' |F| = 230 different identifiers were found this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, let ti be the largest time in seconds that any solver who did not produce wrong results on instance i spend on instance i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Finally, let S be the number of instances that could be solved by at least one solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To ensure that instances with a varying amount of integer variables and nonlinearity are included, the interval [0, 1] was split once at breakpoints 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='9 and once at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Let D and E be the resulting partitions of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For every interval from D and E, the aim is to have roughly the same number of instances with di and ei in the respective intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For the choice of breakpoints that define D and E, the distribution of di and ei, i ∈ I, have been taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For example, MINLPLib contains many purely continuous and purely discrete instances, but not many instances that are mostly linear or completely nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To avoid including too many instances originating from the same model, including more than two instances for each identifier in F is discouraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, instances that seem trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', which are solved by all solvers in no more than five seconds, or could not be solved by any solver are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Introducing penalty terms, the following 20 optimization problem for instance selection is obtained: min � d∈D λ2 d + � e∈E λ2 e + 10 � f∈F λ2 f such that � i∈I:di∈d zi = � N |D| � + λd ∀d ∈ D, � i∈I:ei∈e zi = � N |E| � + λe ∀e ∈ E, � i∈I:fi=f zi ≤ 2 + λf ∀f ∈ F, zi = 0 ∀i ∈ I : ti ≤ 5, zi = 0 ∀i ∈ I : i ̸∈ S, z ∈ {0, 1}|I|, λ ∈ Z|D|+|E|+|F | This problem was solved for N varying between 180 and 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For N = 208, this yield a selection of 200 instances with an acceptable penalty value of 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' See Section A for a list of all selected instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Table 1 shows the number of instances for each element of D × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For five identifiers from F, three instead of two instances were selected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', λf = 1 for five f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' E ↓ | D → [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='05) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='25) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='9) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='9, 1] [0, 1] [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1) 3 7 19 15 6 50 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='25) 8 22 9 7 4 50 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5) 8 8 6 10 18 50 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5, 1] 25 2 5 7 11 50 [0, 1] 44 39 39 39 39 200 Table 1: Number of instances selected with “discreteness” di and “nonlinearity” ei in intervals from D and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 Parameter Settings 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Missing Variable Bounds To compute a lower bound on the optimal value of a minimization problem, all solvers considered here construct a convex relaxation of the given problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For nonconvex constraints, this often relies on the computation of valid convex underestimators or concave overestimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' As these typically depend on variables’ bounds (recall the McCormick underestimators (1)), missing or very large bounds on variables in nonconvex terms can mean that an instance will be very hard or impossible to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Even when the user forgot to specify some variable bounds, the solver may still be able to derive bounds via domain propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, once a feasible solution ˆx has been found, additional bounds may be derived from the inequality c⊤x ≤ c⊤ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, as there are always cases where bounds are still missing after presolve, solvers invented different ways to deal with this obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If SCIP cannot construct an under- or overestimator because of missing variable bounds, it continues by branching on an unbounded variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This way, there will eventually be a node in the branch-and-bound tree where all variables are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Nodes that still contain unbounded variable domains may be pruned due to a derived lower bound on the objective function exceeding the incumbents objective function 21 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' But it may also be the case that pruning will not be possible and SCIP does not terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, variable bounds after branching cannot grow indefinitely in SCIP, but are limited by ±1020 by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' That is, SCIP does not search for solutions with variable values beyond this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The other solvers considered here add variable bounds based on a heuristic decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If BARON is still missing bounds on variables in nonconvex terms after presolve, it sets the bound to a value that depends on the type of nonlinearity involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Typically, this value is around ±1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' BARON also prints a warning to the log and no longer claim to have solved a problem to global optimality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', it does not return a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Lindo API adjusts the bounds for all variables that are involved in convexification to be within [−1010, 1010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' At termination, it returns the lower bound for the restricted problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Octeract proceeds similarly and introduces a bound of ±107 for every missing bound and returns the lower bound for the restricted problem at termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Evidently, just passing an instance with unbounded variables to a solver with default settings may mean that each solver solves a different subproblem of the actual problem and often also reports a lower bound that corresponds to the solved subproblem only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Fortunately, for every solver considered here, parameters are available to adjust the treatment of unbounded variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A first impulse could be to tell all solvers to set missing bounds to infinity, but this is not possible as each solver treats values beyond a certain finite value as “infinity” (BARON: 1050, Octeract: 10308, SCIP: 1020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Changing this value is either not possible or not advisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' We therefore decided to aim for ±1012 as replacement for a missing variable bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For BARON and SCIP, the GAMS interface can replace any missing bound by ±1012 before the instance is passed to the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' BARON will hence also return a lower bound for this restricted problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For Lindo API, a solver parameter can be changed so that bounds for all variables subject to convexification are bounded by ±1012 (instead of ±1010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Finally, also for Octeract, all missing bounds are set to ±1012 (instead of ±107) by changing of a solver parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Note, that this still does not ensure that all solvers solve the same instance, since Lindo API would still change initial finite bounds beyond 1012 and may also not set any bounds for variables that are not involved in convexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Next to missing bounds on problem variables, also singularities in functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', 1/x, log(x)) can prevent finite under- or overestimators from being available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Unfortunately, there are no parameters available to ensure a uniform treatment of this case in all solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SCIP ensures that a variable x in xp, p < 0, or log(x) is bounded away from zero by 10−9, and terminates with a lower bound for this modified problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' BARON applies the same method as the one for missing bounds on problem variables to choose a suitable bound on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' No lower bound is returned at termination then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The methods in Lindo API and Octeract are not known to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 Solution Quality To ensure that all solvers return solutions of the same quality, constraints of (MINLP) are required to be satisfied with an absolute tolerance of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This applies to linear and nonlinear equations, variable bounds, and integrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition, a tolerance on the proof of optimality is set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For this purpose, typically, solvers are allowed to stop when the absolute or relative gap between lower and upper bounds on the optimal value are sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since the test set is diverse and has optimal values of varying magnitude, setting only a relative gap limit and no absolute gap limit would be preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Unfortunately, Octeract does not permit different values for these limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' As a compromise, BARON, Lindo API, and SCIP are run with 10−4 as relative gap limit and 10−6 as absolute gap limit, while for Octeract, 10−6 is used for both the absolute and relative gap limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Below, the impact of using a tighter optimality tolerance for Octeract is analyzed in a separate comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 Working Limits As working limits, a time limit of two hours is used and the jobs on the cluster are restricted to 50 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, the amount parallelization (multiple threads or processes) that a solver is allowed to use is limited in varying degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To simplify the presentation, the term “threads” is used also for Octeract, even though it uses multiple processes instead of threads to parallelize its solving process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4 Summary To summarize,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' the following parameters are used: GAMS (applied to all solvers): optcr=1e-4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' optca=1e-6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' reslim=7200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' workspace= 50000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' threads ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 16} BARON: InfBnd=1e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' AbsConFeasTol=1e-6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' AbsIntFeasTol=1e-6 Lindo API: GOP BNDLIM=1e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' SOLVER FEASTOL=1e-6 Octeract: INFINITY=1e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' INTEGRALITY VIOLATION TOLERANCE=1e-6 SCIP: gams/infbound=1e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' constraints/nonlinear/linearizeheursol=o (this undoes a change in the algorithmic settings of SCIP that is part of the GAMS/SCIP interface) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 Correctness Checks The GAMS/Examiner 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 tool is used to evaluate the violation of constraints, bounds, and integrality in the solutions reported by the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Examiner generates for each solver a file that contains for each instance the solving time, returned lower and upper bound, and solution infeasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A run of a solver on an instance is marked as failed if the solver terminated abnormally, the solution is not feasible with respect to the feasibility tolerance, or the lower or upper bound contradicts with the bounds on the optimal value that are specified on the MINLPLib page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Note, that the primal and dual bounds on the MINLPLib page were calculated without enforcing the ±1012 limit on unbounded variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, in order for an instance to be accepted into the test set, one of the solvers considered here must have solved the instance and found an optimal value that fits within the lower and upper bounds given at MINLPLib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' It is therefore acceptable to use these bounds for checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A run that has not failed is marked as solved if the relative or absolute limits on the gap between lower and upper bound are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If a solver stopped without closing the gap before the time limit, then the solver time is changed to the time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The only exception here is BARON, which stops on two instances before the time limit without reporting a lower bound due to singularities in functions (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To be consistent with the treatment of other solvers, these two instances were accounted as solved by BARON with the original solver time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4 Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Serial Mode For the main comparison, all parallelization features in the solvers were disabled, that is, GAMS was run with option threads set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition to the solver itself, results for the virtual best and virtual worst solver are reported, which are obtained by picking for each instance the fastest or the slowest solver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Table 2 shows for each solver the number of instances that could be solved, the number of times the time limit was reached, and the number of runs that were marked as failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, the shifted geometric mean of the running time of the solver is 23 provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The shift has been set to 1 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Here, instances that failed are accounted with the time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The performance profile [23] in Figure 4 shows the number of instances a solver solved within a time that is at most a factor of the fastest solvers time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 provides detailed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' solved timeout fail time BARON 790 183 27 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4 Lindo API 538 323 139 489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Octeract 671 279 50 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 SCIP 776 183 41 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 virt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' worst 368 405 227 1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 virt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' best 967 33 0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='7 Table 2: Aggregated performance data for all solvers on test set of 1000 instances with parallelization disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 100 101 102 103 104 0 200 400 600 800 1,000 time factor to best (τ) # instances solved BARON Lindo API Octeract SCIP Figure 4: Performance profile comparing all solvers with parallelization disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The results show a small lead of BARON before SCIP with respect to both number of instances solved and average time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since the number of timeouts is almost equal, one could argue that it is the higher stability of BARON that moves it onto the first place here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In fact, the 41 fails of SCIP are due to returning a wrong optimal value 16 times, returning an infeasible solution 23 times, and aborts due to numerical troubles for two instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For BARON, fails are due to returning a wrong optimal value 26 times and an infeasible solution only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' While SCIP 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 has made a large step forward in ensuring that nonlinear constraints are satisfied in the non-presolved problem, violations in linear constraints or variable bounds still occur for a few instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' These are typically due to variables being aggregated during presolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Even though Octeract and Lindo API solved considerably fewer instances than BARON and SCIP, which also results in an increased mean time, it is noteworthy that each of the two is also the fastest solver on 270 and 66 instances, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Octeract also produced correct results for 95% of the test set, while for Lindo API a relatively high number of wrong optimal values, infeasible solutions, or aborts is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The large differences between the real and virtual solvers show that none of the solvers dominates all others or is dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Next, the effect of changing the gap limit for Octeract has been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Recall from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 that relative and absolute gap limits of 10−6 and 10−4, respectively, were used for all solvers except for Octeract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since Octeract does not allow choosing 24 these limits separately, it had been run with the tighter relative gap limit of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' To check whether this lead to a considerable disadvantage for this solver, the solver was rerun on the 200 non-permuted instances with both relative and absolute gap limit set to 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The table in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 shows that the change in the convergence tolerance had essentially no effect on the solver’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In both cases, the same 134 instances could be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The mean time changed from 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='6 for a limit of 10−6 to 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 for a limit of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 Parallel Mode In the next comparison, each solver is allowed to use multiple threads or processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since SCIP’s use of multiple threads is limited to presolving MIPs, checking quadratic functions for convexity, and the linear algebra in Ipopt, FiberSCIP [64] is used to run SCIP in parallel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' FiberSCIP is a shared-memory instantiation of the UG framework [62] for the parallelization of branch-and-bound based solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The framework parallelizes the search of the branch-and-bound tree by collecting and distributing open problems between independent instances of SCIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition, the first seconds of the solving process are used for a “racing ramp-up” phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Here, multiple SCIP instances with differing parameter sets are run concurrently, and the one with the best lower bound is used for the remaining solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The UG version was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 beta3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For the runs in serial mode, reaching the memory limit of 50 GB was not observed for any solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' But since parallelization often increases memory requirements, a memory limit of 100 GB has been used for the runs in parallel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since this meant a reduction in available computing resources, only the 200 non-permuted instances are used for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Table 3 shows, for an increasing number of threads, the number of instances that could be solved by each solver and the mean time spent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In addition, Figure 5 provides a performance profile that compares SCIP and FiberSCIP only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 gives detailed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 1 thread 4 threads 8 threads 16 threads solved time solved time solved time solved time BARON 161 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 160 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 160 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 158 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='6 Lindo API 114 423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='6 114 379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 106 459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5 107 456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='4 Octeract 134 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='6 133 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='9 138 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 135 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2 (Fiber)SCIP 161 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='9 145 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='3 147 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='8 152 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='8 Table 3: Aggregated performance data for all solvers on test set of 200 instances when run with parallelization allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Apparently, enabling parallelization seldom has a considerable advantage on this test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For Octeract, where parallelization was part of its original design, a small increase in the number of instances that could be solved and a reduction in time by 34% when using up to 8 parallel processes is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' As far as we know, BARON’s use of multiple threads is currently limited to enabling this feature in the solver for a MIP relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' As a consequence, only moderate improvement of running time by up to 11% are seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For Lindo API, an improvement due to parallelization seems to be impeded by a further increase in fails when using multiple threads (1 thread: 24, 4: 28, 8: 35, 16: 43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Finally, for SCIP/FiberSCIP the additional overhead due to the parallelization being build on top of the solver instead of being tightly integrated is not compensated by the use of multiple threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, in contrast to other solvers, a monotonous improvement in both number of instances solved and mean solving time when increasing from 4 to 16 threads is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, the virtual solvers in the 25 100 101 102 103 0 50 100 150 200 time factor to best (τ) # instances solved 1 thread 4 threads 8 threads 16 threads virt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' best virt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' worst Figure 5: Performance profile comparing SCIP and FiberSCIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' performance profile show that FiberSCIP can solve instances that SCIP on one thread couldn’t solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Finally, note that a benefit due to parallelization can usually only be expected for rather challenging instances because of the additional overhead in duplicating and synchronizing data and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, the test set deliberately included only instances that could already be solved by some solver in serial mode, and only instances that were trivial for all solvers, though they may be solved quickly by some, were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' As a small experiment, for each solver only those instances that required at least 10 or 100 seconds to solve in serial mode were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Unfortunately, this essentially repeated the trends shown in Table 3, so details are omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' A more thorough analysis of the parallelization capabilities of MINLP solvers using a set of challenging instances only would be necessary, but exceeds the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' 4 Conclusion The development of the MINLP solver in SCIP has come a long way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In a recent version- to-version comparison [57, slides 49-51], a steady improvement in the performance of SCIP on MINLP over the last ten years has been measured, resulting in SCIP 8 solving twice as many instances as SCIP 3 and a speed-up of factor three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Partially, this improvement has been achieved by improving and adding features particular for MINLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' However, due to the generality of SCIP as a CIP solver, also many developments that targeted MIP solving were immediately available for MINLP solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' With version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0, the MINLP solving capabilities of SCIP have been largely reworked and extended, which resulted in a considerable improvement in both robustness and performance [14, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' As a result, SCIP’s performance is currently on par with the state-of-the-art commercial solver BARON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In contrast to the commercial solvers considered here, SCIP offers a variety of possibilities for a user, developer, or researcher to interact with the solving process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' In particular, the newly added “nonlinear handler” plugin type sets SCIP apart from most other MINLP solvers, as it allows focusing on experimenting with new algorithms to handle certain structures in nonlinear functions without modifications to the solver’s code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The rather large number of features that are disabled by default shows that tuning and improving the existing code base has become increasingly necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Future work will of course also include the addition of new features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=', improved separation for signomial functions [77], the use of alternative relaxations for polynomial functions [17], 26 or monoidal strengthening of intersection cuts for quadratic constraints [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The increasing number of cores in present-day CPUs means that to fully utilize an ordinary desktop computer, a solver needs to be parallelized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' While the UG framework provides such a possibility for SCIP in both shared and distributed memory environments, the experiments with FiberSCIP on up to 16 threads show that more tuning is necessary to ensure that the additional overhead can be compensated by the use of additional computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Since the development of UG was initially motivated and has focused primarily on the use of large-scale parallel computing environments [63], an investigation on using UG with SCIP to solve challenging MINLPs in distributed memory environments with many CPU cores could be interesting as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Acknowledgments We are very much in all SCIP developers’ debt – the extensions to support nonlinear constraints and solve MINLPs would not have been possible without the framework’s existence and the powerful MIP solver that we could build upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' While the authors of this paper are the main developers of the new MINLP features in SCIP 8, many have contributed to the MINLP capabilities in previous releases of SCIP, namely Martin Ballerstein, Timo Berthold, Tobias Fischer, Thorsten Gellermann, Ambros Gleixner, Renke Kuhlmann, Dennis Michaels, Marc Pfetsch, and Stefan Weltge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Further, we thank Yuji Shinano for the development of FiberSCIP and swiftly reacting to our request for the possibility to set gap limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Last but not least, we are very grateful to Franziska Schl¨osser for the setup and maintenance of benchmarking and testing facilities for the infamous “consexpr” development branch of SCIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' The work for this article has been conducted within the Research Campus Modal funded by the German Federal Ministry of Education and Research (BMBF grant numbers 05M14ZAM, 05M20ZBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Additional funding has been received from the German Federal Ministry for Economic Affairs and Energy within the project EnBA-M (ID: 03ET1549D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' References [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Achterberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Constraint Integer Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' PhD thesis, Technische Universit¨at Berlin, 2007.' metadata={'source': 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Hendel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Koch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Maher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Miltenberger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' M¨uller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Koch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Miltenberger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' M¨uller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Pfetsch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Puchert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For each instance, the number of variables (n), the number of discrete variables (|I|), the number of constraints (m + ˜m), the number of nonzeros in the Jacobian and objective function gradient (nz), and the number of nonzeros that correspond to nonlinear terms (nlnz) is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='instance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='|I| ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='waterund01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='152 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='78 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='Detailed Computational Results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='The following tables show the outcome from running each solver on instances from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If an instance has been solved to optimality, the time spend is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Note that due to differences in formulas for the relative gap in the various solvers, an instance may be accounted as solved even though the solver stopped at the time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' If a run has been flagged as failed, the reason for this decision is given: “abort” if the solver did not return with a result, “nonopt” if the reported upper or lower bound were not consistent with those given by MINLPLib, and “infeas” if the reported solution is not feasible with respect to the feasibility tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' Otherwise, the relative gap at termination is reported, which is ∞ if no feasible solution or lower bound has been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' An exception here is BARON, where an instance is considered as solved if the solver only decided to not return a lower bound due to singularities in functions (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' This is the case for instances mhw4d and multiplants mtg2 and 35 their permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' For each instance, a time or gap that is at most 10% worse than the one from the best solver on this instance is printed in bold font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='1 Serial Mode The following table shows the outcome from running each solver on the test set of 200 instances and their permutations in serial mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content=' instance perm BARON Lindo API Octeract SCIP alan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='89% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='19 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='13 autocorr bern20-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='29 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='36 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='34 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='18 90.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='55 2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='21 infeas 2425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='34 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='74 3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='60 infeas 6572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='33 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='83 4 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='84 eg all s infeas abort 102% 3095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='77 1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='7% abort 109% 4758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='26 2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='9% abort 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='9% 6829.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='5% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='7% elf 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='91 nonopt 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='01 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='09 infeas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='67 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='47 infeas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='42 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='39 infeas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='28 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='05 infeas infeas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='74 eniplac 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='54 3% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='19 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfs_mP/content/2301.00587v1.pdf'} +page_content='7% 2.' 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NOCHETTO, AND SHUO YANG +Abstract. Bilayer plates are slender structures made of two thin layers of dif- +ferent materials. They react to environmental stimuli and undergo large bending +deformations with relatively small actuation. The reduced model is a constrained +minimization problem for the second fundamental form, with a given spontaneous +curvature that encodes material properties, subject to an isometry constraint. We +design a local discontinuous Galerkin (LDG) method which imposes a linearized +discrete isometry constraint and controls deformation gradients at barycenters of +elements. We prove Γ-convergence of LDG, design a fully practical gradient flow, +which gives rise to a linear scheme at every step, and show energy stability and +control of the isometry defect. We extend the Γ-convergence analysis to piecewise +quadratic creases. We also illustrate the performance of the LDG method with +several insightful simulations of large deformations, one including a curved crease. +1. Introduction +Bilayer plates are slender structures made of two thin layers of different materials +glued together. These layers react differently to non-mechanical stimuli, such as +thermal, electrical, and chemical actuation [30, 43, 31]. Bilayer plates can undergo +large bending deformations using a small amount of energy, which makes them +appealing at small and large scales alike. Amongst the many and broad applications +of bilayer materials in engineering and biomedical science, we list drug delivery +vesicles [28, 44], cell encapsulation devices [45], sensors [34] and self-deployable sun +sails [33]. +We model bilayer plates as thin 3d hyper-elastic bodies as depicted in Fig. 1. +Exploiting their relatively small thickness, two dimensional plate models for the +mid-plane deformation y(Ω), Ω ⊂ R2, are derived and analyzed in [40, 41]; we +also refer to [8] for a formal dimension reduction argument. The plates equilibria +(Andrea Bonito) Department of Mathematics, Texas A&M University, College Sta- +tion, TX 77845, USA. AB was partially supported by the NSF Grants DMS 2110811. +(Ricardo H. Nochetto) Department of Mathematics and Institute for Physical Science +and Technology, University of Maryland, College Park, Maryland 20742, USA. +(Shuo Yang) Yanqi Lake Beijing Institute of Mathematical Sciences and Applica- +tions, Beijing 101408, China, and Yau Mathematical Sciences Center, Tsinghua Uni- +versity, Beijing 100084, China +E-mail addresses: bonito@tamu.edu, rhn@umd.edu, shuoyang@bimsa.cn. +Date: January 10, 2023. +1 +arXiv:2301.03151v1 [math.NA] 9 Jan 2023 + +2 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +are characterized as solutions to a nonlinear minimization problem with a noncon- +vex constraint expressing the plates ability to bend without stretching or shearing. +Therefore, distances within the midplane remain unchanged thereby resulting in +isometric deformations. +Figure 1. Bilayer plates: Ω × (−s/2,s/2), Ω ⊂ R2 is the mid-plane +(bounded Lipschitz domain) and s is the thickness parameter. The +sets Ω×(−s/2,0) and Ω×(0,s/2) represent the two undeformed layers +of different materials. +1.1. Problem statement. The plate deformation y ∶ Ω → R3 must belong to the +following admissible set A, which prevents shearing and stretching within the surface +y(Ω) and imposes possible boundary conditions: +(1) +A ∶= {y ∈ [H2(Ω)]3 ∶ +I[y] = I2 in Ω, +y = ϕ, ∇y = Φ on ΓD}, +where I2 is the 2 × 2 identity matrix and +(2) +I[y] ∶= ∇yT ∇y +is the first fundamental form of y(Ω). We assume that ΓD ⊂ ∂Ω is nonempty and +open and ϕ ∈ [H2(Ω)]3 and Φ ∈ [H1(Ω)]3×2 are given and are compatible with the +isometry constraint, namely Φ = ∇ϕ and ΦT Φ = I2 on ΓD; thus A is non-empty. +Moreover, condition (2) entails that {∂iy}2 +i=1 is an orthonormal basis of the tangent +plane to y(Ω) and its unit normal ν can be written as +(3) +ν ∶= ∂1y × ∂2y +∣∂1y × ∂2y∣ = ∂1y × ∂2y. +Although we will present simulations in Section 6 for both Dirichlet boundary +conditions (i.e. +ΓD ≠ ∅) and free boundary conditions (i.e. +ΓD = ∅), we focus +our presentation on the former for convenience. We emphasize that the analysis of +the latter follows from that in this paper. The modifications are in the spirit of +[14], where we analyze the LDG method for prestrained plates with free boundary +conditions. Consequently, we do not include details to avoid repetitions. +Equilibrium configurations of bilayer plates are solutions y ∈ A of the following +constrained minimization problem +(4) +min +y∈A E [y] ∶= min +y∈A +1 +2 ∫Ω ∣II[y] − Z∣ +2, +where II[y] is the second fundamental form of y(Ω) +(5) +II[y] ∶= (∂ijy ⋅ ν) +2 +ij=1 = (∂ijy ⋅ (∂1y × ∂2y)) +2 +ij=1, + +apQ +y +S +E QLDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +3 +and Z ∈ [L∞(Ω)]2×2 is a spontaneous curvature which encodes the material prop- +erties of the bilayer plates. In fact, Z forces the plate y(Ω) to bend so that II[y] +gets as close as possible to Z. If the material is homogenous and isotropic, then the +spontaneous curvature is diagonal, i.e. Z = αI2 with a constant α depending on the +materials parameters. In particular, when the two layers are identical, Z = 0 and +the model reduces to a single layer plate [4, 17], which coincides with the classical +(nonlinear) Kirchhoff plate theory. +Thanks to the isometry constraint I[y] = I2, the energy functional E [y] can be +further simplified. Recall that for isometries, there holds [5] +(6) +∣II[y]∣ +2 = ∣D2y∣ +2 = ∣∆y∣ +2 = (tr II[y]) +2, +whence expanding the square in (4) and using (5) and (6) yields +(7) +E [y] = 1 +2 ∫Ω ∣D2y∣ +2 − +2 +∑ +i,j=1∫Ω ∂ijy ⋅ (∂1y × ∂2y)Zij + 1 +2 ∫Ω ∣Z∣ +2. +Furthermore, since 1 +2 ∫Ω ∣Z∣ +2 does not depend on y, minimizing the energy in (7) +over A is equivalent to minimizing the reduced energy +(8) +E [y] ∶= 1 +2 ∫Ω ∣D2y∣ +2 − +2 +∑ +i,j=1∫Ω ∂ijy ⋅ (∂1y × ∂2y)Zij, +over A; we keep the same notation for the energies in (7) and (8) for simplicity. +The effect of the layers mismatch appears in the cubic term leading to a nonlinear +Euler-Lagrange equation for the equilibrium deformation y. For latter use, we also +introduce a notation for the single layer bending energy +(9) +B[y] ∶= 1 +2 ∫Ω ∣D2y∣ +2 +and the cubic term +(10) +C[y] ∶= +2 +∑ +i,j=1∫Ω ∂ijy ⋅ (∂1y × ∂2y)Zij, +so that +E[y] = B[y] − C[y]. +We emphasize that the cubic term C satisfies +∣C[y]∣ ≤ ∥y∥H2(Ω)∥∇y∥L2(Ω)∥∇y∥L∞(Ω)∥Z∥L∞(Ω) +and I[y] = I2 implies ∥∇y∥L∞(Ω) ≲ ∥I[y]∥L∞(Ω) ≲ 1, whence +∣C[y]∣ ≲ ∥y∥2 +H2(Ω). +Since the discrete deformation yh is piecewise polynomial, our numerical method +cannot guarantee that yh satisfies the isometry constraint I[yh] = I2 everywhere in +Ω. We choose to enforce a slight violation of this constraint solely at the barycenter +of elements, and still retain control of the ℓ∞-norm of ∇yh at barycenters. This is +a chief ingredient of our LDG method and is inspired by Bartels and Palus [9]. + +4 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +1.2. Previous numerical methods. There are several finite element methods +available for the numerical simulation of bilayers plates [8, 7, 9, 16]. In all of them, +the isometry constraint I[y] = I2 is linearized at y +(11) +L[v;y] ∶= ∇vT ∇y + ∇yT ∇v = 0, +and tangential variations v are evolved within a gradient flow that decreases the +energy E[y] and is favored for its robustness. +The gradients of Kirchhoff finite elements are uniquely defined at the mesh ver- +tices, which is where (11) is imposed in [7, 8]. The discrete gradient flow in [8] treats +the cubic energy C[y] implicitly to get an energy decreasing scheme but requires +the normalization (3) of the discrete normal, which renders the algorithm nonlin- +ear. Discrete energies are shown to Γ-converge in [8]. In contrast, the scheme of +[7] is linear and much more efficient, but stability and Γ-convergence are still open. +Recently, Bartels and Palus [9] reformulated the discretization of C[y] making it +fully explicit and the ensuing algorithm linear, and were also able to show an energy +decreasing property for the explicit gradient flow with a mild time-step constraint +and Γ-convergence of the discrete energies. +On the other hand, interior penalty discontinuous Galerkin (IPDG) finite element +methods are proposed and studied in [16] because they require a lower polynomial +degree (2 instead of 3), are easier to find in existing software platforms, are more +flexible in imposing boundary conditions as well as the linearized isometry constraint +(11), and are amenable to subdivisions containing curved boundaries which is cru- +cial to deal with creases. The linearized constraint (11) is enforced in average on all +elements of the subdivision. Furthermore, the cubic energy C[y] is treated explic- +itly at each step of the discrete gradient flow and the ensuing algorithm is linear. +However, Γ-convergence and energy decreasing properties remain open problems. +We note that the bilayer model (7) reduces to single layer plates endowed with +the bending energy B[y] for y ∈ A provided the upper and lower layers are identical, +i.e. Z = 0. We refer to [4, 17] for the design and analysis of Kirchhoff and IPDG +methods in this simpler context. +1.3. LDG-discretization and our contribution. We propose a local discontin- +uous Galerkin (LDG) method for the approximation of the minimization problem +(4) along the lines of [13, 14]. LDG method was originally introduced in [23], and +further explored in [10, 20, 21, 24, 25]. Our discrete energy Eh[yh] is obtained (up +to stabilization terms) by simply replacing the Hessian D2y in (8) by a discrete +Hessian Hh[yh], which is constructed and analyzed in [13, 14] in terms of the dis- +continuous Galerkin solution yh. This is conceptually simpler than IPDG methods, +which are based on integration by parts and are harder to design for intricate nonlin- +ear systems. In contrast to IPDG, LDG is also stable for any positive stabilization +parameters, and exhibits better convergence properties at the expense of a slightly +worse sparsity pattern [13, 14]. + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +5 +Our treatment of the cubic term hinges on the mid-point quadrature. If Th is a +mesh made of shape-regular triangles or quadrilaterals T with barycenter xT , let +(12) +Ch[yh] ∶= +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣(Hh[yh]ij ⋅ (∂1yh × ∂2yh)Zij)(xT ) +where Hh[yh] = +1 +∣T∣ ∫T Hh[yh] for all T ∈ Th is the piecewise constant reduced discrete +Hessian. Moreover, we also control the isometry defect at barycenters, namely given +a parameter δ > 0 we impose +(13) +Dh[yh](xT ) ∶= ∣[∇yT +h ∇yh − I2](xT )∣ ≤ δ +∀T ∈ Th. +We enforce the Dirichlet condition upon augmenting the discrete energy Eh[yh] via +a Nitsche method. Therefore, we say that discrete functions satisfying (13) belong +to the discrete admissible set Ah, the discrete counterpart of (1). We prove that +Ah is non-empty, and derive convergence of global minimizers yh of Eh within Ah +towards global minimizers y of (4). +Solving the nonconvex discrete minimization counterpart of (4) is a highly non- +trivial task. We resort to a discrete gradient flow that enforces the linearized isom- +etry constraint (11) at the barycenters +(14) +L[vh;yh](xT ) ∶= [∇vT +h ∇yh + ∇yT +h ∇vh](xT ) = 0 +∀T ∈ Th. +and solve a discrete minimization problem for a tangential variation vh of yh, in +the sense (14), with the cubic term (12) treated explicitly. This clever idea, due +to Bartels and Palus [9], renders the problem linear at each step of the gradient +flow. We show that this procedure is energy decreasing, convergent, and preserves +the isometry defect (13) provided δ is proportional to h, which entails a linear +relation between the time step τ of the gradient flow and h. Moreover, we derive a +(suboptimal) discrete inf-sup condition for the Lagrange multiplier approach to the +linear constraint (14), which seems to be the first such result for this type of matrix +constraint and is consistent with computations. +The rest of this article is organized as follows. +Section 2 is about LDG. We +introduce the (broken) finite element spaces in Section 2.2. We examine the discrete +Hessian operator and its reduced counterpart in Subsection 2.3, together with their +boundedness and convergence properties. In Subsections 2.4 and 2.5, we define the +discrete problem and investigate consistency of the cubic discrete energy Ch. The +proof of Γ-convergence of the discrete energy to the exact one is the content of +Section 3, and its extension to a bilayer model with piecewise quadratic creases is +included in Section 4. In Section 5, we introduce the gradient flow scheme used +to solve the discrete problem, prove its conditional stability and show how the +constraint violation (13) is controlled throughout the flow. Moreover, we derive a +suboptimal inf-sup condition for (14) at each step of the flow. We present several +insightful simulations in Section 6 to illustrate the performance of LDG, including +folding across a curved crease. + +6 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +2. LDG Discretization +2.1. Subdivisions. From now on, we assume that Ω ⊂ R2 is a polygonal domain +and denote by {Th}h>0 a shape-regular sequence of conforming partitions of Ω made +of either triangles or quadrilaterals T with diameter hT ∶= diam(T) ≤ h. The set of +edges Eh ∶= E0 +h ∪Eb +h is decomposed into the interior edges E0 +h and boundary edges Eb +h. +For e ∈ Eh, we define he ∶= diam(e) and note that he ≤ h, and thus +(15) +h−1 ≤ h−1 +e +∀e ∈ Eh. +We assume a compatible representation of the Dirichlet boundary ΓD = ∪{e ∶ e ∈ ED +h }, +and let Ea +h ∶= E0 +h ∪ ED +h be the set of active edges on which jumps and averages will be +computed. The union of these edges gives rise to the corresponding skeletons of Th +(16) +Γ0 +h ∶= ∪{e ∶ e ∈ E0 +h}, +ΓD +h ∶= ΓD, +Γa +h ∶= Γ0 +h ∪ ΓD +h . +We use the notation (⋅,⋅)L2(Ω) and (⋅,⋅)L2(Γa +h) to denote the L2 inner products over +Ω and Γa +h, and a similar notation for subsets of Ω and Γa +h. We denote by h a mesh +density function, locally equivalent to hT and he, and utilize it as a weight in the +preceding norms. +We often write f ≲ g to indicate that there exists a constant +C independent of discretization parameters such that f ≤ Cg. Finally, we make +the simplifying assumption that the spontaneous curvature Z in (4) is piecewise +constant over all partitions Th, h > 0. +2.2. Broken spaces and operators. For an integer r ≥ 0, we denote by Pr the +space of polynomials of total degree at most r when the subdivision is made of +triangles and by Qr the space of polynomials of degree at most r in each variable +when quadrilaterals are used. We also use the same notation, ̂T, to denote either +the unit triangle or the unite square depending on the type of subdivision used. We +let FT ∶ ̂T → T ∈ [Q1]2 be the generic map from the reference element to the physical +element. It is affine only when the subdivision is made of triangles. +We fix k ≥ 2. The (broken) finite element space Vk +h to approximate each compo- +nent of the deformation y reads +(17) +Vk +h ∶= {vh ∈ L2(Ω) ∶ vh T ○ FT ∈ Qk +∀T ∈ Th}, +when the subdivision is made of quadrilaterals, and we replace Qk by Pk if we +have triangular elements. We define the broken gradient ∇hvh of vh ∈ Vk +h to be the +elementwise gradient, and use similar notation for other differential operators. For +instance D2 +hvh = ∇h∇hvh stands for the broken Hessian, and ∂ivh ∶= ∂i,hvh denotes +the components i = 1,2 of the broken gradient ∇hvh. +We now introduce the jump and average operators. For every e ∈ E0 +h, fix ne to +be one of the two unit normals to e (the choice is arbitrary but does not affect the +formulation). For a boundary edge e ∈ Eb +h, we set ne = n, the outward unit normal +vector to ∂Ω. The jump of vh ∈ Vk +h and ∇hvh across e ∈ E0 +h are given by +(18) +[vh] e ∶= v− +h − v+ +h, +[∇hvh] e ∶= ∇hv− +h − ∇hv+ +h, +where v± +h(x) ∶= lims→0+ vh(x±sne) for x ∈ e. The jumps of a vector or matrix valued +function is computed componentwise. + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +7 +In order to incorporate the Dirichlet boundary conditions y = ϕ, ∇y = Φ on ΓD, +we resort to a Nitsche’s approach which does not impose essential restrictions on +the discrete space [Vk +h]3 but rather modifies the discrete formulation by including +boundary jumps defined for vh ∈ [Vk +h]3 +(19) +[vh] e ∶= [vh] e(ϕ) ∶= vh − ϕ, +[∇hvh] e ∶= [∇hvh] e(Φ) ∶= ∇hvh − Φ, +for all e ∈ ED +h . However, to simplify the notation, it is convenient to introduce the +discrete set Vk +h(ϕ,Φ) +(20) +Vk +h(ϕ,Φ) ∶= {vh ∈ [Vk +h]3 ∶ [vh] e, [∇hvh] e given by (19) for all e ∈ ED +h }, +which coincide with [Vk +h]3 but carries the notion of boundary jump (19) for its +elements. We define the average of vh ∈ Vk +h across an edge e ∈ Eh as +(21) +{vh} e ∶= { +1 +2(v+ +h + v− +h) +e ∈ E0 +h +v− +h +e ∈ Eb +h, +and apply (21) componentwise to vector and matrix-valued functions. +We let ⟨⋅,⋅⟩H2 +h(Ω) be the following mesh-dependent form defined, for any vh,wh ∈ +Vk +h(ϕ,Φ), by +(22) +⟨vh,wh⟩H2 +h(Ω) ∶= (D2 +hvh,D2 +hwh)L2(Ω) ++ (h−1[∇hvh],[∇hwh])L2(Γa +h) + (h−3[vh],[wh])L2(Γa +h). +We emphasize that (22) is not bilinear in Vk +h(ϕ,Φ) because of the presence of (ϕ,Φ) +in the boundary jump terms, unless ϕ = 0,Φ = 0. Moreover, we set +(23) +∥vh∥2 +H2 +h(Ω) ∶= ⟨vh,vh⟩H2 +h(Ω) +∀vh ∈ Vk +h(ϕ,Φ), +and observe the validity of the following Friedrichs-type inequality [17, (2.27)] +(24) ∥vh∥L2(Ω) +∥∇hvh∥L2(Ω) ≲ ∥vh∥H2 +h(Ω) +∥ϕ∥H1(Ω) +∥Φ∥H1(Ω) +∀vh ∈ Vk +h(ϕ,Φ). +Once restricted to Vk +h(0,0), the form ⟨⋅,⋅⟩H2 +h(Ω) turns out to be a scalar product, +according to (24), which corresponds to the discrete counterpart of ⟨⋅,⋅⟩H2(Ω). +2.3. Discrete Hessians. The central ingredient in the proposed LDG approxima- +tion is the reconstructed Hessian Hh[yh] ∈ [L2(Ω)] +3×2×2 defined in [13, 14]. Let +l1,l2 ≥ 0 be integers and consider two local lifting operators re ∶ [L2(e)]2 → [Vl1 +h ]2×2 +and be ∶ L2(e) → [Vl2 +h ]2×2 defined for e ∈ Ea +h by +re(φ) ∈ [Vl1 +h ]2×2 ∶ ∫ωe +re(φ) ∶ τh = ∫e {τh}ne ⋅ φ +∀τh ∈ [Vl1 +h ]2×2, +(25) +be(φ) ∈ [Vl2 +h ]2×2 ∶ ∫ωe +be(φ) ∶ τh = ∫e {div τh} ⋅ neφ +∀τh ∈ [Vl2 +h ]2×2 , +(26) +where ωe is the union of the two elements of Th sharing e ∈ Γ0 +h or the element of Th +having e ∈ Γb +h as part of its boundary. The definitions extend to [[L2(e)]2] +3 and + +8 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +[L2(e)]3 by component-wise application.The corresponding global lifting operators +are then given by +Rh ∶= ∑ +e∈Ea +h +re ∶ [L2(Γa +h)]2 → [Vl1 +h ]2×2, +Bh ∶= ∑ +e∈Ea +h +be ∶ L2(Γa +h) → [Vl2 +h ]2×2. +(27) +Their purpose is to lift inter-element information to the cells so that once added +to the piecewise Hessian D2 +h, they constitute a weakly convergent approximation of +the exact Hessian (see Lemma 1). In fact, we define the discrete Hessian operator +Hh ∶ Vk +h(ϕ,Φ) → [L2(Ω)] +3×2×2 by +(28) +Hh[vh] ∶= D2 +hvh − Rh([∇hvh]) + Bh([vh]). +We point out the implicit dependence on data (ϕ,Φ) and that we will later compute +Hh[vh] for vh ∈ Vk +h(0,0), i.e. ϕ = 0, Φ = 0, slightly abusing notation. Thanks to +the relation between the edge and cell diameter (15), we have the following a priori +upper bounds for lifting operators +(29) +∥Hh[vh]∥L2(Ω) ≲ ∣∣vh∣∣H2 +h(Ω). +Moreover, we have the following properties of the discrete Hessian Hh[vh]. +Lemma 1 (weak convergence of Hh). Let k ≥ 2 and vh ∈ Vk +h(ϕ,Φ). If ∣∣vh∣∣H2 +h(Ω) ≲ 1 +and vh → v ∈ [H2(Ω)]3 in [L2(Ω)]3 as h → 0, then for any polynomial degree +l1,l2 ≥ 0 we have +(30) +Hh[vh] ⇀ D2v +in [L2(Ω)] +3×2×2 +as h → 0. +Proof. See [14, Lemma 2.4 and Appendix B]. +□ +Lemma 2 (strong convergence of Hh). Let v ∈ [H2(Ω)]3 be any function such that +v = ϕ and ∇v = Φ on ΓD. Moreover, let vh ∈ Vk +h(ϕ,Φ) satisfy +(31) +∥D2vh∥L2(T) ≲ ∥v∥H2(T) ∀T ∈ Th, +∑ +T∈Th +∥vh − v∥2 +H2(T) → 0 as h → 0+. +Then for any polynomial degree l1,l2 ≥ 0 we have as h → 0+ +(32) +Hh[vh] → D2v +strongly in [L2(Ω)]3×2×2. +Proof. This is a minor modification of [14, Lemma 2.5 and Appendix B], which +assume that vh is the Lagrange interpolant of v ∈ H2(Ω). +□ +For later use, we now discuss properties of the reduced discrete Hessian defined +the local L2 projection onto the space of piecewise constants, i.e. +(33) +Hh[vh]∣T ∶= 1 +∣T∣ ∫T Hh[vh] +∀T ∈ Th. +We start with the stability of Hh[yh]. + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +9 +Lemma 3 (stability of Hh[vh]). For any vh ∈ Vk +h(ϕ,Φ), there holds +(34) +∥Hh[vh]∥L2(Ω) ≤ cstab∥vh∥H2 +h(Ω), +where the constant cstab is independent of h. +Proof. This result is a direct consequence of the stability of the reconstructed Hes- +sian (29) and the local L2 projection (33). +□ +The reduced discrete Hessian is also weakly converging. +Lemma 4 (weak convergence of Hh[vh]). Let vh ∈ Vk +h(ϕ,Φ) be a sequence of +discrete deformations satisfying ∥vh∥H2 +h(Ω) ≲ 1 for all h and such that vh → v +in [L2(Ω)]3 for some v ∈ [H2(Ω)]3. Then, Hh[vh] converges weakly to D2v in +[L2(Ω)]3. +Proof. For any φ ∈ [C∞ +0 (Ω)]3×2×2, we have +∫Ω Hh[vh] ∶ φ = ∑ +T∈Th +∫T Hh[vh] ∶ φ = ∑ +T∈Th +∫T Hh[vh] ∶ φ + Hh[vh] ∶ (φ − φ), +where φ ∶= +1 +∣T∣ ∫T φ. Lemma 1 (weak convergence of Hh[yh]) implies +∫Ω Hh[vh] ∶ φ → ∫Ω D2v ∶ φ. +On the other hand, the uniform boundedness (29) and the assumption ∥vh∥H2 +h(Ω) ≲ 1 +guarantee that +∣ ∑ +T∈Th +Hh[vh] ∶ (φ − φ)∣ ≲ h∥Hh[vh]∥L2(Ω)∥∇φ∥L2(Ω) ≲ h∥∇φ∥L2(Ω) → 0 +as h → 0+. Combining these two estimates yields the desired result. +□ +2.4. Discrete admissible set. We introduce the discrete counterpart of the ad- +missible set A. Given a parameter δ > 0 to be related later to h, we recall the discrete +isometry defect Dh[yh] from (13) and define the discrete admissible set Ah,δ as +(35) +Ah,δ ∶= {yh ∈ Vk +h(ϕ,Φ) ∶ +∣Dh[yh](xT )∣ ≤ δ +∀T ∈ Th}, +where the polynomial degree is k ≥ 2 and xT is the barycenter of T ∈ Th. +The +Dirichlet boundary conditions are hidden within the definition (20) of Vk +h(ϕ,Φ) and +imposed in the weak formulation; hence they do not contribute to any essential +restriction in Ah,δ. The following two lemmas are simple consequences of (35). +Lemma 5 (Ah,δ is non-empty). For all h > 0 there exists yh ∈ Vk +h(ϕ,Φ) such that +Dh[yh](xT ) = 0 for all T ∈ Th. +Proof. Let yh(x) ∶= x for x ∈ Ω. We see that yh ∈ [Vk +h]3, and therefore yh ∈ Vk +h(ϕ,Φ) +because the Dirichlet boundary conditions are not imposed essentially in the space +Vk +h(ϕ,Φ) defined in (20). Moreover, I[yh](xT ) = I2, whence Dh[yh](xT ) = 0. +□ + +10 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +Note that this implies Ah,δ is non-empty for any δ > 0, because Ah,0 ⊂ Ah,δ. +We postpone until Theorem 9 the hard question whether Ah,δ is sufficiently rich +to approximate A: for any y ∈ A there is yh ∈ Ah,δ close to y in a suitable sense. +The following lemma provides an estimate on the amount of local stretch and shear +associated with functions in Ah,δ. +Lemma 6 (pointwise isometry constraint). If yh ∈ Ah,δ, then for all T ∈ Th +(36) +1 − δ ≤ ∣∂1yh(xT )∣2,∣∂2yh(xT )∣2 ≤ 1 + δ, +∣∂1y(xT ) ⋅ ∂2y(xT )∣ ≤ δ, +Proof. From definition (35), we deduce that for any i,j = 1,2 +∣∂iy(xT ) ⋅ ∂jy(xT ) − δij∣ ≤ δ, +where δij is the Kronecker delta. The assertion thus follows. +□ +The pointwise control of isometry defect in (35) is inspired by the algorithms based +on Kirchhoff finite elements developed in [8, 9], where this constraint is imposed at +the element vertices. Dealing with element barycenters is novel in the context of DG +methods in that previous schemes impose this constraint in average over elements +[16, 14]. Having control at barycenters does not imply control of ∇hyh anywhere +else, and dictates the use of mid-point quadrature for the discretization of the cubic +nonlinear energy Ch. We discuss this next. +2.5. Discrete energy. The LDG approximation of the energy E[.] reads +(37) +Eh[yh] ∶= Bh[yh] − Ch[yh] +where Bh[.] approximates the bending energy (9) and Ch[.] approximates the cubic +interaction energy in (10). The energy Bh[yh] is defined by +(38) +Bh[yh] ∶= 1 +2 ∫Ω ∣Hh[yh]∣ +2 + Sh[yh], +where +(39) +Sh[yh] ∶= γ1∥h− 1 +2 [∇hyh]∥2 +L2(Γa +h) + γ0∥h− 3 +2 [yh]∥2 +L2(Γa +h) +is a stabilization term with parameters γ0,γ1 > 0, whereas Ch[yh] is given by (12) +(40) +Ch[yh] ∶= +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣(Hh[yh]ij ⋅ (∂1yh × ∂2yh)Zij)(xT ). +With these notations the discrete minimization problem reads +(41) +min +yh∈Ah,δ +Eh[yh]. +We devote the rest of this section to examine the cubic energy (40). Combining +Lemma 6 (pointwise isometry constraint) with Lemma 3 (stability of Hh[yh]) yields +∣Ch[yh]∣ ≲ (1 + δ)∥yh∥H2 +h(Ω)∥Z∥L2(Ω), +whence ∣Ch[yh]∣ is uniformly bounded whenever ∥yh∥H2 +h(Ω) is. Another crucial as- +pect of (40) is the convergence of Ch towards the continuous energy C within the + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +11 +basic H2-regularity framework. +This requires dealing with the reduced Hessian +Hh[yh] as we show next. +Lemma 7 (convergence of cubic energy). Let Z be piecewise constant over Th. Let +yh ∈ Ah,δ be a sequence of discrete deformations satisfying +(42) +∥yh∥H2 +h(Ω) ≲ 1 +∀h > 0 +and such that yh → y in [L2(Ω)]3, ∇hyh → ∇y in [L2(Ω)]3×2 for y ∈ [H2(Ω) ∩ +W 1 +∞(Ω)]3 as h → 0+. Then +(43) +lim +h→0+ Ch[yh] = C[y]. +Proof. For any ϵ > 0, it suffices to show that +(44) +limsup +h→0+ ∣C[y] − Ch[yh]∣ ≲ ϵ. +We need a regularization argument to deal with the effect of quadrature. Since Ω +is Lipschitz we can regularize y, say by convolution, in such a manner that the +approximate deformation yϵ ∈ [H3(Ω)]3 satisfies +(45) +∥yϵ∥H2(Ω) + ∥yϵ∥W 1 +∞(Ω) ≲ 1, +∥y − yϵ∥H2(Ω) ≲ ϵ; +we recall the convention that constants hidden in ≲ are independent of h and ϵ. We +point out that this procedure is simpler than the regularization due to Hornung [29] +in that yϵ need not be an isometry. We first observe that the energies C[y] and +C[yϵ] can be made arbitrarily close because +∣C[y] − C[yϵ]∣ ≲ ∥y − yϵ∥H2(Ω)∥∂1y∥L2(Ω)∥∂2y∥L∞(Ω)∥Z∥L∞(Ω) ++ ∥yϵ∥H2(Ω)∥y − yϵ∥H1(Ω)(∥∂2y∥L∞(Ω) + ∥∂1yϵ∥L∞(Ω))∥Z∥L∞(Ω) ≲ ϵ. +We next write Ch[yh] − C[yϵ] = ∑2 +i,j=1 ∑T∈Th R1(T) + R2(T) + R3(T), where +R1(T) ∶= ∫T (Hh[yh]ij − ∂ijyϵ) ⋅ (∂1yϵ × ∂2yϵ)Zij, +R2(T) ∶= ∣T∣[Hh[yh]ij ⋅ (∂1yh × ∂2yh − ∂1yϵ × ∂2yϵ)Zij](xT ), +R3(T) ∶= ∣T∣[Hh[yh]ij ⋅ (∂1yϵ × ∂2yϵ)Zij](xT ) − ∫T Hh[yh]ij ⋅ (∂1yϵ × ∂2yϵ)Zij, +and disregard the non critical dependence on i,j = 1,2 in the notation. Lemma 4 +(weak convergence of Hh[yh]) in conjunction with (45) implies that +limsup +h→0+ +2 +∑ +i,j=1 +∑ +T∈Th +∣R1(T)∣ ≲ ϵ. +For R2, we note that +∣(∂1yh × ∂2yh − ∂1yϵ × ∂2yϵ)(xT )∣ ≤ ∣∇(yh − yϵ)(xT )∣(∣∇yh(xT )∣ + ∣∇yϵ(xT )∣). +By Lemma 6 (pointwise isometry constraint), the fact that yh ∈ Ah,δ and (45), we +have the uniform bound ∣∇yh(xT )∣+∣∇yϵ(xT )∣ ≲ 1 for all xT . If Ih∇yϵ indicates the + +12 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +standard P1-Lagrange interpolant of ∇yϵ, applying approximating properties of Ih +together with an inverse inequality for polynomials, we conclude +∣∇(yh − yϵ)(xT )∣ ≤ ∣(∇yh − Ih∇yϵ)(xT )∣ + ∣(Ih∇yϵ − ∇yϵ)(xT )∣ +≲ h−1 +T ∥∇yh − Ih∇yϵ∥L2(T) + hT ∥D3yϵ∥L2(T). +We next add and subtract ∇yϵ in the first term of the right-hand side and apply +again an interpolation estimate of Ih to derive +∣T∣1/2∣∇(yh − yϵ)(xT )∣ ≲ ∥∇(yh − yϵ)∥L2(T) + h2 +T ∥D3yϵ∥L2(T). +Moreover, since ∣T∣1/2∣Hh[yh]ij(xT )∣ = ∥Hh[yh]ij∥L2(T) because Hh[yh] is piecewise +constant, we obtain +∣R2(T)∣ ≲ ∥Hh[yh]ij∥L2(T)(∥∇(yh − y)∥L2(T) + ∥∇(y − yϵ)∥L2(T) + h2 +T ∥D3yϵ∥L2(T)), +where the hidden constant is proportional to ∥Z∥L∞(Ω). After summing over ele- +ments, Lemma 3 (stability of Hh[yh]), together with the assumption ∇hyh → ∇y +in [L2(Ω)]3×2, (42) and (45), yields +limsup +h→0+ +2 +∑ +i,j=1 +∑ +T∈Th +∣R2(T)∣ ≲ ϵ. +It remains to deal with R3 which entails the effect of quadrature. Since Z and +Hh[yh] are constant in T, which is the chief reason for utilizing the reduced discrete +Hessian, we can equivalently rewrite R3(T) as follows: +R3(T) = Hh[yh]ijZij ∫T (f(xT ) − f) +with f = ∂1yϵ ×∂2yϵ. The Bramble-Hilbert Lemma, in conjunction with the Sobolev +embedding W 2 +1 (T) ⊂ C(T) (cf. [19, Lemma 4.3.4]), implies the existence of a linear +polynomial p ∈ [P1(T)]3 such that ∥f − p∥L∞(T) ≲ ∥D2f∥L1(T). Since the mid-point +quadrature is exact for linears, we deduce +∣∫T (f(xT ) − f)∣ = ∣∫T {(f − p)(xT ) + (p − f)}∣ ≤ 2∣T∣∥f − p∥L∞(T) ≲ h2 +T ∥D2f∥L1(T). +Moreover, invoking (45), +∥D2f∥L1(T) ≲ ∥D3yϵ∥L2(T)∥∇yϵ∥L2(T) + ∥D2yϵ∥2 +L2(T) ≲ ∥yϵ∥H3(T). +Inserting this back into R3(T) and adding we end up with +limsup +h→0+ +2 +∑ +i,j=1 +∑ +T∈Th +∣R3(T)∣ ≲ limsup +h→0+ (h∥Hh[yh]∥L2(Ω))∥yϵ∥H3(Ω) = 0, +because of Lemma 3. Altogether, we arrive at +limsup +h→0+ ∣Ch[yh] − C[yϵ]∣ ≲ ϵ +which implies the desired estimate (44). +□ + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +13 +It is worth realizing the role of the reduced discrete Hessian Hh[yh] in the pre- +ceding proof, namely that it factors out the integral defining R3(T). If we had used +the discrete Hessian Hh[yh] instead, then there would have been a term of the form +h2 +T ∥D2Hh[yh]∥L2(T) that could only be handled via an inverse inequality within the +H2-regularity setting. This in turn would have gotten rid of the factor h2 +T and the +proof of (43) would have failed. +3. Γ-convergence +The reduced energy (8) consists of a bending energy B[y] and a cubic term C[y], +and so does its discrete counterpart (37), namely Bh[yh] and Ch[yh]. Compactness +and Γ-convergence of the bending energy part, being similar to the single layer +model, could be deduced from the results in [14]. For instance, we have that for any +γ0,γ1 > 0, there exists a constant ccoer such that [14, (37) and (38)] +(46) +c−1 +coer∥yh∥2 +H2 +h(Ω) ≤ Bh[yh] ≤ ccont∥yh∥2 +H2 +h(Ω) +∀yh ∈ Vk +h(ϕ,Φ), +and the constant ccoer → ∞ if either γ0 or γ1 → 0+. In spite of that, [14] enforces +the isometry constraint in average and constructs the recovery sequence needed +for Γ-convergence via standard nodal interpolation. Therefore, the analysis below +incorporates new ideas which do not follow from [14]. +We start with the equicoercivity of energy Eh. The difficulty is dealing with Ch. +Lemma 8 (coercivity of total energy). Let δ > 0 and yh ∈ Ah,δ. There exists a +constant ˜ccoer > 0 independent of δ, but depending on the given data Z and Th only +through its shape regularity constant, such that +(47) +(2ccoer)−1∥yh∥2 +H2 +h(Ω) ≤ Eh[yh] + ˜ccoer(1 + δ)2. +Proof. We write Bh = Eh + Ch and employ (46) for Bh to obtain +c−1 +coer∥yh∥2 +H2 +h(Ω) ≤ Eh[yh] + Ch[yh]. +It remains to estimate the cubic term Ch[yh]. +Combining Lemma 6 (pointwise +isometry constraint) with the Cauchy-Schwarz inequality yields +Ch[yh] ≤ +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣∣Hh[yh]ij ⋅ (∂1yh × ∂2yh)Zij∣(xT ) +≤ +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣ +1 +2 ∥Hh[yh]ij∥L2(T)∣∂1yh(xT )∣∣∂2yh(xT )∣∥Z∥L∞(T) +≤ 2(1 + δ)∥Z∥L∞(Ω)∣Ω∣ +1 +2 ∥Hh[yh]∥L2(Ω). +Invoking Lemma 3 (stability of Hh[yh]) and Young’s inequality yields +(48) +1 +2ccoer +∥yh∥2 +H2 +h(Ω) ≤ Eh[yh] + 2ccoerc2 +stab∣Ω∣∥Z∥2 +L∞(Ω)(1 + δ)2, +which is the desired estimate (47) with ˜ccoer = 2ccoerc2 +stab∣Ω∣∥Z∥2 +L∞(Ω). +□ + +14 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +We now prove Γ-convergence of Eh towards E, which consists of a liminf and a +limsup property. +Theorem 9 (Γ-convergence of Eh). Assume that δ = δ(h) → 0 as h → 0+. Then +(i) Lim-inf property: Let yh ∈ Ah,δ be a sequence such that Eh[yh] is uniformly +bounded in h. Then there exists y ∈ A such that yh → y in [L2(Ω)]3 for a +subsequence (not relabeled) and E[y] ≤ liminf +h→0+ Eh[yh]. +(ii) Lim-sup property: For any y ∈ A there exists yh ∈ Ah,δ such that yh → y in +[L2(Ω)]3 and E[y] ≥ limsup +h→0+ Eh[yh]. +Proof. We prove properties (i) and (ii) separately. +(i) lim-inf property. Lemma 8 (coercivity of total energy) and (24) imply +∥yh∥L2(Ω) + ∥∇hyh∥L2(Ω) + ∥yh∥H2 +h(Ω) ≲ 1. +Proceeding as in [17, Proposition 5.2], there exists y ∈ [H2(Ω)]3 satisfying the +Dirichlet boundary conditions in (1) and yh → y in [L2(Ω)]3, ∇hyh → ∇y in +[L2(Ω)]3×2. +In view of Lemma 1 (weak convergence of Hh), we deduce Hh[yh] ⇀ D2y in +[L2(Ω)] +3×2×2. The lower-semicontinuity of the L2-norm under weak-limits together +the fact that the stabilization terms in Bh[yh] are positive guarantee that +B[y] = 1 +2 ∫Ω ∣D2y∣2 ≤ liminf +h→0+ Bh[yh]. +In addition, Lemma 7 (convergence of cubic energy) yields limh→0+ Ch[yh] = C[yh], +and altogether gives E[y] ≤ liminfh→0+ Eh[yh] as asserted. +It just remains to prove the isometry constraint I[y] = I2 a.e. in Ω. To this end, +recall that I[yh] = ∇hyT +h ∇hyh, let T ∈ Th and note that +∥I[yh] − I2∥L1(T) ≤ ∥I[yh] − I[yh](xT )∥L1(T) + ∣T∣∣I[yh](xT ) − I2∣ +≲ hT ∥D2 +hyh∥L2(T)∥∇hyh∥L2(T) + δ∣T∣, +because yh ∈ Ah,δ whence Dh[yh](xT ) = ∣I[yh](xT ) − I2∣ ≤ δ. Adding over T and +employing the uniform boundedness of ∥D2 +hyh∥L2(Ω) and ∥∇hyh∥L2(Ω) results in +∥I[yh] − I2∥L1(Ω) ≲ h + δ → 0 +as h → 0+. +On the other hand, we see that +I[yh] − I[y] = ∇h(yh − y)T ∇hyh + ∇yT ∇h(yh − y) +implies +∥I[yh] − I[y]∥L1(Ω) ≤ (∥∇y∥L2(Ω) + ∥∇hyh∥L2(Ω))∥∇hyh − ∇y∥L2(Ω) → 0, +as h → 0+ because ∥∇hyh∥L2(Ω) ≲ 1. This and the triangle inequality lead to ∥I[y] − +I2∥L1(Ω) = 0 and consequently I[y] = I2 a.e. in Ω, as desired. +(ii) lim-sup property. The difficulty to construct a recovery sequence yh ∈ Ah,δ is +that the regularity y ∈ [H2(Ω)∩W 1 +∞(Ω)]3 is borderline to define pointwise values of + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +15 +∇y and thus enforce the isometry defect Dh[yh](xT ) at every element barycenter +xT . Hence, we invoke the regularization procedure of P. Hornung [29]: given an +isometry y ∈ [H2(Ω)]3 and ϵ > 0, there exists an isometry yϵ ∈ [H3(Ω)]3 such that +(49) +∥y − yϵ∥H2(Ω) ≲ ϵ, +∥D2yϵ∥L2(Ω) ≲ ∥D2y∥L2(Ω). +As usual, the constants hidden in the symbol ≲ are independent of h and ϵ. We +now set yh ∶= Rh[yϵ], where the recovery operator Rh ∶ [H3(Ω)]3 → [Vk +h]3 is the +following quadratic Taylor expansion about xT for every T ∈ Th +(50) Rh[w](x) ∶= w(xT ) + ∇w(xT )(x − xT ) + 1 +2(x − xT )T QT [w](x − xT ) +∀x ∈ T, +where QT [w] ∶= +1 +∣T∣ ∫T D2w. Note that ∇yh(xT ) = ∇yϵ(xT ) and Dh[yh](xT ) = 0, +whence yh ∈ Ah,0 ⊂ Ah,δ. We next show the two convergence properties of yh in (ii). +Since Rh∣T is invariant over the space P1 of polynomials of degree ≤ 1, we have +w − Rh[w] = (w − p) − Rh[w − p] +∀p ∈ [P1]3. +Therefore, combining the stability in W 1 +∞(T) of the linear part of Rh with the +Bramble-Hilbert lemma and the property ∥QT [w]∥L2(T) ≤ ∣w∣H2(T), we deduce +∥w − Rh[w]∥H1(T) ≲ hT ∥∇(w − p)∥W 1 +∞(T) + hT ∥QT [w]∥L2(T) +≲ h2 +T ∥w∥H3(T) + hT ∣w∣H2(T) ≲ hT ∥w∥H3(T). +Notice the presence of the full H3-norm on the right-hand side of the above estimate, +which accounts for possible subdivisions made of quadrilaterals [22, 26, 17]. We next +square and add over T ∈ Th to obtain +∥w − Rh[w]∥L2(Ω) + ∥∇w − ∇hRh[w]∥L2(Ω) ≲ h∥w∥H3(Ω). +This estimate for w = yϵ, in conjunction with (49), yields +∥y − yh∥L2(Ω) + ∥∇y − ∇hyh∥L2(Ω) ≲ ϵ + h∥yϵ∥H3(Ω), +whence ∥y−yh∥L2(Ω) ≲ ϵ provided h is sufficiently small so that h∥yϵ∥H3(Ω) ≤ ϵ. This +shows the asserted convergence yh → y in [L2(Ω)]2 because ϵ is arbitrary. +It remains to show the convergence Eh[yh] → E[y] as h → 0+, which in turn +implies the desired lim-sup property. Since D2yh = QT [yϵ], we infer that +∥D2 +hyh∥2 +L2(Ω) = ∑ +T∈Th +∥QT [yϵ]∥2 +L2(T) ≤ ∑ +T∈Th +∥D2yϵ∥2 +L2(T) ≲ ∥D2y∥2 +L2(Ω), +according to (49). Moreover, +∥D2 +hyh − D2y∥2 +L2(Ω) = ∑ +T∈Th +∥QT [yϵ] − D2y∥2 +L2(T) +≤ ∑ +T∈Th +∥QT [yϵ] − D2yϵ∥2 +L2(T) + ∥D2yϵ − D2y∥2 +L2(T) ≲ h2∥yϵ∥2 +H3(Ω) + ϵ2 +shows that D2 +hyh → D2y and Lemma 2 (strong convergence of Hh) gives +Hh[yh] → D2y +strongly in [L2(Ω)]3×2×2. + +16 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +An argument similar to [14, Appendix B and C], invoking the trace inequality, yields +Sh[yh] ≲ ∑ +T∈Th +∥y − yh∥2 +H2(T) → 0, +as h → 0+ +for the stabilization energy Sh[yh] in (39) and implies convergence of the bending +energy Bh in (38), namely lim +h→0+ Bh[yh] = B[y]. Finally, in view of the preceding dis- +cussion, we see that the assumptions of Lemma 7 (convergence of the cubic energy) +are valid, whence Lemma 7 implies Ch[yh] → C[y] and completes the proof. +□ +The construction of the recovery sequence in Theorem 9 (Γ-convergence of Eh) +is closely related to Lemma 7 (convergence of the cubic energy) and illustrates the +crucial interplay between enforcing the isometry defect Dh[yh] at barycenters and +the mid-point quadrature rule in the cubic energy Ch[yh]. This, however, limits the +accuracy of LDG to that of lowest polynomial degree k = 2. We leave the design of +an LDG method with formal higher accuracy k > 2 open in this paper. +Corollary 10 (convergence of global minimizers). Let yh ∈ Ah,δ be a sequence of +functions such that Eh[yh] is uniformly bounded in h. If yh is an almost global +minimizer of Eh in the sense that +Eh[yh] ≤ +inf +wh∈Ah,δ +Eh[wh] + σ +where σ,δ → 0 as h → 0+, then {yh}h>0 is precompact in [L2(Ω)]3 and every cluster +point y belongs to A and is a global minimizer of E, namely E[y] = infw∈A E[w]. +Moreover, up to a subsequence (not relabeled) the energies converge +E[y] = lim +h→0+ Eh[yh]. +We omit the proof of Corollary 10, which readily follows from Theorem 9 (Γ- +convergence of Eh), and refer instead to [4, 5, 8, 17] for details. +4. Bilayer model with creases +Bartels, Bonito and Hornung have recently developed a reduced single layer model +that allows for folding across creases [6]. The resulting two-dimensional model hinges +on a general hyperelastic material description with appropriate scaling conditions on +the energy, and consists of a piecewise nonlinear Kirchhoff plate bending model with +a continuity condition at the creases. For a prescribed Lipschitz curve C intersecting +the boundary of Ω transversally, the modified bending energy of [6] reads +̃B[y] ∶= 1 +2 ∫Ω∖C ∣II[y]∣ +2 = 1 +2 ∫Ω∖C ∣D2y∣2 +for deformations y ∈ [H2(Ω ∖ C) ∩ W 1,∞(Ω)]3 satisfying the isometry constraint +I[y] = I2 along with possible boundary conditions. Properly designed creases allow +for flapping mechanisms upon actuation at the boundary which are of interest in +engineering and medicine [6]. + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +17 +In this section we explore a similar modification of the elastic energy (4) +(51) +̃E[y] ∶= 1 +2 ∫Ω∖C ∣II[y] − Z∣ +2, +but without justification from 3d hyperelasticity. +Therefore, we leave open the +question whether this energy is the appropriate Γ-limit for bilayer materials. We +also modify the admissible set to be +̃A ∶= {y ∈ [H2(Ω ∖ C) ∩ W 1 +∞(Ω)]3 ∶ +I[y] = I2 in Ω, +y = ϕ, ∇y = Φ on ΓD}. +Our goal is, instead, to investigate the relation between (51) and its fully discrete +counterpart, and demonstrate computationally the crucial role of spontaneous cur- +vature Z to produce plate folding without actuation via boundary conditions. +We extend our LDG method to account for creases as in [6]. We consider iso- +parametric partitions Th made of possibly curved elements, i.e. the mapping FT +used to define the finite element space Vk +h locally is [Q2]2 instead of [Q1]2 (or [P2]2 +instead of [P1]2). We further assume that the crease C is exactly matched by Th: +(52) +C is made of piecewise quadratic edges e1,...,eJ ∈ Eh. +This geometric assumption is restrictive but instrumental for the theory below. +Dealing with more general creases C, just interpolated by Eh, is important and the +subject of current research; we refer to [6, Section 4.4] for some discussion. +The distributional derivative of y ∈ [H2(Ω ∖ C) ∩ W 1 +∞(Ω)]3 reads +D2y = ̃D2y + [∇y] ⊗ nδC, +where ̃D2y stands for the absolutely continuous part of D2y, or restriction of D2y +to Ω ∖ C that happens to be L2, while [∇y] ⊗ nδC is the singular part supported on +C and n is a unit normal vector to C. The first issue to tackle is the construction of +a discrete Hessian ̃ +Hh[yh] that allows for folding across C and mimics ̃D2y. As in +[6], we replace the global lift Rh in (27) by +̃ +Rh ∶= +∑ +e∈Ea +h∖{e1,...,eJ} +re, +where {ej}J +j=1 are defined in (52), and let the modified discrete Hessian be +̃ +Hh[yh] ∶= D2 +hyh − ̃ +Rh([∇hyh]) + Bh([yh]). +We likewise replace (22) by the modified mesh-dependent form ⟨⋅,⋅⟩ ̃ +H2 +h +⟨vh,wh⟩ ̃ +H2 +h ∶= (D2 +hvh,D2 +hwh)L2(Ω) ++ (h−1[∇hvh],[∇hwh])L2(Γa +h∖C) + (h−3[vh],[wh])L2(Γa +h). +In essence, the ability for the plates to fold freely across C is reflected in the absence +of all the contributions related to [∇yh] across C. This is the key to the following +lemma whose proof follows along the lines of [14, Appendix B] and is thus omitted. +Lemma 11 (convergence of ̃ +Hh). Let the crease C satisfy (52). Then there holds + +18 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +(i) Weak convergence: If k ≥ 2 and vh ∈ Vk +h(ϕ,Φ) satisfies ∣∣vh∣∣H2 +h(Ω) ≲ 1 and +vh → v ∈ [H2(Ω ∖ C) ∩ H1(Ω)]3 in [L2(Ω)]3 as h → 0+, then we have +̃ +Hh[vh] ⇀ ̃D2v +in [L2(Ω)] +3×2×2 +as h → 0+. +(ii) Strong convergence: Let v ∈ [H2(Ω ∖ C)]3 be any function such that v = ϕ +and ∇v = Φ on ΓD. Moreover, let vh ∈ Vk +h(ϕ,Φ) satisfy +∥D2vh∥L2(T) ≲ ∥v∥H2(T) ∀T ∈ Th, +∑ +T∈Th +∥vh − v∥2 +H2(T) → 0 as h → 0+. +Then we have as h → 0+ +̃ +Hh[vh] → ̃D2v +strongly in [L2(Ω)]3×2×2. +We are now ready to introduce the LDG approximation of ̃E[y] in (51), namely +̃Eh[yh] ∶= ̃Bh[yh] + ̃Ch[yh], +where +̃Bh[yh] ∶= 1 +2 ∫Ω ∣ ̃ +Hh[yh]∣2 + γ1∥h− 1 +2 [∇hyh]∥2 +L2(Γa +h∖C) + γ0∥h− 3 +2 [yh]∥2 +L2(Γa +h), +and +̃Ch[yh] ∶= +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣Hh[yh]ij ⋅ (∂1yh × ∂2yh)(xT )Zij, +with Hh[yh]∣T ∶= +1 +∣T∣ ∫T ̃ +Hh[yh]. Lemmas 3 and 4 are valid for Hh[yh], as well as +Lemma 7 (convergence of cubic energy) and Lemma 8 (coercivity of total energy). +It remains to examine the convergence of the discrete global minimizers towards +the continuous global minimizers. Assume that the crease C splits Ω into two disjoint +sets Ω1 and Ω2. Since Hornung’s regularization procedure [29] cannot guarantee +general Dirichlet boundary conditions, it is not clear how to regularize in Ω1 and Ω2 +functions that belong to [H2(Ω∖C)∩W 1 +∞(Ω)]3 and yet maintain the location of the +crease C, namely obtain an isometry in [H3(Ω∖C)∩W 1 +∞(Ω)]3. Another obstruction +stems from the use of curved elements necessarily for the subdivisions to match +the crease. When using polynomial mappings from the reference to the physical +elements, the resulting finite element functions are not necessarily polynomial in +the physical element, thereby ruling out the construction of the recovery sequence +proposed to guarantee the limsup property; see Theorem 9(ii). +We circumvent these issues by requiring slightly more smoothness on one of the +global minimizers y, which in turn allows for a different, more generic, construction +of its recovery sequence. Because the additional regularity cannot be derived from +our Γ-convergence theory, we assume the existence of a global minimizer y∗ ∈ ̃A of +̃E with the following property +(53) +y∗∣Ωi ∈ C1(Ωi), +i = 1,2. +Note that the above assumption is consistent with practical configurations. We also +point out that this regularity assumption and the fact that the subdivision matches + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +19 +the crease entail the existence of a modulus of smoothness ω so that +(54) +∣∇y∗(x) − ∇y∗(z)∣ ≤ ω(hT ) +∀x,z ∈ T, ∀T ∈ Th, +with ω(s) → 0 as s → 0+. +The construction of the recovery sequence for deformations satisfying the addi- +tional regularity (53) is then based on a piecewise averaged Taylor polynomial. The +latter does not preserve the isometry constraint pointwise but (53) allows for control +of the isometry defect. +Before embarking on the proofs, we recall a useful result on the averaged Taylor +polynomial [19] defined on the reference element ̂T (see Section 2.2). Until the end +of this section, we consider the case when the reference element is a square and Qk +finite element functions are used. The case where ̂T is the unit simplex is somewhat +simpler and can be dealt with similarly. Let ̂B be a ball centered at the barycenter +of ̂T such that its closure is contained in ̂T and ̂ζ be a cut-off function with unit +mass supported on the closure of ̂B. For ̂w ∈ L1(̂T) let +(55) +Q[̂w](ˆx) ∶= +∑ +∣α∣∞≤2 +∫ ̂ +B +1 +α! +̂Dα ̂w(ˆz)(ˆx − ˆz)α̂ζ(ˆz)dˆz ∈ Q2, +be the averaged Taylor polynomial where α ∶= (α1,α2) is a multi-index with non- +negative integers α1,α2 and ∣α∣∞ ∶= max{α1,α2}. We recall the following useful +properties of Q and refer to [19] for additional details: Q preserves Q2 on ̂T +(56) +Q[̂p] = ̂p, +̂p ∈ Q2, +is stable +(57) +∥Q[̂w]∥W k +∞(̂T) ≲ ∥̂w∥L1( ̂ +B), +∀k ∈ N +and convergent +(58) +∣̂w − Q[̂w]∣Hk(̂T) ≲ ( +2 +∑ +i=1 +∥∂3 +̂xi ̂w∥2 +L2(̂T)) +1/2 +, +0 ≤ k < 3. +We next discuss estimates for isoparametric mappings FT ∶ ̂T → T between the +reference element ̂T and T ∈ Th so that FT ∈ [Q2]2. They establish relationship +between norms on ̂T and T, as well as provide an interpolation estimate in [V2 +h]3. +In fact, for v ∈ H2(T) and ̂v = v ○ FT ∈ H2(̂T), there holds +(59) +∥̂v∥L2(̂T) ≈ h−1 +T ∥v∥L2(T), +∥̂v∥L∞(̂T) ≈ ∥v∥L∞(T) +(60) +∥̂∇̂v∥L2(̂T) ≈ ∥∇v∥L2(T), +∥̂∇̂v∥L∞(̂T) ≈ hT ∥∇v∥L∞(T), +(61) +∥ ̂D2̂v∥L2(̂T) ≲ hT ∥D2v∥L2(T) + ∥∇v∥L2(T), +(62) +∥D2v∥L2(T) ≲ h−1 +T ∥̂v∥H2(̂T), +whence for m = 0,1,2 +(63) +∣v∣Hm(T) ≲ h1−m +T +∥̂v∥H2(̂T). + +20 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +Moreover if v ∈ H3(T), we further obtain +(64) +∥ ̂D3̂v∥L2(̂T) ≲ h2 +T ∥D3v∥L2(T) + hT ∥D2v∥L2(T) + ∥∇v∥L2(T). +Note that the first four estimates are discussed and proved in [17, Appendix], while +one can extend the proof of (61) to show (64). Additionally, as in [17, Lemma A.4], +the local Lagrange interpolant Ihw ∈ [V2 +h]3 for any w ∈ H3(T) satisfies the estimate +(65) +∣w − Ihw∣Hm(T) ≲ h3−m∥v∥H3(T), +for 0 ≤ m ≤ 3. +The next lemma describes the modified limsup property which hinges on the +averaged Taylor polynomial (55); compare with Theorem 9(ii). +Lemma 12 (limsup property with creases). Let y∗ ∈ ̃A satisfy the regularity as- +sumption (53) and let ω be the modulus of smoothness in (54). There is a constant +c and y∗ +h ∈ Ah,cω(h) such that y∗ +h → y∗ in [L2(Ω)]3 and limh→0+ ̃Eh[yh] = ̃E[y∗]. +Proof. As usual, the hat symbol denotes quantities defined on the reference element +̂T. Let y∗ +h ∈ [V2 +h]3 be defined locally by +y∗ +h∣T ∶= ̂y∗ +h∣T ○ F −1 +T , +̂y∗ +h∣T ∶= Q[y∗○ FT ] +∀T ∈ Th, +where Q is given in (55) and is applied component-wise. Note that by construction +we indeed have y∗ +h ∈ [V2 +h]3. The rest of the proof consists of 4 steps. +Step (i): isometry defect. We claim the intermediate estimates +(66) +∥∇y∗ +h∥L∞(T) ≲ ∥∇y∗∥L∞(T), +∥∇(y∗ − y∗ +h)∥L∞(T) ≲ ω(hT ) +∀T ∈ Th. +To show the first estimate, we use (60) and (56) to write +∥∇y∗ +h∥L∞(T) ≲ h−1 +T ∥̂∇̂y∗ +h∥L∞(̂T) ≲ h−1 +T ∥̂∇(̂y∗ +h − c)∥L∞(̂T) = h−1 +T ∥̂∇Q[̂y∗− c]∥L∞(̂T), +where c ∶= ∣T∣−1 ∫T y∗ ∈ R3 is the average of y∗. We then employ the stability (57) +of Q, together with (59) and Poinc´are inequality, to deduce +∥∇y∗ +h∥L∞(T) ≲ h−1 +T ∥̂y∗ − c∥L∞(̂T) ≲ h−1 +T ∥y∗ − c∥L∞(T) ≲ ∥∇y∗∥L∞(T), +which is the first estimate in (66). +To prove the second estimate in (66), we first notice that for any p ∈ [P1]3 we +have ̂p ∶= p ○ FT ∈ [Q2]3. Since ̂p = Q[̂p], according to (56), we proceed as before, +but now using ̂p ∈ Q2 instead of the constant c along with (60), to write +∥∇(y∗ − y∗ +h)∥L∞(T) ≲ h−1 +T (∥̂∇(̂y∗ − ̂p)∥L∞(̂T) + ∥̂∇Q[̂y∗ − ̂p]∥L∞(̂T)) +≲ h−1 +T ∥̂y∗ − ̂p∥W 1 +∞(̂T) +≲ h−1 +T ∥y∗ − p∥L∞(T) + ∥∇(y∗ − p)∥L∞(T). +(67) +We next choose y∗ to take advantage of the piecewise smoothness (53), namely +p(x) ∶= y∗(xT ) + ∇y∗(xT )(x − xT ). +The property (54) of ω implies +∥∇(y∗ − p)∥L∞(T) ≤ ω(hT ) + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +21 +and combined with y∗(x) − y∗(xT ) = ∇y∗(ξ)(x − xT ) for some ξ ∈ T, gives +∥y∗ − p∥L∞(T) ≤ hT ω(hT ). +Inserting these estimates in (67) yields the second estimate in (66). +The estimate on the isometry defect ∥(∇y∗ +h)T ∇y∗ +h−I2∥L∞(T) follows directly from +the intermediate estimates (66) and the assumption I[y∗] = I2 +∥(∇y∗ +h)T ∇y∗ +h − I2∥L∞(T) = ∥(∇y∗ +h)T ∇y∗ +h − (∇y∗)T ∇y∗∥L∞(T) +≤ (∥∇y∗∥L∞(T) + ∥∇y∗ +h∥L∞(T))∥∇(y∗ +h − y∗)∥L∞(T) ≤ cω(hT ). +for a constant c independent of the discretization parameters; hence y∗ +h ∈ Ah,cω(h). +Step (ii): Broken H2− Stability. For p ∈ [P1]3, we set ̂p ∶= p ○ FT ∈ [Q2]3 to get +∥D2y∗ +h∥L2(T) = ∥D2(y∗ +h − p)∥L2(T) ≲ h−1 +T ∥̂y∗ +h − ̂p∥H2(̂T), +in view of (62). Thanks to the invariance (56) and stability (57) of Q, we obtain +∥D2y∗ +h∥L2(T) ≲ h−1 +T ∥Q[̂y∗ − ̂p]∥H2(̂T) ≲ h−1 +T ∥̂y∗ − ̂p∥L2(̂T). +This, together with (59) and a standard interpolation estimate on T, yields +∥D2y∗ +h∥L2(T) ≲ h−2 +T ∥y∗ − p∥L2(T) ≲ ∥D2y∗∥L2(T), +which is the desired stability estimate. +Step (iii): H2−Convergence. We exploit a density argument. For any ϵ > 0, there +exists yϵ ∈ H3(Ω) so that ∥y∗ − yϵ∥H2(Ω) ≤ ϵ; yϵ may not be an isometry. We split +(68) +∑ +T∈Th +∥y∗ −y∗ +h∥2 +H2(T) ≲ ∥y∗ −yϵ∥2 +H2(Ω) + ∑ +T∈Th +∥yϵ −yϵ +h∥2 +H2(T) + ∑ +T∈Th +∥yϵ +h −y∗ +h∥2 +H2(T) +with yϵ +h = ̂yϵ +h ○ F −1 +T +and ̂yϵ +h ∶= Q[yϵ ○ FT ], and estimate each of the three terms +separately. The first term is obviously bounded by ϵ2. +For the second term, we let m = 0,1,2 and combine (58) with (63) to arrive at +∣yϵ +h − yϵ∣Hm(T) ≲ h1−m +T +( +2 +∑ +i=1 +∥∂3 +̂xîyϵ∥2 +L2(̂T)) +1/2 +≲ h1−m +T +( +2 +∑ +i=1 +∥∂3 +̂xi(̂yϵ − ̂ +Ihyϵ)∥2 +L2(̂T)) +1/2 +, +where Ih is the local Lagrange interpolant onto [V2 +h]3 and ̂ +Ihyϵ ∶= Ihyϵ ○F −1 +T +∈ [Q2]3. +As a consequence, using the estimate (64) to map back to the physical element T, +and applying the error estimate (65) for Ih to the ensuing terms, we deduce +∣yϵ +h − yϵ∣Hm(T) ≲ h3−m +T +∥yϵ∥H3(T), +for m = 0,1,2. We thus conclude +∑ +T∈Th +∥yϵ − yϵ +h∥2 +H2(T) ≲ h2∥yϵ∥2 +H3(Ω). +It remains to estimate the third term ∥y∗ +h − yϵ +h∥H2(T). To deal with each term +∣y∗ +h − yϵ +h∣Hm(T) for m = 0,1,2, we let pm−1 ∈ Pm−1 to be chosen later with the + +22 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +convention that p−1 = 0, and use the invariance Q[p ○ FT ] = p ○ FT ∈ [Q2]3 for any +p ∈ [P1]3. Combining (63) with the stability (57) of Q yields +∣y∗ +h − yϵ +h∣Hm(T) = ∣y∗ +h − yϵ +h − pm−1∣Hm(T) ≲ h1−m +T +∥Q[̂y∗ − ̂yϵ − pm−1 ○ FT ]∥H2(̂T) +≲ h1−m +T +∥̂y∗ − ̂yϵ − pm−1 ○ FT ∥L2(̂T) ≲ h−m +T ∥y∗ − yϵ − pm−1∥L2(T) +≲ ∥y∗ − yϵ∥H2(T), +provided pm−1 is an averaged Taylor polynomial of y∗ − yϵ. This in turn implies +∑ +T∈Th +∥yϵ +h − y∗ +h∥2 +H2(T) ≲ ∥y∗ − yϵ∥2 +H2(Ω) ≲ ϵ2. +Therefore, gathering the estimates for the three terms in (68) we obtain +∑ +T∈Th +∥y∗ − y∗ +h∥2 +H2(T) ≲ ϵ2 + h2∥yϵ∥2 +H3(Ω) ≤ 2ϵ2, +provided h∥yϵ∥H3(T) ≤ ϵ for h sufficiently small. Since ϵ is arbitrary, we deduce +∑ +T∈Th +∥y∗ − y∗ +h∥2 +H2(T) → 0, +and in particular y∗ +h → y∗ in [L2(Ω)]3, as h → 0+. +Step (iv): Convergence of ̃Eh[y∗ +h]. Steps (iii) and (iv) show that the conditions +in Lemma 11(ii) (strong convergence of ̃ +Hh) are fulfilled, whence ̃ +Hh[y∗ +h] → ̃D2y∗ +strongly in [L2(Ω)]3×2×2. Consequently, convergence of ̃Eh[y∗ +h] towards ̃E[y∗] re- +duces to the argument given in Theorem 9 (ii) and is not repeated here. +□ +The next theorem guarantees convergence of discrete global minimizers towards +exact global minimizers. but it is not a standard Γ-convergence result because we +assume (53) for one global minimizer. Other minimizers may fail to satisfy (53). +Theorem 13 (convergence of global discrete minimizers with creases). Assume +that a global minimizer y∗ ∈ ̃A of ̃E satisfy the additional regularity (53). +Let +yh ∈ Ah,δ be a sequence of functions such that ̃Eh[yh] is uniformly bounded in h +and let δ = δ(h) ≥ cω(h) with c the constant in Lemma 12 and ω the modulus of +smoothness in (54). If yh is an almost global minimizer of ̃Eh in the sense that +(69) +̃Eh[yh] ≤ +inf +wh∈Ah,δ +̃Eh[wh] + σ +where σ,δ → 0 as h → 0+, then {yh}h>0 is precompact in [L2(Ω)]3 and every clus- +ter point y belongs to widetildeA and is a global minimizer of ̃E, namely ̃E[y] = +infw∈̃A ̃E[w]. Moreover, up to a subsequence (not relabeled) +(70) +̃E[y] = lim +h→0+ ̃Eh[yh]. +Proof. The liminf property follows along the lines of Theorem 9 (i) because it is +based on Lemmas 8 and 7, which remain valid in this context, as well as Lemma +11 (i) (weak convergence of ̃ +Hh) instead of Lemma 1 (weak convergence of Hh) and + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +23 +the weak lower semicontinuity of the L2-norm. Therefore, there is y ∈ ̃A such that +(up to a subsequence not relabelled) yh → y in [L2(Ω)]3 and +̃E[y] ≤ liminf +h→0+ ̃Eh[yh]. +To show that y is a global minimizer of ̃E we resort to the extra regularity (53) of +the global minimizer y∗ ∈ ̃A of ̃E. Let {y∗ +h}h>0 ⊂ [L2(Ω)]3 be the sequence provided +by Lemma 12 (limsup property with creases), which satisfies +y∗ +h ∈ Ah,cω(h), +̃Eh[y∗ +h] → ̃E[y∗] +as h → 0+. In view of (69) and yh ∈ Ah,δ(h), we end up with +̃E[y] ≤ liminf +h→0+ ̃Eh[yh] ≤ limsup +h→0+ ( ̃Eh[y∗ +h] + σ) = ̃E[y∗] = inf +w∈ ̃ +A +̃E[w]. +Therefore, y is indeed a global minimizer of ̃E and (70) is valid. +□ +5. Discrete gradient flow +Solving the minimization problem (41) is a nontrivial task because it entails en- +forcing the nonconvex constraint Dh[yh](xT ) ≤ δ at element barycenters xT . We +now develop a discrete gradient flow with respect to the H2 +h metric (22) that lin- +earizes the isometry constraint according to (14). We refer to [4, 7, 8, 13, 14, 17, 16] +and especially to S. Bartels and Ch. Palus [9] for similar gradient flows. +We start recalling the notion of linearized isometry constraint for vh,yh ∈ [Vk +h]3 +(71) +L[vh;wh](xT ) = [∇vT +h ∇wh + ∇wT +h ∇vh](xT ) +∀T ∈ Th +and defining a tangent space associated with the isometry constraint for any wh ∈ +Ah,δ +(72) +Fh(wh) ∶= {vh ∈ Vk +h(0,0) ∶ +L[vh;wh](xT ) = 0 +∀T ∈ Th}. +Given y0 +h ∈ Ah,0 (i.e, y0 +h satisfies the isometry constraint I[y0 +h](xT ) = I2 at each +barycenter xT ), the discrete gradient flow consists of seeking recursively δyn+1 +h +∶= +yn+1 +n +− yn +h ∈ Fh(yn +h) such that +(73) 1 +τ ⟨δyn+1 +h +,vh⟩H2 +h(Ω) + ah(δyn+1 +h +,vh) = −ah(yn +h,vh) + ℓ[yn +h](vh) +∀vh ∈ Fh(yn +h). +Here τ > 0 is a pseudo time step and ah is the bilinear form corresponding to the +variational derivative of the bending energy Bh[yh] defined in (38), i.e. +ah(wh,vh) ∶= ∫Ω Hh[wh] ∶ Hh[vh] ++ γ1(h−1[∇hwh],[∇hvh])L2(Γa +h) + γ0(h−3[wh],[vh])L2(Γa +h). +(74) + +24 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +The linear form ℓ[yn +h](vh) on vh is the first variation of the cubic energy Ch[yn +h], +defined in (40), along the direction of the test function vh and is given by +ℓ[yn +h](vh) ∶= +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣Hh[vh]ij ⋅ (∂1yn +h × ∂2yn +h)(xT )Zij ++ +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣Hh[yn +h]ij ⋅ (∂1vh × ∂2yn +h)(xT )Zij ++ +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣Hh[yn +h]ij ⋅ (∂1yn +h × ∂2vh)(xT )Zij; +recall that both Hh[vh] and Z are piecewise constant on Th. The explicit treatment +of yn +h in ℓ[yn +h](vh) is similar to the scheme proposed and analyzed by S. Bartels and +Ch. Palus [9]. For latter use, we note that Lemma 3 (stability of Hh[vh]) yields +∣ℓ[wh](vh)∣ ≲ +√ +1 + δ∥Z∥L∞(Ω)(∥∇hvh∥L2(Ω)∥wh∥H2 +h(Ω) + ∥∇hwh∥L2(Ω)∥vh∥H2 +h(Ω)), +provided wh ∈ Ah,δ because ∣∇wh(xT )∣ ≤ +√ +1 + δ from Lemma 6 (pointwise isometry +control) and the inverse inequality ∣∇vh(xT )∣ ≲ h−1 +T ∥∇vh∥L2(T) applies. In addition, +we rewrite the Friedrichs inequality (24) as follows +(75) +∥∇hwh∥L2(Ω) ≲ ∥wh∥H2 +h(Ω) + Cϕ,Φ, +∀wh ∈ Vk +h(ϕ,Φ), +where Cϕ,Φ = ∥ϕ∥H1(Ω) + ∥Φ∥H1(Ω). From these estimates we deduce the existence +of a constant cnl such that for wh ∈ Ah,δ and vh ∈ Vk +h(0,0) we have +∣ℓ[wh](vh)∣ ≤ cnl +√ +1 + δ∥Z∥L∞(Ω)(∥wh∥H2 +h(Ω)+ Cϕ,Φ)∥vh∥H2 +h(Ω). +(76) +5.1. Energy stability and admissibility. We discuss in this section the en- +ergy reduction property of the gradient flow and, although the isometry constraint +Dh[yn +h](xT ) = 0 is relaxed and linearized in the iterative scheme, the deviation of +Dh[yn +h](xT ) from 0 is controlled by a parameter δ > 0 provided τ is sufficiently small. +These results rely on a discrete inverse inequality on finite dimensional subsets. +Lemma 14 (discrete Sobolev inequality). Let Wh ⊂ ΠT∈ThH1(T) be a generic finite +element space subordinated to the partition Th. For any wh ∈ Wh there holds +∥wh∥L∞(Ω) ≲ (1 + ∣log hmin∣) +1 +2 (∥wh∥L2(Ω) + ∥∇hwh∥L2(Ω) + ∥h− 1 +2 [wh]∥L2(Γ0 +h)), +where hmin ∶= minT∈Th hT . +Proof. We denote by Πh ∶ ΠT∈ThH1(T) → Vk +h ∩ H1(Ω) the smoothing operator from +[15, 17] and recall that it satisfies +∥∇Πhwh∥L2(Ω) + ∥h−1(wh − Πhwh)∥L2(Ω) ≲ ∥∇hwh∥L2(Ω) + ∥h− 1 +2 [wh]∥L2(Γ0 +h), +and +∥Πhwh∥L2(Ω) ≲ ∥wh∥L2(Ω). + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +25 +Therefore, combining the triangle and inverse inequalities implies +∥wh∥L∞(Ω) ≲ ∥wh − Πhwh∥L∞(Ω) + ∥Πhwh∥L∞(Ω) +≲ ∥h−1(wh − Πhwh)∥L2(Ω) + (1 + ∣log hmin∣) +1 +2 ∥Πhwh∥H1(Ω), +in view of the following discrete Sobolev inequality in 2d [18, 19] +∥Πhwh∥L∞(Ω) ≲ (1 + ∣log hmin∣) +1 +2 ∥Πhwh∥H1(Ω). +This leads to the assertion upon applying the preceding estimates for Πh. +□ +We are now in a position to prove the main result of this section, namely that +the gradient flow is energy decreasing and controls the isometry defect. +Theorem 15 (properties of gradient flow). Let {y0 +h}h>0 ⊂ Ah,0 satisfy Eh[y0 +h] ≤ +c0 with c0 a constant independent of h and let all subdivisions Th be such that +∣log hmin∣ ≥ 1. +Let N be the number of iterations of the gradient flow and τ be +its pseudo-time step. There exists a constant α1 = α1(ϕ,Φ,Z) > 0 independent of h +and N such that if τ ≤ (2α1∣log hmin∣)−1, then the energy Eh[yN +h ] satisfies +(77) +Eh[yN +h ] + 1 +2τ +N−1 +∑ +n=0 +∥δyn+1 +h +∥2 +H2 +h(Ω) ≤ Eh[y0 +h]. +In addition, there are constants α2 = α2(ϕ,Φ,Z) > 0 and α3 > 0, both independent +of h and N, such that the isometry defect Dh[yN +h ] satisfies +(78) +∣Dh[yN +h ](xT )∣ ≤ α3τ∣log hmin∣(Eh[y0 +h] + α2) +∀T ∈ Th. +Proof. We proceed by induction. We first note that estimates (77) and (78) hold +trivially for N = 0 and y0 +h ∈ Ah,0. Therefore, we assume that (77) and (78) are valid +for N = 1,...,M with positive constants α1,α2,α3 to be specified below and prove +the validity of the same estimates for N = M +1 with the same constants α1,α2,α3. +We split the proof into four steps with the following roadmap. After deriving an +intermediate estimate in Step (i), we prove (77) in Step (ii) and (78) in Step (iii) +under suitable restrictions on αi, i = 1,2,3. In Step (iv), we show that it is always +possible to find values of these parameters satisfying the desired restrictions. In this +proof, the generic constants hidden in the symbol “≲” are not only independent of +h but also of τ, M and αi, i = 1,2,3. +Step (i): intermediate estimate. We take vh = δyM+1 +h +∈ Fh(yh) in (73) for n = M, +use the elementary relation 2b(b−a) = (b−a)2 +b2 −a2 and discard the positive term +(b − a)2 = ah(δyM+1 +h +,δyM+1 +h +) to write +(79) +∥δyM+1 +h +∥2 +H2 +h(Ω) + τ +2ah(yM+1 +h +,yM+1 +h +) − τ +2ah(yM +h ,yM +h ) ≤ τℓ[yM +h ](δyM+1 +h +). +Using (78) with N = M (induction assumption), together with the restriction on τ +and the uniform bound Eh[y0 +h] ≤ c0, we obtain +(80) +∣Dh[yM +h ](xT )∣ ≤ α3 +2α1 +(c0 + α2) = δ. + +26 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +To simplify the expressions below, we let α = (αi)3 +i=1 and c2 +α = 1 + δ, whence +(81) +cα ∶= +√ α3 +2α1 +(c0 + α2) + 1. +Estimate (80) shows that yM +h ∈ Ah,δ with δ = c2 +α − 1 which in turn implies +∣∂iyM +h (xT )∣ ≤ cα, +i = 1,2, +T ∈ Th, +according to Lemma 6 (pointwise isometry constraint). Substituting into (76) yields +∣ℓ[yM +h ](δyM+1 +h +)∣ ≤ cnlcα∥Z∥L∞(Ω)∥δyM+1 +h +∥H2 +h(Ω)(∥yM +h ∥H2 +h(Ω) + Cϕ,Φ). +Inserting this back into (79) and using Young’s inequality to absorb the term +∥δyk+1 +h +∥2 +H2 +h(Ω) in the left hand side of (79), gives the estimate +1 +2∥δyM+1 +h +∥2 +H2 +h(Ω) + τ +2ah(yM+1 +h +,yM+1 +h +) +≤ τ +2ah(yM +h ,yM +h ) + τ 2c2 +nlc2 +α∥Z∥2 +L∞(Ω)(∥yM +h ∥2 +H2 +h(Ω) + C2 +ϕ,Φ). +(82) +We next improve upon (82) by deriving a uniformly bound for the right-hand +side. According to (80), the isometry defect is controlled by δ = c2 +α − 1. Moreover, +(77) for N = M (induction assumption) implies that Eh[yM +h ] ≤ Eh[y0 +h] ≤ c0. Hence, +the coercivity estimate (47) reads +(83) +(2ccoer)−1∥yM +h ∥2 +H2 +h(Ω) ≤ c0 + ˜ccoerc4 +α. +Estimate (46) can be rewritten in terms of the bilinear form ah as follows +(84) +c−1 +coer∥vh∥2 +H2 +h(Ω) ≤ 1 +2ah(vh,vh) ≤ ccont∥vh∥2 +H2 +h(Ω) +∀vh ∈ Vk +h(ϕ,Φ), +Since ∣log hmin∣ ≥ 1, τ satisfies τ ≤ +1 +2α1 ≤ 1 provided α1 ≥ 1 +2, which is our first +restriction on α1. Combining this with (83) and (84) with vh = yM +h , and replacing +back into (82), gives the desired intermediate estimate +(85) +1 +2τ ∥δyM+1 +h +∥2 +H2 +h(Ω) + 1 +2ah(yM+1 +h +,yM+1 +h +) ≤ ψ1(cα) +where ψ1(cα) ≥ 0 is a positive increasing function of its argument cα but whose +specific expression is irrelevant except that it is independent of h, M and depends +on α = (αi)3 +i=1 only through the variable cα rather than separately on each αi. +Step (ii): proof of (77) for N = M +1. In view of (79), and telescopic cancellation, +this requires dealing with the cubic term ℓ[yM +h ](δyM+1 +h +). Using the identity +aM+1bM+1cM+1 − aMbMcM += (aM+1 − aM)bM+1cM+1 + aM(bM+1 − bM)cM+1 + aMbM(cM+1 − cM), + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +27 +we deduce +(aM+1 − aM)bMcM + aM(bM+1 − bM)cM + aMbM(cM+1 − cM) += aM+1bM+1cM+1 − aMbMcM − (aM+1 − aM)(bM+1 − bM)cM+1 +− (aM+1 − aM)bM(cM+1 − cM) − aM(bM+1 − bM)(cM+1 − cM), +and rewrite ℓ[yM +h ](δyM+1 +h +) as follows: +ℓ[yM +h ](δyM+1 +h +) = +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣Hh[yM+1 +h +]ij ⋅ (∂1yM+1 +h +× ∂2yM+1 +h +)(xT )Zij +− +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣Hh[yM +h ]ij ⋅ (∂1yM +h × ∂2yM +h )(xT )Zij +− +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣Hh[δyM+1 +h +]ij ⋅ (∂1δyM+1 +h +× ∂2yM+1 +h +)(xT )Zij +− +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣Hh[δyM+1 +h +]ij ⋅ (∂1yM +h × ∂2δyM+1 +h +)(xT )Zij +− +2 +∑ +i,j=1 +∑ +T∈Th +∣T∣Hh[yM +h ]ij ⋅ (∂1δyM+1 +h +× ∂2δyM+1 +h +)(xT )Zij. +We note that the first two terms are exactly the cubic energies Ch[yM+1 +h +] and +Ch[yM +h ] and together with the bending energies Bh[yM+1 +h +] = 1 +2ah(yM+1 +h +,yM+1 +h +) and +Bh[yM +h ] = 1 +2ah(yM +h ,yM +h ) in (79) give rise to the full energies Eh[yM+1 +h +] and Eh[yM +h ] +in (77). +In contrast, the last three terms must be estimated and absorbed into +the remaining term ∥δyM+1 +h +∥2 +H2 +h(Ω) in (77). To this end, we combine the Friedrichs +inequality (75) for wh = yM+1 +h +,yM +h +∈ Vk +h(ϕ,Φ) and wh = δyM+1 +h +∈ Vk +h(0,0), and +Lemma 3 (stability of Hh[vh]), to obtain +∥δyM+1 +h +∥2 +H2 +h(Ω) + τEh[yM+1 +h +] − τEh[yM +h ] +≲ τ∥δyM+1 +h +∥H2 +h(Ω)∥∇δyM+1 +h +∥L∞(Ω)(∥yM+1 +h +∥H2 +h(Ω) + ∥yM +h ∥H2 +h(Ω) + Cϕ,Φ), +where the symbol ≲ hides ∥Z∥L∞(Ω). To estimate the L∞-norm on the right-hand +side, we resort to Lemma 14 (discrete Sobolev inequality) +∥δyM+1 +h +∥2 +H2 +h(Ω) + τEh[yM+1 +h +] − τEh[yM +h ] +≲ τ∣log hmin∣∥δyM+1 +h +∥2 +H2 +h(Ω)(∥yM+1 +h +∥H2 +h(Ω) + ∥yM +h ∥H2 +h(Ω) + Cϕ,Φ), +because ∣log hmin∣ ≥ 1. Moreover, the coercivity estimate (46) of Bh, written now as +∥yM+1 +h +∥2 +H2 +h(Ω) ≤ ccoerBh[yM+1 +h +] = ccoer +2 +ah(yM+1 +h +,yM+1 +h +) ≤ ccoerψ1(cα) +according to (85), together with (83) guarantees that +∥yM+1 +h +∥H2 +h(Ω) + ∥yM +h ∥H2 +h(Ω) + Cϕ,Φ ≤ ψ2(cα), + +28 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +where ψ2(cα) is a positive increasing function of the argument cα which is indepen- +dent of h,M, and the individual parameters (αi)3 +i=1. Substituting back yields +∥δyM+1 +h +∥2 +H2 +h(Ω) + τEh[yM+1 +h +] − τEh[yM +h ] ≤ τ∣log hmin∣ψ2(cα)∥δyM+1 +h +∥2 +H2 +h(Ω). +Consequently, since τ ≤ (2α1∣log hmin∣)−1, it remains to choose (αi)3 +i=1 so that +ψ2(cα) ≤ α1, +to derive the desired estimate (77) for N = M + 1. The validity of this estimate will +be justified in Step (iv). +Step (iii): proof of (78) for N = M + 1. +Since δyn+1 +h +∈ Fh(yn +h), expanding +I[yn+1 +h +](xT ) = [(∇yn+1 +h +)T ∇yn+1 +h +](xT ) and using the definition (72) of Fh(yn +h) yields +(86) +Dh[yn+1 +h +](xT ) = Dh[yN +h ](xT ) + I[δyn+1 +h +](xT ) +∀T ∈ Th. +Applying Lemma 14 (discrete Sobolev inequality), followed by the discrete Friedrichs +inequality (75) to estimate ∥∇hδyn+1 +h +∥L2(Ω), implies +(87) +∣I[δyn+1 +h +](xT )∣ ≤ ∥∇hδyn+1 +h +∥2 +L∞(T) ≲ ∣log hmin∣∥δyn+1 +h +∥2 +H2 +h(Ω), +because ∣log(hmin)∣ ≥ 1 and δyn+1 +h +∈ Vk +h(0,0). Summing over 0 ≤ n ≤ M, and using +telescopic cancellation along with y0 +h ∈ Ah,0, yields +(88) +∣Dh[yM+1 +h +](xT )∣ ≲ ∣log hmin∣ +M +∑ +n=0 +∥δyn+1 +h +∥2 +H2 +h(Ω). +Exploiting the energy decay (77), proved for N = M + 1 in Step (ii), gives +(89) +M +∑ +n=0 +∥δyn+1 +h +∥2 +H2 +h(Ω) ≤ 2τ(Eh[y0 +h] − Eh[yM+1 +h +]). +We now need a lower bound for the energy Eh[yM+1 +h +], which is a consequence of +(47) provided yM+1 +h +∈ Ah,ϵ for some ϵ > 0. To this end, we resort again to (86). We +first bound the second term on the right-hand side upon combining the intermediate +estimate (85) for ∥δyM+1 +h +∥H2 +h(Ω) with (87) +∣I[δyM+1 +h +](xT )∣ ≤ 2τ∣log hmin∣ψ1(cα) ≤ α−1 +1 ψ1(cα), +because τ ≤ (2α1∣log hmin∣)−1. Using this bound in (86), along with the fact that +yM +h ∈ Ah,δ for δ = c2 +α − 1 according to the induction assumption (80), implies +∣Dh[yM+1 +h +]](xT )∣ ≤ c2 +α − 1 + α−1 +1 ψ1(cα) =∶ ϵ, +whence yM+1 +h +∈ Ah,ϵ. Inserting Eh[yM+1 +h +] ≥ −˜ccoer(1 + ϵ)2 from (47) into (89) gives +M +∑ +n=0 +∥δyn+1 +h +∥2 +H2 +h(Ω) ≤ 2τ(Eh[y0 +h] + ˜ccoer(c2 +α + α−1 +1 ψ1(cα)) +2). +Returning to (88), we arrive at +∣Dh[yM+1 +h +](xT )∣ ≤ cIτ∣log hmin∣(Eh[y0 +h] + ˜ccoer(c2 +α + α−1 +1 ψ1(cα)) +2), + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +29 +where cI is a constant independent of h, M and (αi)3 +i=1. The desired control on the +isometry defect (78) is thus guaranteed provided +α3 ≥ cI, +α2 ≥ ˜ccoer(c2 +α + α−1 +1 ψ1(cα)) +2. +Step (iv): choice of parameters. α = (αi)3 +i=1 must satisfy +α1 ≥ 1 +2, +ψ2(cα) ≤ α1, +α3 ≥ cI, +α2 ≥ ˜ccoer(c2 +α + α−1 +1 ψ1(cα)) +2 =∶ ψ3(α1,cα), +where cα is defined in (81) and the functions ψ1,ψ2 are positive and increasing in +their arguments. One admissible set of parameters is +α3 = cI, +α2 = α +1 +2 +1 , +with α1 ≥ 1 +2 sufficiently large. In fact, we note that as α1 → ∞ +cα ↓ 1, +ψ1(cα) ↓ ψ1(1) ≥ 0, +ψ2(cα) ↓ ψ2(1) ≥ 0, +ψ3(α1,cα) ↓ ˜ccoer, +and the condition α1 ≥ max{1 +2,ψ2(cα),ψ3(α1,cα)2} admits a solution. This com- +pletes the induction argument. +□ +It is worth realizing that the ℓ∞-control of the isometry defect (78) implies that +yh ∈ Ah,δ provided τ is so small that +α3τ∣log hmin∣(Eh[y0 +h] + α2) ≤ δ, +where Ah,δ is defined in (35). This property is novel in the context of DG approx- +imations [13, 14, 16, 17, 38], but is inspired by a similar one at element vertices +shown by S. Bartels and Ch. Palus for Kirchhoff elements [9]. It is responsible +for the explicit treatment of the cubic term in (73), which in turn converts (73) +into a linear system to solve for δyn+1 +h +. The fact that H2(Ω) does not embed in +W 1 +∞(Ω) in two dimensions, but is borderline instead, explains the critical nature +of the estimates (77) and (78). The discrete H2-metric of the gradient flow (73), +combined with Lemma 14 (discrete Sobolev inequality), makes it possible to exploit +this borderline structure discretely at the expense of a log term. No weaker metric +for the gradient flow than H2 would allow for ℓ∞-control of the isometry defect. +5.2. Lagrange multipliers for the isometry constraint. We enforce tangen- +tial variations δyn+1 +h +∈ Fh(yn +h) at each step of the gradient flow using Lagrange +multipliers within the space of symmetric piecewise constant tensors +Λh ∶= {λh ∶ Ω → R2×2 ∶ λT +h = λh, λh ∈ [V0 +h] +2×2}. +For any wh ∈ Vk +h(ϕ,Φ), we define the bilinear form bh(wh;⋅,⋅) on Vk +h(0,0) × Λh as +(90) +bh(wh;vh,µh) ∶= ∑ +T∈Th +∣T∣L[vh;wh](xT ) ∶ µh, +where the linearized isometry constraint L[vh;wh] is given in (71). Note that bh is +continuous with a continuity constant uniform in h +(91) +∣bh(wh;vh,µh)∣ ≲ ∥wh∥H2 +h(Ω)∥vh∥H2 +h(Ω)∥µh∥L2(Ω), + +30 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +thanks to the inverse inequality ∣L[vh;wh](xT )∣ ≲ h−1 +T ∥L[vh;wh]∥L2(T) and the dis- +crete Sobolev inequality ∥∇hwh∥L4(Ω) ≲ ∥wh∥H2 +h(Ω) valid for all wh ∈ [Vk +h]3, see [14, +(6.9)]. We also observe that bh(wh;vh,µh) = 0 for all µh ∈ Λh implies vh ∈ Fh(wh) +according to (72). Therefore, in each step of the gradient flow augmented with the +linearized metric constraint, we seek (δyn+1 +h +,λn+1 +h +) ∈ Vk +h(0,0) × Λh such that +τ −1(δyn+1 +h +,vh)H2 +h(Ω)+ah(δyn+1 +h +,vh)+bh(yn +h;vh,λn+1 +h +) ++ bh(yn +h;δyn+1 +h +,µh)=ℓ[yn +h](vh)−ah(yn +h,vh), +(92) +for all (vh,µh) ∈ Vk +h(0,0) × Λh. +The proposed strategy is summarized in Algorithm 1. +Algorithm 1: (discrete-H2 gradient flow with Lagrange multipliers) +Given a pseudo-time step τ > 0 and a target tolerance tol; +Choose an initial guess y0 +h ∈ Ah,0; +while τ −1∣Eh[yn+1 +h +] − Eh[yn +h]∣ >tol do +Solve (92) for (δyn+1 +h +,λn+1 +h +) ∈ Vk +h(0,0) × Λh; +Update yn+1 +h += yn +h + δyn+1 +h +; +end +It is worth pointing out that utilizing Lagrange multipliers is ubiquitous to enforce +linearized metric constraints [4, 7, 8, 9, 13, 14, 17, 16]. In particular, the system (92) +is solved using the Schur complement approach, whose performance depends on the +inf-sup stability of bh; see e.g. [38, 11] and refer to Section 6.1 for additional details +on the practical implementation. Unfortunately, there are no results available in the +literature guaranteeing a uniform inf-sup not even for the continuous problem. In +this section, we make a first step towards a better understanding of the situation +in that we derive a sub-optimal estimate of the inf-sup constant. We start with a +linear algebra lemma. +Lemma 16 (solvability of a matrix equation). Given a 2 × 2 symmetric matrix C +and a full-rank 3 ×2 matrix B, there exists a 3 ×2 matrix A that solves the equation +(93) +(AT B + BT A) ∶ C = ∣C∣2 +and satisfies ∣A∣ ≤ +∣C∣ +2σ2(B), where ∣ ⋅ ∣ denotes the Frobenius norm of matrices and +σ2(B) > 0 is the smallest singular value of B. +Proof. Using the cyclic properties of the trace operator yields +AT B ∶ C = tr(BT AC) = tr(CAT B) = tr(AT BC) = BT A ∶ C = A ∶ BC, +whence (93) is equivalent to +A ∶ BC = 1 +2∣C∣2. +Let B = UΣV T be the singular value decomposition of B, where U ∈ R3×3 and +V = R2×2 are orthogonal matrices, and Σ = [σ1(B),0;0,σ2(B);0,0] ∈ R3×2 carries + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +31 +the singular values σ1(B) ≥ σ2(B) ≥ 0 of B. Since B is full-rank, we deduce that +σ2(B) > 0 and +∣BC∣2 = ∣UΣV T C∣2 = ∣ΣC∣2 ≥ σ2(B)2∣C∣2 = σ2(B)2∣C∣2, +where C = V T C and thus ∣C∣ = ∣C∣. We can now assume that C ≠ 0, for otherwise +A = 0 solves (93). We then realize that ∣BC∣ > 0 and +A = (BC)∣C∣2 +2∣BC∣2 +is clearly a solution to (93) as well as +∣A∣ = ∣C∣2 +2∣BC∣ ≤ +∣C∣ +2σ2(B), +which is the desired estimate ∣A∣ ≲ ∣C∣. +□ +The following sub-optimal estimate of the discrete inf-sup constant is a conse- +quence of the previous lemma. Since only the gradient of vh ∈ Vk +h(0,0) appears in +(90), but the underlying norm of Vk +h(0,0) is the discrete H2-norm, it seems natural +to consider a negative Sobolev norm of order −1 for the space of Lagrange multipli- +ers Λh. However, the fact that ∇hvh is discontinuous makes it problematic to pair +it with a distribution in a negative Sobolev space of order −1. This leads to the +embedding of Λh into [L2(Ω)]2×2, which is somehow responsible for suboptimality. +Theorem 17 (discrete inf-sup constant). For any n ≥ 0 and yn +h ∈ Ah,δ, there exists +a constant β independent of n and h such that βh = βhmin > 0 satisfies +(94) +inf +µh∈Λh +sup +vh∈Vk +h(0,0) +bh(yn +h;vh,µh) +∥vh∥H2 +h(Ω)∥µh∥L2(Ω) +≥ βh. +Proof. We proceed in two steps: we first construct a suitable vh and next show (94). +Step 1: Construction of vh. Given µh ∈ Λh, let µh,T = µh∣T be the constant symmet- +ric 2 × 2 restriction of µh to any element T ∈ Th. Thanks to Lemma 16 (solvability +of a matrix equation), there exists a 3 × 2 constant matrix AT such that +L[AT ;yn +h](xT ) ∶ µh,T = (AT +T ∇yn +h(xT ) + ∇yn +h(xT )T AT ) ∶ µh,T = ∣µh,T ∣2, +and ∣AT ∣ ≤ +∣µh,T ∣ +2σmin(∇yn +h(xT )). Let λmin ∶ M2×2 → R be the smallest eigenvalue function +defined over the space of symmetric matrices M2×2 into R, which turns out to be +continuous with respect to any norm. In particular, because yn +h ∈ Ah,δ we have +Dh[yn +h](xT ) = ∣I[yn +h](xT ) − I2∣ ≤ δ. +and there is a constant c independent of h and n so that ∣λmin(I[yn +h](xT )−I2)∣ ≤ cδ, +or +λmin(I[∇yn +h](xT )) ≥ 1 − cδ, +Consequently, for δ sufficiently small we deduce +σmin((∇yn +h(xT )) = (λmin(I[yn +h](xT ))) +1 +2 ≥ (1 − cδ) +1 +2 , + +32 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +is bounded away from 0 and we have ∣AT ∣ ≤ +∣µh,T ∣ +2(1−cδ) +1 +2 . We finally define vh(x)∣T ∶= +AT (x − xT ) on each T ∈ Th, where xT is the barycenter of T, and observe that +vh ∈ [Vk +h]3 for k ≥ 2 and ∇vh∣T = AT . +Step 2: Discrete inf-sup property. We first compute +bh(yn +h;vh,µh) = ∑ +T∈Th +∣T∣L[vh;yn +h](xT ) ∶ µh,T = ∑ +T∈Th +∣µh,T ∣2∣T∣ = ∥µh∥2 +L2(Ω). +we don’tSince D2 +hvh = 0 for vh piecewise linear, combining a trace inequality with +the Poincar´e inequality on each element T gives +∥vh∥2 +H2 +h(Ω) = ∑ +e∈Ea +h +∥h−3/2[vh]∥2 +L2(e) + ∥h−1/2[∇vh]∥2 +L2(e) +≲ ∑ +T∈Th +h−4∥vh∥2 +L2(T) + h−2∥∇vh∥2 +L2(T) ≲ ∑ +T∈Th +h−2∥∇vh∥2 +L2(T) +due to the the fact that vh ∈ Vk +h(0,0) has vanishing mean value on T. Therefore, +(95) +∥vh∥2 +H2 +h(Ω) ≲ ∑ +T∈Th +h−2 ∫T ∣AT ∣2 ≲ h−2 +min(1 − cδ)−1∥µh∥2 +L2(Ω). +In summary, we have shown that for every µh ∈ Λh, there exists vh ∈ Vk +h(0,0) such +that bh(yn +h;vh,µh) = ∥µh∥2 +L2(Ω) and ∥vh∥H2 +h(Ω) ≲ h−1 +min∥µh∥L2(Ω). This is the desired +inf-sup condition in disguised. +□ +6. Numerical experiments +In this section we present several numerical experiments, some motivated by com- +putations [8, 7, 9, 16] and other by lab experiments [1, 30, 36, 32, 35, 37, 39, 42]. +We carry out simulations with several spontaneous curvature matrices Z and both +Dirichlet and free boundary conditions, so as to capture a variety of insightful config- +urations exhibiting large bending deformations. We consider the effect of different +aspect ratios of rectangular domains. We also explore properties of a novel model +inspired by [6], which allows folding across curved creases (bilayer origami). Our +numerical simulations illustrate the computational performance of our algorithm. +6.1. Implementation. We start with a few comments on the implementation of +the gradient flow (92) and Algorithm 1. +Saddle-point structure. We resort to a Schur complement method to solve the dis- +crete problem (92). We refer to [13] for full implementation details of a similar linear +algebra structure, but emphasize here how Theorem 17 (inf-sup stability) guarantees +solvability and affects the solver efficiency in the spirit of [11, Lemma 3.1]. +To make explicit the Schur complement matrix and deduce its condition number, +we denote by {ϕi}N +i=1 a basis for Vk +h(0,0) and by {ψi}M +i=1 an orthonormal basis for +Λh. The matrix representations of the bilinear forms Ah(⋅,⋅) ∶= τ −1(⋅,⋅)H2 +h(Ω)+ah(⋅,⋅) +and bh(yn +h;⋅,⋅) used to define the gradient flow (92) are thus given by +A ∶= (Ah(ϕj,ϕi)) +N +i,j=1, +Bn ∶= (bh(yn +h;ϕj,ψi)) +M,N +i=1,j=1. + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +33 +With this notation, the Schur complement matrix reads Sn ∶= BnA−1BT +n and satisfies +(Snm,m) = (A−1/2BT +nm,A−1/2BT +nm) = sup +w∈RN ((w,A−1/2BT +nm) +∥w∥2 +) +2 += sup +v∈RN ((v,BT +nm) +∥A1/2v∥2 +) +2 += +sup +vh∈Vk +h(0,0) +bh(yn +h;vh,µh)2 +Ah(vh,vh) +, +where µh ∶= ∑M +i=1 miψi ∈ Λh,vh ∶= ∑N +j=1 vjϕj ∈ Vk +h(0,0) with m = (mi)M +i=1,v = (vj)N +j=1, +and ∥ ⋅ ∥2 = (⋅,⋅)1/2 is the Euclidean norm in RN. On one hand, the continuity (91) +of bh and the coercivity estimate (46) for Bh[vh] = 1 +2ah(vh,vh) yield +(Snm,m) ≲ τ∥yn +h∥2 +H2 +h(Ω)∥µh∥2 +L2(Ω) ≲ τ∥µh∥2 +L2(Ω) = τ∥m∥2 +2, +because ∥yn +h∥H2 +h(Ω) ≲ 1 in view of the energy stability (77) satisfied by yn +h and the +coercivity of total energy (47). On the other hand, the inf-sup stability (94) and +the continuity estimate (46) for Bh[vh] = 1 +2ah(vh,vh) imply +τh2 +min∥m∥2 +2 = τh2 +min∥µh∥2 +L2(Ω) ≲ (Snm,m). +Combining these two inequalities yields an estimate for the condition number of Sn +(96) +κ(Sn) ∶= max +m∈RM +(Snm,m) +∥m∥2 +2 +( min +m∈RM +(Snm,m) +∥m∥2 +2 +) +−1 +≲ h−2 +min. +Estimate (96) guarantees that the saddle-point system is invertible but ill-conditioned. +We use a conjugate gradient (CG) iterative solver for the numerical experiments be- +low. Classical convergence theory for CG asserts that the number of iterations to +achieve a desired accuracy is of order +√ +κ(Sn) [27, Theorem 3.1.1]. Our numerical +experiments reveal that the number of iterations needed in the CG solver roughly +behaves like h−1 +min, which is consistent with (96). +We emphasize that solving the linear system (92) by the Schur complement +method for several steps of the gradient flow remains the bottle neck in terms of +computing time. We leave the design of suitable preconditioners open. +Assembly. Since the scalar product ⟨⋅,⋅⟩H2 +h(Ω) and bilinear form ah do not change in +the course of the gradient flow, we assemble them once for all before the main loop. +In contrast, we assemble the bilinear form bh(yn +h;⋅,⋅) and right hand side ℓ[yn +h](.) +at each step of the loop as they depend on the previous iterate yn +h. Computing +the discrete Hessian Hh[yh] is the most expensive part in the assembly process, as +it requires solving the linear systems (25) and (26) for lifting operators. In order +to save computing time, we find the discrete Hessian of each basis function at the +beginning of the simulation and store its values for later use; this pre-processing +drastically decreases the assembly time. +Software and data. We implement our LDG method within the software platform +deal.ii [3] and visualize the outcome with paraview [2]. For all the simulations, we + +34 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +fix the polynomial degree k of the deformation yh and the two liftings l1,l2 of the +discrete Hessian Hh[yh], as well as the stabilization parameters γ1,γ2 to be +k = l1 = l2 = 2, +γ0 = γ1 = 1. +Recall that LDG is stable for any positive choice of parameters γ1,γ2, which con- +trasts with IPDG that requires γ1,γ2 large for stability purposes [17, 16]. +In the following numerical simulations, we consider both clamped Dirichlet (ΓD ≠ +∅) and free boundary conditions (ΓD = ∅). For the latter, the discrete equation (73) +is no longer well-defined. To fix the system kernel, we add an L2-term to the metric +(⋅,⋅)H2 +h(Ω), while all other implementation aspects are similar to the case ΓD ≠ ∅. +We refer to [13] for implementation details of free boundary conditions. +In either situation, a natural choice of initial deformation is that of a flat plate +y0 +h(x1,x2) = (x1,x2,0) +∀(x1,x2) ∈ Ω, +and satisfies clamped boundary conditions and the isometry constraint I[y0 +h] = I2 +everywhere. This is much simpler than prestrained plates [13, 14], which require +preprocessing of both boundary condition and metric constraint to construct suitable +initial deformations for LDG to start. +6.2. Clamped plate: Isotropic curvature. We consider a rectangular plate Ω = +(−5,5) × (−2,2), clamped on the side {−5} × [−2,2], with isotropic spontaneous +curvature Z = I2. The deformation with minimal energy corresponds to a cylinder +of radius 1 and energy 20 [40]. This is confirmed by our simulations in Fig.2, which +displays iterations of the discrete gradient flow with number of elements is 1024 +(30720 dofs), τ = 5 × 10−3 and tol = 10−4 in Algorithm 1, as in [8]. +We notice that surface self-intersecting develop during the relaxation dynamics +of Algorithm 1; this is similar to [9] but different from [16]. Moreover, it takes fewer +iterations for Algorithm 1 to reach the cylindrical equilibrium configuration, with +the same or even smaller time step τ, than FEMs in [8, 9, 16]. +Moreover, we keep the time step τ = 5×10−3 fixed and consider two quasi-uniform +meshes with 256 and 1024 elements. We obtain bending energies Eh = 16.8627 and +Eh = 17.8038 (16% and 11% relative error) respectively, which exhibit smaller errors +than the corresponding ones Eh = 15.961 and Eh = 16.544 with the Kirchhoff FEM +of [8] for the same mesh-size and time step. In addition, the energy error compares +favorably with the new Kirchhoff FEM in [9], which computes with Z = 2.5I2 and +produces a 36% relative error even with a finer mesh of 5120 triangular elements. +6.3. Free plate: Anisotropic curvature. We now explore a cigar-type configu- +ration motivated by experiments [30] and computations [8, 16]. The plate is again +the rectangle Ω = (−5,5) × (−2,2), but now we impose no boundary condition (free +boundary) along with the anisotropic spontaneous curvature +(97) +Z = [ 3 +−2 +−2 +3 ]. +We observe that the eigenpairs of Z are (1,[1,1]T ) and (5,[1,−1]T ). +We thus +expect that the plate deforms at −45 degrees with respect to the Cartesian axes in + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +35 +Figure 2. Isotropic curvature: Relaxation dynamics of Algorithm 1 to- +wards the cylinder equilibrium shape of a clamped rectangular plate with +the isotropic spontaneous curvature Z = I. The bilayer plate is depicted +at times 0,50,1000,9000,18000,36050,48100,56050,72100 of the gradient +flow (counter-clockwise). +a symmetric way and eventually reaches a cigar-like configuration, as in [16]. We +confirm this in Figure 3, that displays computations with 1024 elements (30720 dofs) +and τ = 5 × 10−3. The final energy is Eh = 46.3898. Remarkably, Algorithm 1 takes +fewer iterations to reach the equilibrium configuration than [16]. +Figure 3. Anisotropic curvature: Relaxation dynamics of Algorithm 1 +towards the cigar-type equilibrium of a free rectangular plate with the +anisotropic spontaneous curvature of (97). The bilayer plate is depicted +at times 0,50,200,1000,10000,30000 of the gradient (counter-clockwise). + +36 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +Figure 4. Anisotropic indefinite curvature: Relaxation dynamics of Al- +gorithm 1 with spontaneous curvature (98) towards a DNA-like equilibrium +configuration of a free rectangular strip with large aspect ratio. The bilayer +plate is depicted at times 0,100,200,1000,4000,12600 of the gradient flow +(left to right). +6.4. Free plate: +Helix shape. We present a helix-type shape motivated by a +DNA-like configuration [42]. We consider a high aspect ratio plate Ω = (−8,8) × +(−0.5,0.5), with free boundary condition and anisotropic spontaneous curvature +(98) +Z = [ 1 +−3/2 +−3/2 +1 ]. +We point out that the eigenpairs of Z are (−1 +2,[1,1]T ) and (5 +2,[1,−1]T ), which +again correspond to principal directions that form an angle of 45 degrees with the +coordinate axes. This, together with eigenvalues of opposite sign and high aspect +ratio, leads to a deformation that resembles the twisting of DNA molecules, as in +[16]. We display several snapshots of the relaxation dynamics of Algorithm 1 in +Figure 4. The simulation is carried out with 1024 elements and τ = 10−2, and yields +a final energy Eh = 3.2507. Moreover, it again takes fewer iterations for LDG to +reach the equilibrium configuration than the DG method of [16]. +6.5. Climate responsive architectures. Bilayer devices can be used to control +the temperature or moisture inside a room. +The HygroSkin project [36, 37, 39, +32, 35] exploits this technology by designing visually appealing humidity responsive +apertures to a pavilion. Heat and moisture are thus dynamically controlled without +any high-tech equipment owing to the dominant orientation of fibers in plywood. +To simulate this device with our bilayer model, we consider an equilateral triangle +with side length 1 and vertices (0,0), (1,0) and (1 +2, +√ +3 +2 ). The actual climate respon- +sive device consists of 6 of these triangular shapes suitably rotated and arranged + +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +37 +together as to form a flat regular hexagon, with the exterior edge of each triangle +clamped; we refer to Figure 5. To mimic the effect of different relative humidity +values, we choose several anisotropic spontaneous curvatures +(99) +Z = (0 +0 +0 +α), +with α = 0,1,2,3,4,5. +for the triangle with exterior edge parallel to the x-axis and suitably rotated for the +other triangles. This matrix favors bending along the y-axis exclusively. +Figure 5. Climate responsive device. The undeformed plate is made of +6 equilateral triangles that together form a regular hexagon (right). Finite +element partition of each triangle into trapezoids (left). +Upon actuation, the climate device automatically opens as depicted in Figure 6. +The matching of the computed (left) and actual (right) equilibrium shapes in Fig- +ure 6 is quite remarkable for a model with just one parameter α within Z. We run +this simulation with time step τ = 1 and stopping tolerance tol = 10−4. +6.6. Folding Model: Bilayer Origami. We finally explore computationally the +combined effect of spontaneous curvature, as driving mechanism, and folding across a +preassigned crease. The corresponding bilayer model and LDG method are discussed +in Section 4. We consider below the setting from [6, Section 5.2] and refer to [12] +for additional numerical simulations. +The computational domain is a rectangle Ω = (0,9.6) × (0,15) and the folding +crease is a quadratic curve C passing through the points (0,2), (9.6,2), and (4.8,6), +which can be exactly represented by the isoparametric mesh Th; see Fig. 7. +In +order to generate a configuration similar to the flapping mechanism in [6], which is +obtained by compression of the lateral boundary, we set the spontaneous curvatures +Z = (0 +0 +0 +1), +Z = (0 +0 +0 +−1 +2 +), +below the folding arc and above of it, respectively, and do not impose any boundary +condition. The resulting equilibrium shape is displayed in Fig. 7. We point out the +crucial role played by the sign of principal curvatures λ = 1,−1 +2 corresponding to +the same coordinate eigendirection: bending of the lower and upper plates occurs in +opposite directions which gives rise to folding across the crease and yields a rather +large compatible deformation. + +38 +LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +Figure 6. Climate responsive device. +(Left) Approximate deforma- +tion for different values of spontaneous curvature (99) with parameter +α = 0,1,2,3,4,5. (from left to right and top to bottom) (Right) Exper- +imental deformations of a device made of plywood. Picture taken from [35] +(courtesy of Prof. Achim Menges); see also [36]. The matching is remark- +able. +Figure 7. Bilayer origami: Flapping mechanism generated by folding of +a bilayer plate across a crease. (Left) Conforming subdivision of Ω with +quadratic crease. (Right) Different perspectives of the resulting very large +deformation. +7. Conclusions +In this article, we present a new LDG method for large bending isometric defor- +mations of bilayer plates. We summarize our contributions in this section. +1. LDG discretization. It consists of replacing the Hessian D2y by a reconstructed +Hessian Hh[yh] in the bending energy Bh[yh], and by a reduced (piecewise constant) + +30%RH +36%RH +43% RH +49%·RH +BS%RH +62%RH +69%·RH +Z5%·RHLDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES +39 +discrete Hessian Hh[yh] in the cubic energy Ch[yh], which encodes the interaction +with spontaneous curvature. We use the mid-point quadrature to integrate Ch[yh]. +2. Linearized isometry constraint. This allows for a slight violation of the isometry +constraint I[yh] = I2 while providing control of the ℓ∞-norm of the isometry defect +∣I[yh] − I2∣ at element barycenters. This turns out to be a significant improvement +over previous DG methods that enforce such defect as sum of averages over elements +[13, 14, 17, 16]. +3. Γ-convergence. The key novelty of the Γ-convergence of discrete energies is the +construction of the recovery sequence of any admissible deformation y ∈ [H2(Ω) ∩ +W 1 +∞(Ω)]3. It hinges on a quadratic Taylor expansion at element barycenters of a +suitable regularization of y, and exploits that both the reduced quadrature of Ch[yh] +and the isometry defect are imposed at element barycenters. +4. Bilayer model with foldings. We extend the LDG method to deal with a piece- +wise quadratic crease C and prove its convergence. The construction of a recovery +sequence for one absolute minimizer y∗ ∈ [H2(Ω/C)∩W 1 +∞(Ω)]3 requires the slightly +stronger assumption that y∗ is C1 in each subdomain created by C. +5. Fully linear solver. We design a semi-implicit discrete gradient flow that treats +Bh[yh] implicitly and Ch[yh] explicitly. 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ACS nano 6, 5 (2012), 3925–3934. + diff --git a/NNE1T4oBgHgl3EQfZgSA/content/tmp_files/load_file.txt b/NNE1T4oBgHgl3EQfZgSA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5901860980ee613d2f80fa6b430324b2f21bf16 --- /dev/null +++ b/NNE1T4oBgHgl3EQfZgSA/content/tmp_files/load_file.txt @@ -0,0 +1,1066 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf,len=1065 +page_content='GAMMA-CONVERGENT LDG METHOD FOR LARGE BENDING DEFORMATIONS OF BILAYER PLATES ANDREA BONITO, RICARDO H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' NOCHETTO, AND SHUO YANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bilayer plates are slender structures made of two thin layers of dif- ferent materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' They react to environmental stimuli and undergo large bending deformations with relatively small actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The reduced model is a constrained minimization problem for the second fundamental form, with a given spontaneous curvature that encodes material properties, subject to an isometry constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We design a local discontinuous Galerkin (LDG) method which imposes a linearized discrete isometry constraint and controls deformation gradients at barycenters of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We prove Γ-convergence of LDG, design a fully practical gradient flow, which gives rise to a linear scheme at every step, and show energy stability and control of the isometry defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We extend the Γ-convergence analysis to piecewise quadratic creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We also illustrate the performance of the LDG method with several insightful simulations of large deformations, one including a curved crease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Introduction Bilayer plates are slender structures made of two thin layers of different materials glued together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' These layers react differently to non-mechanical stimuli, such as thermal, electrical, and chemical actuation [30, 43, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bilayer plates can undergo large bending deformations using a small amount of energy, which makes them appealing at small and large scales alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Amongst the many and broad applications of bilayer materials in engineering and biomedical science, we list drug delivery vesicles [28, 44], cell encapsulation devices [45], sensors [34] and self-deployable sun sails [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We model bilayer plates as thin 3d hyper-elastic bodies as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Exploiting their relatively small thickness, two dimensional plate models for the mid-plane deformation y(Ω), Ω ⊂ R2, are derived and analyzed in [40, 41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' we also refer to [8] for a formal dimension reduction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The plates equilibria (Andrea Bonito) Department of Mathematics, Texas A&M University, College Sta- tion, TX 77845, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' AB was partially supported by the NSF Grants DMS 2110811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (Ricardo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Nochetto) Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (Shuo Yang) Yanqi Lake Beijing Institute of Mathematical Sciences and Applica- tions, Beijing 101408, China, and Yau Mathematical Sciences Center, Tsinghua Uni- versity, Beijing 100084, China E-mail addresses: bonito@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='edu, rhn@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='edu, shuoyang@bimsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='03151v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='NA] 9 Jan 2023 2 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES are characterized as solutions to a nonlinear minimization problem with a noncon- vex constraint expressing the plates ability to bend without stretching or shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, distances within the midplane remain unchanged thereby resulting in isometric deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bilayer plates: Ω × (−s/2,s/2), Ω ⊂ R2 is the mid-plane (bounded Lipschitz domain) and s is the thickness parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The sets Ω×(−s/2,0) and Ω×(0,s/2) represent the two undeformed layers of different materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The plate deformation y ∶ Ω → R3 must belong to the following admissible set A, which prevents shearing and stretching within the surface y(Ω) and imposes possible boundary conditions: (1) A ∶= {y ∈ [H2(Ω)]3 ∶ I[y] = I2 in Ω, y = ϕ, ∇y = Φ on ΓD}, where I2 is the 2 × 2 identity matrix and (2) I[y] ∶= ∇yT ∇y is the first fundamental form of y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We assume that ΓD ⊂ ∂Ω is nonempty and open and ϕ ∈ [H2(Ω)]3 and Φ ∈ [H1(Ω)]3×2 are given and are compatible with the isometry constraint, namely Φ = ∇ϕ and ΦT Φ = I2 on ΓD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' thus A is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, condition (2) entails that {∂iy}2 i=1 is an orthonormal basis of the tangent plane to y(Ω) and its unit normal ν can be written as (3) ν ∶= ∂1y × ∂2y ∣∂1y × ∂2y∣ = ∂1y × ∂2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Although we will present simulations in Section 6 for both Dirichlet boundary conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' ΓD ≠ ∅) and free boundary conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' ΓD = ∅), we focus our presentation on the former for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We emphasize that the analysis of the latter follows from that in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The modifications are in the spirit of [14], where we analyze the LDG method for prestrained plates with free boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Consequently, we do not include details to avoid repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Equilibrium configurations of bilayer plates are solutions y ∈ A of the following constrained minimization problem (4) min y∈A E [y] ∶= min y∈A 1 2 ∫Ω ∣II[y] − Z∣ 2, where II[y] is the second fundamental form of y(Ω) (5) II[y] ∶= (∂ijy ⋅ ν) 2 ij=1 = (∂ijy ⋅ (∂1y × ∂2y)) 2 ij=1, apQ y S E QLDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 3 and Z ∈ [L∞(Ω)]2×2 is a spontaneous curvature which encodes the material prop- erties of the bilayer plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In fact, Z forces the plate y(Ω) to bend so that II[y] gets as close as possible to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' If the material is homogenous and isotropic, then the spontaneous curvature is diagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Z = αI2 with a constant α depending on the materials parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In particular, when the two layers are identical, Z = 0 and the model reduces to a single layer plate [4, 17], which coincides with the classical (nonlinear) Kirchhoff plate theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Thanks to the isometry constraint I[y] = I2, the energy functional E [y] can be further simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Recall that for isometries, there holds [5] (6) ∣II[y]∣ 2 = ∣D2y∣ 2 = ∣∆y∣ 2 = (tr II[y]) 2, whence expanding the square in (4) and using (5) and (6) yields (7) E [y] = 1 2 ∫Ω ∣D2y∣ 2 − 2 ∑ i,j=1∫Ω ∂ijy ⋅ (∂1y × ∂2y)Zij + 1 2 ∫Ω ∣Z∣ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Furthermore, since 1 2 ∫Ω ∣Z∣ 2 does not depend on y, minimizing the energy in (7) over A is equivalent to minimizing the reduced energy (8) E [y] ∶= 1 2 ∫Ω ∣D2y∣ 2 − 2 ∑ i,j=1∫Ω ∂ijy ⋅ (∂1y × ∂2y)Zij, over A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' we keep the same notation for the energies in (7) and (8) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The effect of the layers mismatch appears in the cubic term leading to a nonlinear Euler-Lagrange equation for the equilibrium deformation y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For latter use, we also introduce a notation for the single layer bending energy (9) B[y] ∶= 1 2 ∫Ω ∣D2y∣ 2 and the cubic term (10) C[y] ∶= 2 ∑ i,j=1∫Ω ∂ijy ⋅ (∂1y × ∂2y)Zij, so that E[y] = B[y] − C[y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We emphasize that the cubic term C satisfies ∣C[y]∣ ≤ ∥y∥H2(Ω)∥∇y∥L2(Ω)∥∇y∥L∞(Ω)∥Z∥L∞(Ω) and I[y] = I2 implies ∥∇y∥L∞(Ω) ≲ ∥I[y]∥L∞(Ω) ≲ 1, whence ∣C[y]∣ ≲ ∥y∥2 H2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since the discrete deformation yh is piecewise polynomial, our numerical method cannot guarantee that yh satisfies the isometry constraint I[yh] = I2 everywhere in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We choose to enforce a slight violation of this constraint solely at the barycenter of elements, and still retain control of the ℓ∞-norm of ∇yh at barycenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This is a chief ingredient of our LDG method and is inspired by Bartels and Palus [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 4 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Previous numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' There are several finite element methods available for the numerical simulation of bilayers plates [8, 7, 9, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In all of them, the isometry constraint I[y] = I2 is linearized at y (11) L[v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='y] ∶= ∇vT ∇y + ∇yT ∇v = 0, and tangential variations v are evolved within a gradient flow that decreases the energy E[y] and is favored for its robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The gradients of Kirchhoff finite elements are uniquely defined at the mesh ver- tices, which is where (11) is imposed in [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The discrete gradient flow in [8] treats the cubic energy C[y] implicitly to get an energy decreasing scheme but requires the normalization (3) of the discrete normal, which renders the algorithm nonlin- ear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Discrete energies are shown to Γ-converge in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In contrast, the scheme of [7] is linear and much more efficient, but stability and Γ-convergence are still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Recently, Bartels and Palus [9] reformulated the discretization of C[y] making it fully explicit and the ensuing algorithm linear, and were also able to show an energy decreasing property for the explicit gradient flow with a mild time-step constraint and Γ-convergence of the discrete energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' On the other hand, interior penalty discontinuous Galerkin (IPDG) finite element methods are proposed and studied in [16] because they require a lower polynomial degree (2 instead of 3), are easier to find in existing software platforms, are more flexible in imposing boundary conditions as well as the linearized isometry constraint (11), and are amenable to subdivisions containing curved boundaries which is cru- cial to deal with creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The linearized constraint (11) is enforced in average on all elements of the subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Furthermore, the cubic energy C[y] is treated explic- itly at each step of the discrete gradient flow and the ensuing algorithm is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' However, Γ-convergence and energy decreasing properties remain open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We note that the bilayer model (7) reduces to single layer plates endowed with the bending energy B[y] for y ∈ A provided the upper and lower layers are identical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We refer to [4, 17] for the design and analysis of Kirchhoff and IPDG methods in this simpler context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG-discretization and our contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We propose a local discontin- uous Galerkin (LDG) method for the approximation of the minimization problem (4) along the lines of [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG method was originally introduced in [23], and further explored in [10, 20, 21, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Our discrete energy Eh[yh] is obtained (up to stabilization terms) by simply replacing the Hessian D2y in (8) by a discrete Hessian Hh[yh], which is constructed and analyzed in [13, 14] in terms of the dis- continuous Galerkin solution yh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This is conceptually simpler than IPDG methods, which are based on integration by parts and are harder to design for intricate nonlin- ear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In contrast to IPDG, LDG is also stable for any positive stabilization parameters, and exhibits better convergence properties at the expense of a slightly worse sparsity pattern [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 5 Our treatment of the cubic term hinges on the mid-point quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' If Th is a mesh made of shape-regular triangles or quadrilaterals T with barycenter xT , let (12) Ch[yh] ∶= 2 ∑ i,j=1 ∑ T∈Th ∣T∣(Hh[yh]ij ⋅ (∂1yh × ∂2yh)Zij)(xT ) where Hh[yh] = 1 ∣T∣ ∫T Hh[yh] for all T ∈ Th is the piecewise constant reduced discrete Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, we also control the isometry defect at barycenters, namely given a parameter δ > 0 we impose (13) Dh[yh](xT ) ∶= ∣[∇yT h ∇yh − I2](xT )∣ ≤ δ ∀T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We enforce the Dirichlet condition upon augmenting the discrete energy Eh[yh] via a Nitsche method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, we say that discrete functions satisfying (13) belong to the discrete admissible set Ah, the discrete counterpart of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We prove that Ah is non-empty, and derive convergence of global minimizers yh of Eh within Ah towards global minimizers y of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Solving the nonconvex discrete minimization counterpart of (4) is a highly non- trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We resort to a discrete gradient flow that enforces the linearized isom- etry constraint (11) at the barycenters (14) L[vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='yh](xT ) ∶= [∇vT h ∇yh + ∇yT h ∇vh](xT ) = 0 ∀T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' and solve a discrete minimization problem for a tangential variation vh of yh, in the sense (14), with the cubic term (12) treated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This clever idea, due to Bartels and Palus [9], renders the problem linear at each step of the gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We show that this procedure is energy decreasing, convergent, and preserves the isometry defect (13) provided δ is proportional to h, which entails a linear relation between the time step τ of the gradient flow and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, we derive a (suboptimal) discrete inf-sup condition for the Lagrange multiplier approach to the linear constraint (14), which seems to be the first such result for this type of matrix constraint and is consistent with computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Section 2 is about LDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We introduce the (broken) finite element spaces in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We examine the discrete Hessian operator and its reduced counterpart in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='3, together with their boundedness and convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In Subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='5, we define the discrete problem and investigate consistency of the cubic discrete energy Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The proof of Γ-convergence of the discrete energy to the exact one is the content of Section 3, and its extension to a bilayer model with piecewise quadratic creases is included in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In Section 5, we introduce the gradient flow scheme used to solve the discrete problem, prove its conditional stability and show how the constraint violation (13) is controlled throughout the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, we derive a suboptimal inf-sup condition for (14) at each step of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We present several insightful simulations in Section 6 to illustrate the performance of LDG, including folding across a curved crease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 6 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG Discretization 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' From now on, we assume that Ω ⊂ R2 is a polygonal domain and denote by {Th}h>0 a shape-regular sequence of conforming partitions of Ω made of either triangles or quadrilaterals T with diameter hT ∶= diam(T) ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The set of edges Eh ∶= E0 h ∪Eb h is decomposed into the interior edges E0 h and boundary edges Eb h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For e ∈ Eh, we define he ∶= diam(e) and note that he ≤ h, and thus (15) h−1 ≤ h−1 e ∀e ∈ Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We assume a compatible representation of the Dirichlet boundary ΓD = ∪{e ∶ e ∈ ED h }, and let Ea h ∶= E0 h ∪ ED h be the set of active edges on which jumps and averages will be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The union of these edges gives rise to the corresponding skeletons of Th (16) Γ0 h ∶= ∪{e ∶ e ∈ E0 h}, ΓD h ∶= ΓD, Γa h ∶= Γ0 h ∪ ΓD h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We use the notation (⋅,⋅)L2(Ω) and (⋅,⋅)L2(Γa h) to denote the L2 inner products over Ω and Γa h, and a similar notation for subsets of Ω and Γa h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We denote by h a mesh density function, locally equivalent to hT and he, and utilize it as a weight in the preceding norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We often write f ≲ g to indicate that there exists a constant C independent of discretization parameters such that f ≤ Cg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Finally, we make the simplifying assumption that the spontaneous curvature Z in (4) is piecewise constant over all partitions Th, h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Broken spaces and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For an integer r ≥ 0, we denote by Pr the space of polynomials of total degree at most r when the subdivision is made of triangles and by Qr the space of polynomials of degree at most r in each variable when quadrilaterals are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We also use the same notation, ̂T, to denote either the unit triangle or the unite square depending on the type of subdivision used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We let FT ∶ ̂T → T ∈ [Q1]2 be the generic map from the reference element to the physical element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It is affine only when the subdivision is made of triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We fix k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The (broken) finite element space Vk h to approximate each compo- nent of the deformation y reads (17) Vk h ∶= {vh ∈ L2(Ω) ∶ vh T ○ FT ∈ Qk ∀T ∈ Th}, when the subdivision is made of quadrilaterals, and we replace Qk by Pk if we have triangular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We define the broken gradient ∇hvh of vh ∈ Vk h to be the elementwise gradient, and use similar notation for other differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For instance D2 hvh = ∇h∇hvh stands for the broken Hessian, and ∂ivh ∶= ∂i,hvh denotes the components i = 1,2 of the broken gradient ∇hvh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We now introduce the jump and average operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For every e ∈ E0 h, fix ne to be one of the two unit normals to e (the choice is arbitrary but does not affect the formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For a boundary edge e ∈ Eb h, we set ne = n, the outward unit normal vector to ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The jump of vh ∈ Vk h and ∇hvh across e ∈ E0 h are given by (18) [vh] e ∶= v− h − v+ h, [∇hvh] e ∶= ∇hv− h − ∇hv+ h, where v± h(x) ∶= lims→0+ vh(x±sne) for x ∈ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The jumps of a vector or matrix valued function is computed componentwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 7 In order to incorporate the Dirichlet boundary conditions y = ϕ, ∇y = Φ on ΓD, we resort to a Nitsche’s approach which does not impose essential restrictions on the discrete space [Vk h]3 but rather modifies the discrete formulation by including boundary jumps defined for vh ∈ [Vk h]3 (19) [vh] e ∶= [vh] e(ϕ) ∶= vh − ϕ, [∇hvh] e ∶= [∇hvh] e(Φ) ∶= ∇hvh − Φ, for all e ∈ ED h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' However, to simplify the notation, it is convenient to introduce the discrete set Vk h(ϕ,Φ) (20) Vk h(ϕ,Φ) ∶= {vh ∈ [Vk h]3 ∶ [vh] e, [∇hvh] e given by (19) for all e ∈ ED h }, which coincide with [Vk h]3 but carries the notion of boundary jump (19) for its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We define the average of vh ∈ Vk h across an edge e ∈ Eh as (21) {vh} e ∶= { 1 2(v+ h + v− h) e ∈ E0 h v− h e ∈ Eb h, and apply (21) componentwise to vector and matrix-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We let ⟨⋅,⋅⟩H2 h(Ω) be the following mesh-dependent form defined, for any vh,wh ∈ Vk h(ϕ,Φ), by (22) ⟨vh,wh⟩H2 h(Ω) ∶= (D2 hvh,D2 hwh)L2(Ω) + (h−1[∇hvh],[∇hwh])L2(Γa h) + (h−3[vh],[wh])L2(Γa h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We emphasize that (22) is not bilinear in Vk h(ϕ,Φ) because of the presence of (ϕ,Φ) in the boundary jump terms, unless ϕ = 0,Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, we set (23) ∥vh∥2 H2 h(Ω) ∶= ⟨vh,vh⟩H2 h(Ω) ∀vh ∈ Vk h(ϕ,Φ), and observe the validity of the following Friedrichs-type inequality [17, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='27)] (24) ∥vh∥L2(Ω) +∥∇hvh∥L2(Ω) ≲ ∥vh∥H2 h(Ω) +∥ϕ∥H1(Ω) +∥Φ∥H1(Ω) ∀vh ∈ Vk h(ϕ,Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Once restricted to Vk h(0,0), the form ⟨⋅,⋅⟩H2 h(Ω) turns out to be a scalar product, according to (24), which corresponds to the discrete counterpart of ⟨⋅,⋅⟩H2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Discrete Hessians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The central ingredient in the proposed LDG approxima- tion is the reconstructed Hessian Hh[yh] ∈ [L2(Ω)] 3×2×2 defined in [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let l1,l2 ≥ 0 be integers and consider two local lifting operators re ∶ [L2(e)]2 → [Vl1 h ]2×2 and be ∶ L2(e) → [Vl2 h ]2×2 defined for e ∈ Ea h by re(φ) ∈ [Vl1 h ]2×2 ∶ ∫ωe re(φ) ∶ τh = ∫e {τh}ne ⋅ φ ∀τh ∈ [Vl1 h ]2×2, (25) be(φ) ∈ [Vl2 h ]2×2 ∶ ∫ωe be(φ) ∶ τh = ∫e {div τh} ⋅ neφ ∀τh ∈ [Vl2 h ]2×2 , (26) where ωe is the union of the two elements of Th sharing e ∈ Γ0 h or the element of Th having e ∈ Γb h as part of its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The definitions extend to [[L2(e)]2] 3 and 8 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES [L2(e)]3 by component-wise application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='The corresponding global lifting operators are then given by Rh ∶= ∑ e∈Ea h re ∶ [L2(Γa h)]2 → [Vl1 h ]2×2, Bh ∶= ∑ e∈Ea h be ∶ L2(Γa h) → [Vl2 h ]2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (27) Their purpose is to lift inter-element information to the cells so that once added to the piecewise Hessian D2 h, they constitute a weakly convergent approximation of the exact Hessian (see Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In fact, we define the discrete Hessian operator Hh ∶ Vk h(ϕ,Φ) → [L2(Ω)] 3×2×2 by (28) Hh[vh] ∶= D2 hvh − Rh([∇hvh]) + Bh([vh]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We point out the implicit dependence on data (ϕ,Φ) and that we will later compute Hh[vh] for vh ∈ Vk h(0,0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' ϕ = 0, Φ = 0, slightly abusing notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Thanks to the relation between the edge and cell diameter (15), we have the following a priori upper bounds for lifting operators (29) ∥Hh[vh]∥L2(Ω) ≲ ∣∣vh∣∣H2 h(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, we have the following properties of the discrete Hessian Hh[vh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 1 (weak convergence of Hh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let k ≥ 2 and vh ∈ Vk h(ϕ,Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' If ∣∣vh∣∣H2 h(Ω) ≲ 1 and vh → v ∈ [H2(Ω)]3 in [L2(Ω)]3 as h → 0, then for any polynomial degree l1,l2 ≥ 0 we have (30) Hh[vh] ⇀ D2v in [L2(Ω)] 3×2×2 as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' See [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='4 and Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ Lemma 2 (strong convergence of Hh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let v ∈ [H2(Ω)]3 be any function such that v = ϕ and ∇v = Φ on ΓD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, let vh ∈ Vk h(ϕ,Φ) satisfy (31) ∥D2vh∥L2(T) ≲ ∥v∥H2(T) ∀T ∈ Th, ∑ T∈Th ∥vh − v∥2 H2(T) → 0 as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Then for any polynomial degree l1,l2 ≥ 0 we have as h → 0+ (32) Hh[vh] → D2v strongly in [L2(Ω)]3×2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This is a minor modification of [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='5 and Appendix B], which assume that vh is the Lagrange interpolant of v ∈ H2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ For later use, we now discuss properties of the reduced discrete Hessian defined the local L2 projection onto the space of piecewise constants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (33) Hh[vh]∣T ∶= 1 ∣T∣ ∫T Hh[vh] ∀T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We start with the stability of Hh[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 9 Lemma 3 (stability of Hh[vh]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For any vh ∈ Vk h(ϕ,Φ), there holds (34) ∥Hh[vh]∥L2(Ω) ≤ cstab∥vh∥H2 h(Ω), where the constant cstab is independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This result is a direct consequence of the stability of the reconstructed Hes- sian (29) and the local L2 projection (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ The reduced discrete Hessian is also weakly converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 4 (weak convergence of Hh[vh]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let vh ∈ Vk h(ϕ,Φ) be a sequence of discrete deformations satisfying ∥vh∥H2 h(Ω) ≲ 1 for all h and such that vh → v in [L2(Ω)]3 for some v ∈ [H2(Ω)]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Then, Hh[vh] converges weakly to D2v in [L2(Ω)]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For any φ ∈ [C∞ 0 (Ω)]3×2×2, we have ∫Ω Hh[vh] ∶ φ = ∑ T∈Th ∫T Hh[vh] ∶ φ = ∑ T∈Th ∫T Hh[vh] ∶ φ + Hh[vh] ∶ (φ − φ), where φ ∶= 1 ∣T∣ ∫T φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 1 (weak convergence of Hh[yh]) implies ∫Ω Hh[vh] ∶ φ → ∫Ω D2v ∶ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' On the other hand, the uniform boundedness (29) and the assumption ∥vh∥H2 h(Ω) ≲ 1 guarantee that ∣ ∑ T∈Th Hh[vh] ∶ (φ − φ)∣ ≲ h∥Hh[vh]∥L2(Ω)∥∇φ∥L2(Ω) ≲ h∥∇φ∥L2(Ω) → 0 as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Combining these two estimates yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Discrete admissible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We introduce the discrete counterpart of the ad- missible set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Given a parameter δ > 0 to be related later to h, we recall the discrete isometry defect Dh[yh] from (13) and define the discrete admissible set Ah,δ as (35) Ah,δ ∶= {yh ∈ Vk h(ϕ,Φ) ∶ ∣Dh[yh](xT )∣ ≤ δ ∀T ∈ Th}, where the polynomial degree is k ≥ 2 and xT is the barycenter of T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The Dirichlet boundary conditions are hidden within the definition (20) of Vk h(ϕ,Φ) and imposed in the weak formulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' hence they do not contribute to any essential restriction in Ah,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The following two lemmas are simple consequences of (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 5 (Ah,δ is non-empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For all h > 0 there exists yh ∈ Vk h(ϕ,Φ) such that Dh[yh](xT ) = 0 for all T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let yh(x) ∶= x for x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We see that yh ∈ [Vk h]3, and therefore yh ∈ Vk h(ϕ,Φ) because the Dirichlet boundary conditions are not imposed essentially in the space Vk h(ϕ,Φ) defined in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, I[yh](xT ) = I2, whence Dh[yh](xT ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ 10 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES Note that this implies Ah,δ is non-empty for any δ > 0, because Ah,0 ⊂ Ah,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We postpone until Theorem 9 the hard question whether Ah,δ is sufficiently rich to approximate A: for any y ∈ A there is yh ∈ Ah,δ close to y in a suitable sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The following lemma provides an estimate on the amount of local stretch and shear associated with functions in Ah,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 6 (pointwise isometry constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' If yh ∈ Ah,δ, then for all T ∈ Th (36) 1 − δ ≤ ∣∂1yh(xT )∣2,∣∂2yh(xT )∣2 ≤ 1 + δ, ∣∂1y(xT ) ⋅ ∂2y(xT )∣ ≤ δ, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' From definition (35), we deduce that for any i,j = 1,2 ∣∂iy(xT ) ⋅ ∂jy(xT ) − δij∣ ≤ δ, where δij is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The assertion thus follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ The pointwise control of isometry defect in (35) is inspired by the algorithms based on Kirchhoff finite elements developed in [8, 9], where this constraint is imposed at the element vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Dealing with element barycenters is novel in the context of DG methods in that previous schemes impose this constraint in average over elements [16, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Having control at barycenters does not imply control of ∇hyh anywhere else, and dictates the use of mid-point quadrature for the discretization of the cubic nonlinear energy Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We discuss this next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Discrete energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The LDG approximation of the energy E[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='] reads (37) Eh[yh] ∶= Bh[yh] − Ch[yh] where Bh[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='] approximates the bending energy (9) and Ch[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='] approximates the cubic interaction energy in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The energy Bh[yh] is defined by (38) Bh[yh] ∶= 1 2 ∫Ω ∣Hh[yh]∣ 2 + Sh[yh], where (39) Sh[yh] ∶= γ1∥h− 1 2 [∇hyh]∥2 L2(Γa h) + γ0∥h− 3 2 [yh]∥2 L2(Γa h) is a stabilization term with parameters γ0,γ1 > 0, whereas Ch[yh] is given by (12) (40) Ch[yh] ∶= 2 ∑ i,j=1 ∑ T∈Th ∣T∣(Hh[yh]ij ⋅ (∂1yh × ∂2yh)Zij)(xT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' With these notations the discrete minimization problem reads (41) min yh∈Ah,δ Eh[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We devote the rest of this section to examine the cubic energy (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Combining Lemma 6 (pointwise isometry constraint) with Lemma 3 (stability of Hh[yh]) yields ∣Ch[yh]∣ ≲ (1 + δ)∥yh∥H2 h(Ω)∥Z∥L2(Ω), whence ∣Ch[yh]∣ is uniformly bounded whenever ∥yh∥H2 h(Ω) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Another crucial as- pect of (40) is the convergence of Ch towards the continuous energy C within the LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 11 basic H2-regularity framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This requires dealing with the reduced Hessian Hh[yh] as we show next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 7 (convergence of cubic energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let Z be piecewise constant over Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let yh ∈ Ah,δ be a sequence of discrete deformations satisfying (42) ∥yh∥H2 h(Ω) ≲ 1 ∀h > 0 and such that yh → y in [L2(Ω)]3, ∇hyh → ∇y in [L2(Ω)]3×2 for y ∈ [H2(Ω) ∩ W 1 ∞(Ω)]3 as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Then (43) lim h→0+ Ch[yh] = C[y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For any ϵ > 0, it suffices to show that (44) limsup h→0+ ∣C[y] − Ch[yh]∣ ≲ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We need a regularization argument to deal with the effect of quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since Ω is Lipschitz we can regularize y, say by convolution, in such a manner that the approximate deformation yϵ ∈ [H3(Ω)]3 satisfies (45) ∥yϵ∥H2(Ω) + ∥yϵ∥W 1 ∞(Ω) ≲ 1, ∥y − yϵ∥H2(Ω) ≲ ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' we recall the convention that constants hidden in ≲ are independent of h and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We point out that this procedure is simpler than the regularization due to Hornung [29] in that yϵ need not be an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We first observe that the energies C[y] and C[yϵ] can be made arbitrarily close because ∣C[y] − C[yϵ]∣ ≲ ∥y − yϵ∥H2(Ω)∥∂1y∥L2(Ω)∥∂2y∥L∞(Ω)∥Z∥L∞(Ω) + ∥yϵ∥H2(Ω)∥y − yϵ∥H1(Ω)(∥∂2y∥L∞(Ω) + ∥∂1yϵ∥L∞(Ω))∥Z∥L∞(Ω) ≲ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We next write Ch[yh] − C[yϵ] = ∑2 i,j=1 ∑T∈Th R1(T) + R2(T) + R3(T), where R1(T) ∶= ∫T (Hh[yh]ij − ∂ijyϵ) ⋅ (∂1yϵ × ∂2yϵ)Zij, R2(T) ∶= ∣T∣[Hh[yh]ij ⋅ (∂1yh × ∂2yh − ∂1yϵ × ∂2yϵ)Zij](xT ), R3(T) ∶= ∣T∣[Hh[yh]ij ⋅ (∂1yϵ × ∂2yϵ)Zij](xT ) − ∫T Hh[yh]ij ⋅ (∂1yϵ × ∂2yϵ)Zij, and disregard the non critical dependence on i,j = 1,2 in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 4 (weak convergence of Hh[yh]) in conjunction with (45) implies that limsup h→0+ 2 ∑ i,j=1 ∑ T∈Th ∣R1(T)∣ ≲ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For R2, we note that ∣(∂1yh × ∂2yh − ∂1yϵ × ∂2yϵ)(xT )∣ ≤ ∣∇(yh − yϵ)(xT )∣(∣∇yh(xT )∣ + ∣∇yϵ(xT )∣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' By Lemma 6 (pointwise isometry constraint), the fact that yh ∈ Ah,δ and (45), we have the uniform bound ∣∇yh(xT )∣+∣∇yϵ(xT )∣ ≲ 1 for all xT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' If Ih∇yϵ indicates the 12 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES standard P1-Lagrange interpolant of ∇yϵ, applying approximating properties of Ih together with an inverse inequality for polynomials, we conclude ∣∇(yh − yϵ)(xT )∣ ≤ ∣(∇yh − Ih∇yϵ)(xT )∣ + ∣(Ih∇yϵ − ∇yϵ)(xT )∣ ≲ h−1 T ∥∇yh − Ih∇yϵ∥L2(T) + hT ∥D3yϵ∥L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We next add and subtract ∇yϵ in the first term of the right-hand side and apply again an interpolation estimate of Ih to derive ∣T∣1/2∣∇(yh − yϵ)(xT )∣ ≲ ∥∇(yh − yϵ)∥L2(T) + h2 T ∥D3yϵ∥L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, since ∣T∣1/2∣Hh[yh]ij(xT )∣ = ∥Hh[yh]ij∥L2(T) because Hh[yh] is piecewise constant, we obtain ∣R2(T)∣ ≲ ∥Hh[yh]ij∥L2(T)(∥∇(yh − y)∥L2(T) + ∥∇(y − yϵ)∥L2(T) + h2 T ∥D3yϵ∥L2(T)), where the hidden constant is proportional to ∥Z∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' After summing over ele- ments, Lemma 3 (stability of Hh[yh]), together with the assumption ∇hyh → ∇y in [L2(Ω)]3×2, (42) and (45), yields limsup h→0+ 2 ∑ i,j=1 ∑ T∈Th ∣R2(T)∣ ≲ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It remains to deal with R3 which entails the effect of quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since Z and Hh[yh] are constant in T, which is the chief reason for utilizing the reduced discrete Hessian, we can equivalently rewrite R3(T) as follows: R3(T) = Hh[yh]ijZij ∫T (f(xT ) − f) with f = ∂1yϵ ×∂2yϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The Bramble-Hilbert Lemma, in conjunction with the Sobolev embedding W 2 1 (T) ⊂ C(T) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' [19, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='4]), implies the existence of a linear polynomial p ∈ [P1(T)]3 such that ∥f − p∥L∞(T) ≲ ∥D2f∥L1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since the mid-point quadrature is exact for linears, we deduce ∣∫T (f(xT ) − f)∣ = ∣∫T {(f − p)(xT ) + (p − f)}∣ ≤ 2∣T∣∥f − p∥L∞(T) ≲ h2 T ∥D2f∥L1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, invoking (45), ∥D2f∥L1(T) ≲ ∥D3yϵ∥L2(T)∥∇yϵ∥L2(T) + ∥D2yϵ∥2 L2(T) ≲ ∥yϵ∥H3(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Inserting this back into R3(T) and adding we end up with limsup h→0+ 2 ∑ i,j=1 ∑ T∈Th ∣R3(T)∣ ≲ limsup h→0+ (h∥Hh[yh]∥L2(Ω))∥yϵ∥H3(Ω) = 0, because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Altogether, we arrive at limsup h→0+ ∣Ch[yh] − C[yϵ]∣ ≲ ϵ which implies the desired estimate (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 13 It is worth realizing the role of the reduced discrete Hessian Hh[yh] in the pre- ceding proof, namely that it factors out the integral defining R3(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' If we had used the discrete Hessian Hh[yh] instead, then there would have been a term of the form h2 T ∥D2Hh[yh]∥L2(T) that could only be handled via an inverse inequality within the H2-regularity setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This in turn would have gotten rid of the factor h2 T and the proof of (43) would have failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Γ-convergence The reduced energy (8) consists of a bending energy B[y] and a cubic term C[y], and so does its discrete counterpart (37), namely Bh[yh] and Ch[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Compactness and Γ-convergence of the bending energy part, being similar to the single layer model, could be deduced from the results in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For instance, we have that for any γ0,γ1 > 0, there exists a constant ccoer such that [14, (37) and (38)] (46) c−1 coer∥yh∥2 H2 h(Ω) ≤ Bh[yh] ≤ ccont∥yh∥2 H2 h(Ω) ∀yh ∈ Vk h(ϕ,Φ), and the constant ccoer → ∞ if either γ0 or γ1 → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In spite of that, [14] enforces the isometry constraint in average and constructs the recovery sequence needed for Γ-convergence via standard nodal interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, the analysis below incorporates new ideas which do not follow from [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We start with the equicoercivity of energy Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The difficulty is dealing with Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 8 (coercivity of total energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let δ > 0 and yh ∈ Ah,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' There exists a constant ˜ccoer > 0 independent of δ, but depending on the given data Z and Th only through its shape regularity constant, such that (47) (2ccoer)−1∥yh∥2 H2 h(Ω) ≤ Eh[yh] + ˜ccoer(1 + δ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We write Bh = Eh + Ch and employ (46) for Bh to obtain c−1 coer∥yh∥2 H2 h(Ω) ≤ Eh[yh] + Ch[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It remains to estimate the cubic term Ch[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Combining Lemma 6 (pointwise isometry constraint) with the Cauchy-Schwarz inequality yields Ch[yh] ≤ 2 ∑ i,j=1 ∑ T∈Th ∣T∣∣Hh[yh]ij ⋅ (∂1yh × ∂2yh)Zij∣(xT ) ≤ 2 ∑ i,j=1 ∑ T∈Th ∣T∣ 1 2 ∥Hh[yh]ij∥L2(T)∣∂1yh(xT )∣∣∂2yh(xT )∣∥Z∥L∞(T) ≤ 2(1 + δ)∥Z∥L∞(Ω)∣Ω∣ 1 2 ∥Hh[yh]∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Invoking Lemma 3 (stability of Hh[yh]) and Young’s inequality yields (48) 1 2ccoer ∥yh∥2 H2 h(Ω) ≤ Eh[yh] + 2ccoerc2 stab∣Ω∣∥Z∥2 L∞(Ω)(1 + δ)2, which is the desired estimate (47) with ˜ccoer = 2ccoerc2 stab∣Ω∣∥Z∥2 L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ 14 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES We now prove Γ-convergence of Eh towards E, which consists of a liminf and a limsup property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Theorem 9 (Γ-convergence of Eh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Assume that δ = δ(h) → 0 as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Then (i) Lim-inf property: Let yh ∈ Ah,δ be a sequence such that Eh[yh] is uniformly bounded in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Then there exists y ∈ A such that yh → y in [L2(Ω)]3 for a subsequence (not relabeled) and E[y] ≤ liminf h→0+ Eh[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (ii) Lim-sup property: For any y ∈ A there exists yh ∈ Ah,δ such that yh → y in [L2(Ω)]3 and E[y] ≥ limsup h→0+ Eh[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We prove properties (i) and (ii) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (i) lim-inf property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 8 (coercivity of total energy) and (24) imply ∥yh∥L2(Ω) + ∥∇hyh∥L2(Ω) + ∥yh∥H2 h(Ω) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proceeding as in [17, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2], there exists y ∈ [H2(Ω)]3 satisfying the Dirichlet boundary conditions in (1) and yh → y in [L2(Ω)]3, ∇hyh → ∇y in [L2(Ω)]3×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In view of Lemma 1 (weak convergence of Hh), we deduce Hh[yh] ⇀ D2y in [L2(Ω)] 3×2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The lower-semicontinuity of the L2-norm under weak-limits together the fact that the stabilization terms in Bh[yh] are positive guarantee that B[y] = 1 2 ∫Ω ∣D2y∣2 ≤ liminf h→0+ Bh[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In addition, Lemma 7 (convergence of cubic energy) yields limh→0+ Ch[yh] = C[yh], and altogether gives E[y] ≤ liminfh→0+ Eh[yh] as asserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It just remains to prove the isometry constraint I[y] = I2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To this end, recall that I[yh] = ∇hyT h ∇hyh, let T ∈ Th and note that ∥I[yh] − I2∥L1(T) ≤ ∥I[yh] − I[yh](xT )∥L1(T) + ∣T∣∣I[yh](xT ) − I2∣ ≲ hT ∥D2 hyh∥L2(T)∥∇hyh∥L2(T) + δ∣T∣, because yh ∈ Ah,δ whence Dh[yh](xT ) = ∣I[yh](xT ) − I2∣ ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Adding over T and employing the uniform boundedness of ∥D2 hyh∥L2(Ω) and ∥∇hyh∥L2(Ω) results in ∥I[yh] − I2∥L1(Ω) ≲ h + δ → 0 as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' On the other hand, we see that I[yh] − I[y] = ∇h(yh − y)T ∇hyh + ∇yT ∇h(yh − y) implies ∥I[yh] − I[y]∥L1(Ω) ≤ (∥∇y∥L2(Ω) + ∥∇hyh∥L2(Ω))∥∇hyh − ∇y∥L2(Ω) → 0, as h → 0+ because ∥∇hyh∥L2(Ω) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This and the triangle inequality lead to ∥I[y] − I2∥L1(Ω) = 0 and consequently I[y] = I2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' in Ω, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (ii) lim-sup property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The difficulty to construct a recovery sequence yh ∈ Ah,δ is that the regularity y ∈ [H2(Ω)∩W 1 ∞(Ω)]3 is borderline to define pointwise values of LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 15 ∇y and thus enforce the isometry defect Dh[yh](xT ) at every element barycenter xT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Hence, we invoke the regularization procedure of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Hornung [29]: given an isometry y ∈ [H2(Ω)]3 and ϵ > 0, there exists an isometry yϵ ∈ [H3(Ω)]3 such that (49) ∥y − yϵ∥H2(Ω) ≲ ϵ, ∥D2yϵ∥L2(Ω) ≲ ∥D2y∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' As usual, the constants hidden in the symbol ≲ are independent of h and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We now set yh ∶= Rh[yϵ], where the recovery operator Rh ∶ [H3(Ω)]3 → [Vk h]3 is the following quadratic Taylor expansion about xT for every T ∈ Th (50) Rh[w](x) ∶= w(xT ) + ∇w(xT )(x − xT ) + 1 2(x − xT )T QT [w](x − xT ) ∀x ∈ T, where QT [w] ∶= 1 ∣T∣ ∫T D2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Note that ∇yh(xT ) = ∇yϵ(xT ) and Dh[yh](xT ) = 0, whence yh ∈ Ah,0 ⊂ Ah,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We next show the two convergence properties of yh in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since Rh∣T is invariant over the space P1 of polynomials of degree ≤ 1, we have w − Rh[w] = (w − p) − Rh[w − p] ∀p ∈ [P1]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, combining the stability in W 1 ∞(T) of the linear part of Rh with the Bramble-Hilbert lemma and the property ∥QT [w]∥L2(T) ≤ ∣w∣H2(T), we deduce ∥w − Rh[w]∥H1(T) ≲ hT ∥∇(w − p)∥W 1 ∞(T) + hT ∥QT [w]∥L2(T) ≲ h2 T ∥w∥H3(T) + hT ∣w∣H2(T) ≲ hT ∥w∥H3(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Notice the presence of the full H3-norm on the right-hand side of the above estimate, which accounts for possible subdivisions made of quadrilaterals [22, 26, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We next square and add over T ∈ Th to obtain ∥w − Rh[w]∥L2(Ω) + ∥∇w − ∇hRh[w]∥L2(Ω) ≲ h∥w∥H3(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This estimate for w = yϵ, in conjunction with (49), yields ∥y − yh∥L2(Ω) + ∥∇y − ∇hyh∥L2(Ω) ≲ ϵ + h∥yϵ∥H3(Ω), whence ∥y−yh∥L2(Ω) ≲ ϵ provided h is sufficiently small so that h∥yϵ∥H3(Ω) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This shows the asserted convergence yh → y in [L2(Ω)]2 because ϵ is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It remains to show the convergence Eh[yh] → E[y] as h → 0+, which in turn implies the desired lim-sup property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since D2yh = QT [yϵ], we infer that ∥D2 hyh∥2 L2(Ω) = ∑ T∈Th ∥QT [yϵ]∥2 L2(T) ≤ ∑ T∈Th ∥D2yϵ∥2 L2(T) ≲ ∥D2y∥2 L2(Ω), according to (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, ∥D2 hyh − D2y∥2 L2(Ω) = ∑ T∈Th ∥QT [yϵ] − D2y∥2 L2(T) ≤ ∑ T∈Th ∥QT [yϵ] − D2yϵ∥2 L2(T) + ∥D2yϵ − D2y∥2 L2(T) ≲ h2∥yϵ∥2 H3(Ω) + ϵ2 shows that D2 hyh → D2y and Lemma 2 (strong convergence of Hh) gives Hh[yh] → D2y strongly in [L2(Ω)]3×2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 16 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES An argument similar to [14, Appendix B and C], invoking the trace inequality, yields Sh[yh] ≲ ∑ T∈Th ∥y − yh∥2 H2(T) → 0, as h → 0+ for the stabilization energy Sh[yh] in (39) and implies convergence of the bending energy Bh in (38), namely lim h→0+ Bh[yh] = B[y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Finally, in view of the preceding dis- cussion, we see that the assumptions of Lemma 7 (convergence of the cubic energy) are valid, whence Lemma 7 implies Ch[yh] → C[y] and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ The construction of the recovery sequence in Theorem 9 (Γ-convergence of Eh) is closely related to Lemma 7 (convergence of the cubic energy) and illustrates the crucial interplay between enforcing the isometry defect Dh[yh] at barycenters and the mid-point quadrature rule in the cubic energy Ch[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This, however, limits the accuracy of LDG to that of lowest polynomial degree k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We leave the design of an LDG method with formal higher accuracy k > 2 open in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Corollary 10 (convergence of global minimizers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let yh ∈ Ah,δ be a sequence of functions such that Eh[yh] is uniformly bounded in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' If yh is an almost global minimizer of Eh in the sense that Eh[yh] ≤ inf wh∈Ah,δ Eh[wh] + σ where σ,δ → 0 as h → 0+, then {yh}h>0 is precompact in [L2(Ω)]3 and every cluster point y belongs to A and is a global minimizer of E, namely E[y] = infw∈A E[w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, up to a subsequence (not relabeled) the energies converge E[y] = lim h→0+ Eh[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We omit the proof of Corollary 10, which readily follows from Theorem 9 (Γ- convergence of Eh), and refer instead to [4, 5, 8, 17] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bilayer model with creases Bartels, Bonito and Hornung have recently developed a reduced single layer model that allows for folding across creases [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The resulting two-dimensional model hinges on a general hyperelastic material description with appropriate scaling conditions on the energy, and consists of a piecewise nonlinear Kirchhoff plate bending model with a continuity condition at the creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For a prescribed Lipschitz curve C intersecting the boundary of Ω transversally, the modified bending energy of [6] reads ̃B[y] ∶= 1 2 ∫Ω∖C ∣II[y]∣ 2 = 1 2 ∫Ω∖C ∣D2y∣2 for deformations y ∈ [H2(Ω ∖ C) ∩ W 1,∞(Ω)]3 satisfying the isometry constraint I[y] = I2 along with possible boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Properly designed creases allow for flapping mechanisms upon actuation at the boundary which are of interest in engineering and medicine [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 17 In this section we explore a similar modification of the elastic energy (4) (51) ̃E[y] ∶= 1 2 ∫Ω∖C ∣II[y] − Z∣ 2, but without justification from 3d hyperelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, we leave open the question whether this energy is the appropriate Γ-limit for bilayer materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We also modify the admissible set to be ̃A ∶= {y ∈ [H2(Ω ∖ C) ∩ W 1 ∞(Ω)]3 ∶ I[y] = I2 in Ω, y = ϕ, ∇y = Φ on ΓD}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Our goal is, instead, to investigate the relation between (51) and its fully discrete counterpart, and demonstrate computationally the crucial role of spontaneous cur- vature Z to produce plate folding without actuation via boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We extend our LDG method to account for creases as in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We consider iso- parametric partitions Th made of possibly curved elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' the mapping FT used to define the finite element space Vk h locally is [Q2]2 instead of [Q1]2 (or [P2]2 instead of [P1]2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We further assume that the crease C is exactly matched by Th: (52) C is made of piecewise quadratic edges e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=',eJ ∈ Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This geometric assumption is restrictive but instrumental for the theory below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Dealing with more general creases C, just interpolated by Eh, is important and the subject of current research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' we refer to [6, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='4] for some discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The distributional derivative of y ∈ [H2(Ω ∖ C) ∩ W 1 ∞(Ω)]3 reads D2y = ̃D2y + [∇y] ⊗ nδC, where ̃D2y stands for the absolutely continuous part of D2y, or restriction of D2y to Ω ∖ C that happens to be L2, while [∇y] ⊗ nδC is the singular part supported on C and n is a unit normal vector to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The first issue to tackle is the construction of a discrete Hessian ̃ Hh[yh] that allows for folding across C and mimics ̃D2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' As in [6], we replace the global lift Rh in (27) by ̃ Rh ∶= ∑ e∈Ea h∖{e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=',eJ} re, where {ej}J j=1 are defined in (52), and let the modified discrete Hessian be ̃ Hh[yh] ∶= D2 hyh − ̃ Rh([∇hyh]) + Bh([yh]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We likewise replace (22) by the modified mesh-dependent form ⟨⋅,⋅⟩ ̃ H2 h ⟨vh,wh⟩ ̃ H2 h ∶= (D2 hvh,D2 hwh)L2(Ω) + (h−1[∇hvh],[∇hwh])L2(Γa h∖C) + (h−3[vh],[wh])L2(Γa h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In essence, the ability for the plates to fold freely across C is reflected in the absence of all the contributions related to [∇yh] across C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This is the key to the following lemma whose proof follows along the lines of [14, Appendix B] and is thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 11 (convergence of ̃ Hh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let the crease C satisfy (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Then there holds 18 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES (i) Weak convergence: If k ≥ 2 and vh ∈ Vk h(ϕ,Φ) satisfies ∣∣vh∣∣H2 h(Ω) ≲ 1 and vh → v ∈ [H2(Ω ∖ C) ∩ H1(Ω)]3 in [L2(Ω)]3 as h → 0+, then we have ̃ Hh[vh] ⇀ ̃D2v in [L2(Ω)] 3×2×2 as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (ii) Strong convergence: Let v ∈ [H2(Ω ∖ C)]3 be any function such that v = ϕ and ∇v = Φ on ΓD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, let vh ∈ Vk h(ϕ,Φ) satisfy ∥D2vh∥L2(T) ≲ ∥v∥H2(T) ∀T ∈ Th, ∑ T∈Th ∥vh − v∥2 H2(T) → 0 as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Then we have as h → 0+ ̃ Hh[vh] → ̃D2v strongly in [L2(Ω)]3×2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We are now ready to introduce the LDG approximation of ̃E[y] in (51), namely ̃Eh[yh] ∶= ̃Bh[yh] + ̃Ch[yh], where ̃Bh[yh] ∶= 1 2 ∫Ω ∣ ̃ Hh[yh]∣2 + γ1∥h− 1 2 [∇hyh]∥2 L2(Γa h∖C) + γ0∥h− 3 2 [yh]∥2 L2(Γa h), and ̃Ch[yh] ∶= 2 ∑ i,j=1 ∑ T∈Th ∣T∣Hh[yh]ij ⋅ (∂1yh × ∂2yh)(xT )Zij, with Hh[yh]∣T ∶= 1 ∣T∣ ∫T ̃ Hh[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemmas 3 and 4 are valid for Hh[yh], as well as Lemma 7 (convergence of cubic energy) and Lemma 8 (coercivity of total energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It remains to examine the convergence of the discrete global minimizers towards the continuous global minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Assume that the crease C splits Ω into two disjoint sets Ω1 and Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since Hornung’s regularization procedure [29] cannot guarantee general Dirichlet boundary conditions, it is not clear how to regularize in Ω1 and Ω2 functions that belong to [H2(Ω∖C)∩W 1 ∞(Ω)]3 and yet maintain the location of the crease C, namely obtain an isometry in [H3(Ω∖C)∩W 1 ∞(Ω)]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Another obstruction stems from the use of curved elements necessarily for the subdivisions to match the crease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' When using polynomial mappings from the reference to the physical elements, the resulting finite element functions are not necessarily polynomial in the physical element, thereby ruling out the construction of the recovery sequence proposed to guarantee the limsup property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' see Theorem 9(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We circumvent these issues by requiring slightly more smoothness on one of the global minimizers y, which in turn allows for a different, more generic, construction of its recovery sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Because the additional regularity cannot be derived from our Γ-convergence theory, we assume the existence of a global minimizer y∗ ∈ ̃A of ̃E with the following property (53) y∗∣Ωi ∈ C1(Ωi), i = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Note that the above assumption is consistent with practical configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We also point out that this regularity assumption and the fact that the subdivision matches LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 19 the crease entail the existence of a modulus of smoothness ω so that (54) ∣∇y∗(x) − ∇y∗(z)∣ ≤ ω(hT ) ∀x,z ∈ T, ∀T ∈ Th, with ω(s) → 0 as s → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The construction of the recovery sequence for deformations satisfying the addi- tional regularity (53) is then based on a piecewise averaged Taylor polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The latter does not preserve the isometry constraint pointwise but (53) allows for control of the isometry defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Before embarking on the proofs, we recall a useful result on the averaged Taylor polynomial [19] defined on the reference element ̂T (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Until the end of this section, we consider the case when the reference element is a square and Qk finite element functions are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The case where ̂T is the unit simplex is somewhat simpler and can be dealt with similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let ̂B be a ball centered at the barycenter of ̂T such that its closure is contained in ̂T and ̂ζ be a cut-off function with unit mass supported on the closure of ̂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For ̂w ∈ L1(̂T) let (55) Q[̂w](ˆx) ∶= ∑ ∣α∣∞≤2 ∫ ̂ B 1 α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' ̂Dα ̂w(ˆz)(ˆx − ˆz)α̂ζ(ˆz)dˆz ∈ Q2, be the averaged Taylor polynomial where α ∶= (α1,α2) is a multi-index with non- negative integers α1,α2 and ∣α∣∞ ∶= max{α1,α2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We recall the following useful properties of Q and refer to [19] for additional details: Q preserves Q2 on ̂T (56) Q[̂p] = ̂p, ̂p ∈ Q2, is stable (57) ∥Q[̂w]∥W k ∞(̂T) ≲ ∥̂w∥L1( ̂ B), ∀k ∈ N and convergent (58) ∣̂w − Q[̂w]∣Hk(̂T) ≲ ( 2 ∑ i=1 ∥∂3 ̂xi ̂w∥2 L2(̂T)) 1/2 , 0 ≤ k < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We next discuss estimates for isoparametric mappings FT ∶ ̂T → T between the reference element ̂T and T ∈ Th so that FT ∈ [Q2]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' They establish relationship between norms on ̂T and T, as well as provide an interpolation estimate in [V2 h]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In fact, for v ∈ H2(T) and ̂v = v ○ FT ∈ H2(̂T), there holds (59) ∥̂v∥L2(̂T) ≈ h−1 T ∥v∥L2(T), ∥̂v∥L∞(̂T) ≈ ∥v∥L∞(T) (60) ∥̂∇̂v∥L2(̂T) ≈ ∥∇v∥L2(T), ∥̂∇̂v∥L∞(̂T) ≈ hT ∥∇v∥L∞(T), (61) ∥ ̂D2̂v∥L2(̂T) ≲ hT ∥D2v∥L2(T) + ∥∇v∥L2(T), (62) ∥D2v∥L2(T) ≲ h−1 T ∥̂v∥H2(̂T), whence for m = 0,1,2 (63) ∣v∣Hm(T) ≲ h1−m T ∥̂v∥H2(̂T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 20 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES Moreover if v ∈ H3(T), we further obtain (64) ∥ ̂D3̂v∥L2(̂T) ≲ h2 T ∥D3v∥L2(T) + hT ∥D2v∥L2(T) + ∥∇v∥L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Note that the first four estimates are discussed and proved in [17, Appendix], while one can extend the proof of (61) to show (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Additionally, as in [17, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='4], the local Lagrange interpolant Ihw ∈ [V2 h]3 for any w ∈ H3(T) satisfies the estimate (65) ∣w − Ihw∣Hm(T) ≲ h3−m∥v∥H3(T), for 0 ≤ m ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The next lemma describes the modified limsup property which hinges on the averaged Taylor polynomial (55);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' compare with Theorem 9(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 12 (limsup property with creases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let y∗ ∈ ̃A satisfy the regularity as- sumption (53) and let ω be the modulus of smoothness in (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' There is a constant c and y∗ h ∈ Ah,cω(h) such that y∗ h → y∗ in [L2(Ω)]3 and limh→0+ ̃Eh[yh] = ̃E[y∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' As usual, the hat symbol denotes quantities defined on the reference element ̂T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let y∗ h ∈ [V2 h]3 be defined locally by y∗ h∣T ∶= ̂y∗ h∣T ○ F −1 T , ̂y∗ h∣T ∶= Q[y∗○ FT ] ∀T ∈ Th, where Q is given in (55) and is applied component-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Note that by construction we indeed have y∗ h ∈ [V2 h]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The rest of the proof consists of 4 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step (i): isometry defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We claim the intermediate estimates (66) ∥∇y∗ h∥L∞(T) ≲ ∥∇y∗∥L∞(T), ∥∇(y∗ − y∗ h)∥L∞(T) ≲ ω(hT ) ∀T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To show the first estimate, we use (60) and (56) to write ∥∇y∗ h∥L∞(T) ≲ h−1 T ∥̂∇̂y∗ h∥L∞(̂T) ≲ h−1 T ∥̂∇(̂y∗ h − c)∥L∞(̂T) = h−1 T ∥̂∇Q[̂y∗− c]∥L∞(̂T), where c ∶= ∣T∣−1 ∫T y∗ ∈ R3 is the average of y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We then employ the stability (57) of Q, together with (59) and Poinc´are inequality, to deduce ∥∇y∗ h∥L∞(T) ≲ h−1 T ∥̂y∗ − c∥L∞(̂T) ≲ h−1 T ∥y∗ − c∥L∞(T) ≲ ∥∇y∗∥L∞(T), which is the first estimate in (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To prove the second estimate in (66), we first notice that for any p ∈ [P1]3 we have ̂p ∶= p ○ FT ∈ [Q2]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since ̂p = Q[̂p], according to (56), we proceed as before, but now using ̂p ∈ Q2 instead of the constant c along with (60), to write ∥∇(y∗ − y∗ h)∥L∞(T) ≲ h−1 T (∥̂∇(̂y∗ − ̂p)∥L∞(̂T) + ∥̂∇Q[̂y∗ − ̂p]∥L∞(̂T)) ≲ h−1 T ∥̂y∗ − ̂p∥W 1 ∞(̂T) ≲ h−1 T ∥y∗ − p∥L∞(T) + ∥∇(y∗ − p)∥L∞(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (67) We next choose y∗ to take advantage of the piecewise smoothness (53), namely p(x) ∶= y∗(xT ) + ∇y∗(xT )(x − xT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The property (54) of ω implies ∥∇(y∗ − p)∥L∞(T) ≤ ω(hT ) LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 21 and combined with y∗(x) − y∗(xT ) = ∇y∗(ξ)(x − xT ) for some ξ ∈ T, gives ∥y∗ − p∥L∞(T) ≤ hT ω(hT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Inserting these estimates in (67) yields the second estimate in (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The estimate on the isometry defect ∥(∇y∗ h)T ∇y∗ h−I2∥L∞(T) follows directly from the intermediate estimates (66) and the assumption I[y∗] = I2 ∥(∇y∗ h)T ∇y∗ h − I2∥L∞(T) = ∥(∇y∗ h)T ∇y∗ h − (∇y∗)T ∇y∗∥L∞(T) ≤ (∥∇y∗∥L∞(T) + ∥∇y∗ h∥L∞(T))∥∇(y∗ h − y∗)∥L∞(T) ≤ cω(hT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' for a constant c independent of the discretization parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' hence y∗ h ∈ Ah,cω(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step (ii): Broken H2− Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For p ∈ [P1]3, we set ̂p ∶= p ○ FT ∈ [Q2]3 to get ∥D2y∗ h∥L2(T) = ∥D2(y∗ h − p)∥L2(T) ≲ h−1 T ∥̂y∗ h − ̂p∥H2(̂T), in view of (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Thanks to the invariance (56) and stability (57) of Q, we obtain ∥D2y∗ h∥L2(T) ≲ h−1 T ∥Q[̂y∗ − ̂p]∥H2(̂T) ≲ h−1 T ∥̂y∗ − ̂p∥L2(̂T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This, together with (59) and a standard interpolation estimate on T, yields ∥D2y∗ h∥L2(T) ≲ h−2 T ∥y∗ − p∥L2(T) ≲ ∥D2y∗∥L2(T), which is the desired stability estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step (iii): H2−Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We exploit a density argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For any ϵ > 0, there exists yϵ ∈ H3(Ω) so that ∥y∗ − yϵ∥H2(Ω) ≤ ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' yϵ may not be an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We split (68) ∑ T∈Th ∥y∗ −y∗ h∥2 H2(T) ≲ ∥y∗ −yϵ∥2 H2(Ω) + ∑ T∈Th ∥yϵ −yϵ h∥2 H2(T) + ∑ T∈Th ∥yϵ h −y∗ h∥2 H2(T) with yϵ h = ̂yϵ h ○ F −1 T and ̂yϵ h ∶= Q[yϵ ○ FT ], and estimate each of the three terms separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The first term is obviously bounded by ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For the second term, we let m = 0,1,2 and combine (58) with (63) to arrive at ∣yϵ h − yϵ∣Hm(T) ≲ h1−m T ( 2 ∑ i=1 ∥∂3 ̂xîyϵ∥2 L2(̂T)) 1/2 ≲ h1−m T ( 2 ∑ i=1 ∥∂3 ̂xi(̂yϵ − ̂ Ihyϵ)∥2 L2(̂T)) 1/2 , where Ih is the local Lagrange interpolant onto [V2 h]3 and ̂ Ihyϵ ∶= Ihyϵ ○F −1 T ∈ [Q2]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' As a consequence, using the estimate (64) to map back to the physical element T, and applying the error estimate (65) for Ih to the ensuing terms, we deduce ∣yϵ h − yϵ∣Hm(T) ≲ h3−m T ∥yϵ∥H3(T), for m = 0,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We thus conclude ∑ T∈Th ∥yϵ − yϵ h∥2 H2(T) ≲ h2∥yϵ∥2 H3(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It remains to estimate the third term ∥y∗ h − yϵ h∥H2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To deal with each term ∣y∗ h − yϵ h∣Hm(T) for m = 0,1,2, we let pm−1 ∈ Pm−1 to be chosen later with the 22 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES convention that p−1 = 0, and use the invariance Q[p ○ FT ] = p ○ FT ∈ [Q2]3 for any p ∈ [P1]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Combining (63) with the stability (57) of Q yields ∣y∗ h − yϵ h∣Hm(T) = ∣y∗ h − yϵ h − pm−1∣Hm(T) ≲ h1−m T ∥Q[̂y∗ − ̂yϵ − pm−1 ○ FT ]∥H2(̂T) ≲ h1−m T ∥̂y∗ − ̂yϵ − pm−1 ○ FT ∥L2(̂T) ≲ h−m T ∥y∗ − yϵ − pm−1∥L2(T) ≲ ∥y∗ − yϵ∥H2(T), provided pm−1 is an averaged Taylor polynomial of y∗ − yϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This in turn implies ∑ T∈Th ∥yϵ h − y∗ h∥2 H2(T) ≲ ∥y∗ − yϵ∥2 H2(Ω) ≲ ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, gathering the estimates for the three terms in (68) we obtain ∑ T∈Th ∥y∗ − y∗ h∥2 H2(T) ≲ ϵ2 + h2∥yϵ∥2 H3(Ω) ≤ 2ϵ2, provided h∥yϵ∥H3(T) ≤ ϵ for h sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since ϵ is arbitrary, we deduce ∑ T∈Th ∥y∗ − y∗ h∥2 H2(T) → 0, and in particular y∗ h → y∗ in [L2(Ω)]3, as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step (iv): Convergence of ̃Eh[y∗ h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Steps (iii) and (iv) show that the conditions in Lemma 11(ii) (strong convergence of ̃ Hh) are fulfilled, whence ̃ Hh[y∗ h] → ̃D2y∗ strongly in [L2(Ω)]3×2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Consequently, convergence of ̃Eh[y∗ h] towards ̃E[y∗] re- duces to the argument given in Theorem 9 (ii) and is not repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ The next theorem guarantees convergence of discrete global minimizers towards exact global minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' but it is not a standard Γ-convergence result because we assume (53) for one global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Other minimizers may fail to satisfy (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Theorem 13 (convergence of global discrete minimizers with creases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Assume that a global minimizer y∗ ∈ ̃A of ̃E satisfy the additional regularity (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let yh ∈ Ah,δ be a sequence of functions such that ̃Eh[yh] is uniformly bounded in h and let δ = δ(h) ≥ cω(h) with c the constant in Lemma 12 and ω the modulus of smoothness in (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' If yh is an almost global minimizer of ̃Eh in the sense that (69) ̃Eh[yh] ≤ inf wh∈Ah,δ ̃Eh[wh] + σ where σ,δ → 0 as h → 0+, then {yh}h>0 is precompact in [L2(Ω)]3 and every clus- ter point y belongs to widetildeA and is a global minimizer of ̃E, namely ̃E[y] = infw∈̃A ̃E[w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, up to a subsequence (not relabeled) (70) ̃E[y] = lim h→0+ ̃Eh[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The liminf property follows along the lines of Theorem 9 (i) because it is based on Lemmas 8 and 7, which remain valid in this context, as well as Lemma 11 (i) (weak convergence of ̃ Hh) instead of Lemma 1 (weak convergence of Hh) and LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 23 the weak lower semicontinuity of the L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, there is y ∈ ̃A such that (up to a subsequence not relabelled) yh → y in [L2(Ω)]3 and ̃E[y] ≤ liminf h→0+ ̃Eh[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To show that y is a global minimizer of ̃E we resort to the extra regularity (53) of the global minimizer y∗ ∈ ̃A of ̃E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let {y∗ h}h>0 ⊂ [L2(Ω)]3 be the sequence provided by Lemma 12 (limsup property with creases), which satisfies y∗ h ∈ Ah,cω(h), ̃Eh[y∗ h] → ̃E[y∗] as h → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In view of (69) and yh ∈ Ah,δ(h), we end up with ̃E[y] ≤ liminf h→0+ ̃Eh[yh] ≤ limsup h→0+ ( ̃Eh[y∗ h] + σ) = ̃E[y∗] = inf w∈ ̃ A ̃E[w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, y is indeed a global minimizer of ̃E and (70) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Discrete gradient flow Solving the minimization problem (41) is a nontrivial task because it entails en- forcing the nonconvex constraint Dh[yh](xT ) ≤ δ at element barycenters xT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We now develop a discrete gradient flow with respect to the H2 h metric (22) that lin- earizes the isometry constraint according to (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We refer to [4, 7, 8, 13, 14, 17, 16] and especially to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bartels and Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Palus [9] for similar gradient flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We start recalling the notion of linearized isometry constraint for vh,yh ∈ [Vk h]3 (71) L[vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='wh](xT ) = [∇vT h ∇wh + ∇wT h ∇vh](xT ) ∀T ∈ Th and defining a tangent space associated with the isometry constraint for any wh ∈ Ah,δ (72) Fh(wh) ∶= {vh ∈ Vk h(0,0) ∶ L[vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='wh](xT ) = 0 ∀T ∈ Th}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Given y0 h ∈ Ah,0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e, y0 h satisfies the isometry constraint I[y0 h](xT ) = I2 at each barycenter xT ), the discrete gradient flow consists of seeking recursively δyn+1 h ∶= yn+1 n − yn h ∈ Fh(yn h) such that (73) 1 τ ⟨δyn+1 h ,vh⟩H2 h(Ω) + ah(δyn+1 h ,vh) = −ah(yn h,vh) + ℓ[yn h](vh) ∀vh ∈ Fh(yn h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Here τ > 0 is a pseudo time step and ah is the bilinear form corresponding to the variational derivative of the bending energy Bh[yh] defined in (38), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' ah(wh,vh) ∶= ∫Ω Hh[wh] ∶ Hh[vh] + γ1(h−1[∇hwh],[∇hvh])L2(Γa h) + γ0(h−3[wh],[vh])L2(Γa h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (74) 24 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES The linear form ℓ[yn h](vh) on vh is the first variation of the cubic energy Ch[yn h], defined in (40), along the direction of the test function vh and is given by ℓ[yn h](vh) ∶= 2 ∑ i,j=1 ∑ T∈Th ∣T∣Hh[vh]ij ⋅ (∂1yn h × ∂2yn h)(xT )Zij + 2 ∑ i,j=1 ∑ T∈Th ∣T∣Hh[yn h]ij ⋅ (∂1vh × ∂2yn h)(xT )Zij + 2 ∑ i,j=1 ∑ T∈Th ∣T∣Hh[yn h]ij ⋅ (∂1yn h × ∂2vh)(xT )Zij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' recall that both Hh[vh] and Z are piecewise constant on Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The explicit treatment of yn h in ℓ[yn h](vh) is similar to the scheme proposed and analyzed by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bartels and Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Palus [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For latter use, we note that Lemma 3 (stability of Hh[vh]) yields ∣ℓ[wh](vh)∣ ≲ √ 1 + δ∥Z∥L∞(Ω)(∥∇hvh∥L2(Ω)∥wh∥H2 h(Ω) + ∥∇hwh∥L2(Ω)∥vh∥H2 h(Ω)), provided wh ∈ Ah,δ because ∣∇wh(xT )∣ ≤ √ 1 + δ from Lemma 6 (pointwise isometry control) and the inverse inequality ∣∇vh(xT )∣ ≲ h−1 T ∥∇vh∥L2(T) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In addition, we rewrite the Friedrichs inequality (24) as follows (75) ∥∇hwh∥L2(Ω) ≲ ∥wh∥H2 h(Ω) + Cϕ,Φ, ∀wh ∈ Vk h(ϕ,Φ), where Cϕ,Φ = ∥ϕ∥H1(Ω) + ∥Φ∥H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' From these estimates we deduce the existence of a constant cnl such that for wh ∈ Ah,δ and vh ∈ Vk h(0,0) we have ∣ℓ[wh](vh)∣ ≤ cnl √ 1 + δ∥Z∥L∞(Ω)(∥wh∥H2 h(Ω)+ Cϕ,Φ)∥vh∥H2 h(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (76) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Energy stability and admissibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We discuss in this section the en- ergy reduction property of the gradient flow and, although the isometry constraint Dh[yn h](xT ) = 0 is relaxed and linearized in the iterative scheme, the deviation of Dh[yn h](xT ) from 0 is controlled by a parameter δ > 0 provided τ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' These results rely on a discrete inverse inequality on finite dimensional subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 14 (discrete Sobolev inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let Wh ⊂ ΠT∈ThH1(T) be a generic finite element space subordinated to the partition Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For any wh ∈ Wh there holds ∥wh∥L∞(Ω) ≲ (1 + ∣log hmin∣) 1 2 (∥wh∥L2(Ω) + ∥∇hwh∥L2(Ω) + ∥h− 1 2 [wh]∥L2(Γ0 h)), where hmin ∶= minT∈Th hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We denote by Πh ∶ ΠT∈ThH1(T) → Vk h ∩ H1(Ω) the smoothing operator from [15, 17] and recall that it satisfies ∥∇Πhwh∥L2(Ω) + ∥h−1(wh − Πhwh)∥L2(Ω) ≲ ∥∇hwh∥L2(Ω) + ∥h− 1 2 [wh]∥L2(Γ0 h), and ∥Πhwh∥L2(Ω) ≲ ∥wh∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 25 Therefore, combining the triangle and inverse inequalities implies ∥wh∥L∞(Ω) ≲ ∥wh − Πhwh∥L∞(Ω) + ∥Πhwh∥L∞(Ω) ≲ ∥h−1(wh − Πhwh)∥L2(Ω) + (1 + ∣log hmin∣) 1 2 ∥Πhwh∥H1(Ω), in view of the following discrete Sobolev inequality in 2d [18, 19] ∥Πhwh∥L∞(Ω) ≲ (1 + ∣log hmin∣) 1 2 ∥Πhwh∥H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This leads to the assertion upon applying the preceding estimates for Πh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ We are now in a position to prove the main result of this section, namely that the gradient flow is energy decreasing and controls the isometry defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Theorem 15 (properties of gradient flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let {y0 h}h>0 ⊂ Ah,0 satisfy Eh[y0 h] ≤ c0 with c0 a constant independent of h and let all subdivisions Th be such that ∣log hmin∣ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let N be the number of iterations of the gradient flow and τ be its pseudo-time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' There exists a constant α1 = α1(ϕ,Φ,Z) > 0 independent of h and N such that if τ ≤ (2α1∣log hmin∣)−1, then the energy Eh[yN h ] satisfies (77) Eh[yN h ] + 1 2τ N−1 ∑ n=0 ∥δyn+1 h ∥2 H2 h(Ω) ≤ Eh[y0 h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In addition, there are constants α2 = α2(ϕ,Φ,Z) > 0 and α3 > 0, both independent of h and N, such that the isometry defect Dh[yN h ] satisfies (78) ∣Dh[yN h ](xT )∣ ≤ α3τ∣log hmin∣(Eh[y0 h] + α2) ∀T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We first note that estimates (77) and (78) hold trivially for N = 0 and y0 h ∈ Ah,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, we assume that (77) and (78) are valid for N = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=',M with positive constants α1,α2,α3 to be specified below and prove the validity of the same estimates for N = M +1 with the same constants α1,α2,α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We split the proof into four steps with the following roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' After deriving an intermediate estimate in Step (i), we prove (77) in Step (ii) and (78) in Step (iii) under suitable restrictions on αi, i = 1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In Step (iv), we show that it is always possible to find values of these parameters satisfying the desired restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In this proof, the generic constants hidden in the symbol “≲” are not only independent of h but also of τ, M and αi, i = 1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step (i): intermediate estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We take vh = δyM+1 h ∈ Fh(yh) in (73) for n = M, use the elementary relation 2b(b−a) = (b−a)2 +b2 −a2 and discard the positive term (b − a)2 = ah(δyM+1 h ,δyM+1 h ) to write (79) ∥δyM+1 h ∥2 H2 h(Ω) + τ 2ah(yM+1 h ,yM+1 h ) − τ 2ah(yM h ,yM h ) ≤ τℓ[yM h ](δyM+1 h ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Using (78) with N = M (induction assumption), together with the restriction on τ and the uniform bound Eh[y0 h] ≤ c0, we obtain (80) ∣Dh[yM h ](xT )∣ ≤ α3 2α1 (c0 + α2) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 26 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES To simplify the expressions below, we let α = (αi)3 i=1 and c2 α = 1 + δ, whence (81) cα ∶= √ α3 2α1 (c0 + α2) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Estimate (80) shows that yM h ∈ Ah,δ with δ = c2 α − 1 which in turn implies ∣∂iyM h (xT )∣ ≤ cα, i = 1,2, T ∈ Th, according to Lemma 6 (pointwise isometry constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Substituting into (76) yields ∣ℓ[yM h ](δyM+1 h )∣ ≤ cnlcα∥Z∥L∞(Ω)∥δyM+1 h ∥H2 h(Ω)(∥yM h ∥H2 h(Ω) + Cϕ,Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Inserting this back into (79) and using Young’s inequality to absorb the term ∥δyk+1 h ∥2 H2 h(Ω) in the left hand side of (79), gives the estimate 1 2∥δyM+1 h ∥2 H2 h(Ω) + τ 2ah(yM+1 h ,yM+1 h ) ≤ τ 2ah(yM h ,yM h ) + τ 2c2 nlc2 α∥Z∥2 L∞(Ω)(∥yM h ∥2 H2 h(Ω) + C2 ϕ,Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (82) We next improve upon (82) by deriving a uniformly bound for the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' According to (80), the isometry defect is controlled by δ = c2 α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, (77) for N = M (induction assumption) implies that Eh[yM h ] ≤ Eh[y0 h] ≤ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Hence, the coercivity estimate (47) reads (83) (2ccoer)−1∥yM h ∥2 H2 h(Ω) ≤ c0 + ˜ccoerc4 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Estimate (46) can be rewritten in terms of the bilinear form ah as follows (84) c−1 coer∥vh∥2 H2 h(Ω) ≤ 1 2ah(vh,vh) ≤ ccont∥vh∥2 H2 h(Ω) ∀vh ∈ Vk h(ϕ,Φ), Since ∣log hmin∣ ≥ 1, τ satisfies τ ≤ 1 2α1 ≤ 1 provided α1 ≥ 1 2, which is our first restriction on α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Combining this with (83) and (84) with vh = yM h , and replacing back into (82), gives the desired intermediate estimate (85) 1 2τ ∥δyM+1 h ∥2 H2 h(Ω) + 1 2ah(yM+1 h ,yM+1 h ) ≤ ψ1(cα) where ψ1(cα) ≥ 0 is a positive increasing function of its argument cα but whose specific expression is irrelevant except that it is independent of h, M and depends on α = (αi)3 i=1 only through the variable cα rather than separately on each αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step (ii): proof of (77) for N = M +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In view of (79), and telescopic cancellation, this requires dealing with the cubic term ℓ[yM h ](δyM+1 h ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Using the identity aM+1bM+1cM+1 − aMbMcM = (aM+1 − aM)bM+1cM+1 + aM(bM+1 − bM)cM+1 + aMbM(cM+1 − cM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 27 we deduce (aM+1 − aM)bMcM + aM(bM+1 − bM)cM + aMbM(cM+1 − cM) = aM+1bM+1cM+1 − aMbMcM − (aM+1 − aM)(bM+1 − bM)cM+1 − (aM+1 − aM)bM(cM+1 − cM) − aM(bM+1 − bM)(cM+1 − cM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' and rewrite ℓ[yM h ](δyM+1 h ) as follows: ℓ[yM h ](δyM+1 h ) = 2 ∑ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='j=1 ∑ T∈Th ∣T∣Hh[yM+1 h ]ij ⋅ (∂1yM+1 h × ∂2yM+1 h )(xT )Zij − 2 ∑ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='j=1 ∑ T∈Th ∣T∣Hh[yM h ]ij ⋅ (∂1yM h × ∂2yM h )(xT )Zij − 2 ∑ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='j=1 ∑ T∈Th ∣T∣Hh[δyM+1 h ]ij ⋅ (∂1δyM+1 h × ∂2yM+1 h )(xT )Zij − 2 ∑ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='j=1 ∑ T∈Th ∣T∣Hh[δyM+1 h ]ij ⋅ (∂1yM h × ∂2δyM+1 h )(xT )Zij − 2 ∑ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='j=1 ∑ T∈Th ∣T∣Hh[yM h ]ij ⋅ (∂1δyM+1 h × ∂2δyM+1 h )(xT )Zij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We note that the first two terms are exactly the cubic energies Ch[yM+1 h ] and Ch[yM h ] and together with the bending energies Bh[yM+1 h ] = 1 2ah(yM+1 h ,yM+1 h ) and Bh[yM h ] = 1 2ah(yM h ,yM h ) in (79) give rise to the full energies Eh[yM+1 h ] and Eh[yM h ] in (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In contrast, the last three terms must be estimated and absorbed into the remaining term ∥δyM+1 h ∥2 H2 h(Ω) in (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To this end, we combine the Friedrichs inequality (75) for wh = yM+1 h ,yM h ∈ Vk h(ϕ,Φ) and wh = δyM+1 h ∈ Vk h(0,0), and Lemma 3 (stability of Hh[vh]), to obtain ∥δyM+1 h ∥2 H2 h(Ω) + τEh[yM+1 h ] − τEh[yM h ] ≲ τ∥δyM+1 h ∥H2 h(Ω)∥∇δyM+1 h ∥L∞(Ω)(∥yM+1 h ∥H2 h(Ω) + ∥yM h ∥H2 h(Ω) + Cϕ,Φ), where the symbol ≲ hides ∥Z∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To estimate the L∞-norm on the right-hand side, we resort to Lemma 14 (discrete Sobolev inequality) ∥δyM+1 h ∥2 H2 h(Ω) + τEh[yM+1 h ] − τEh[yM h ] ≲ τ∣log hmin∣∥δyM+1 h ∥2 H2 h(Ω)(∥yM+1 h ∥H2 h(Ω) + ∥yM h ∥H2 h(Ω) + Cϕ,Φ), because ∣log hmin∣ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, the coercivity estimate (46) of Bh, written now as ∥yM+1 h ∥2 H2 h(Ω) ≤ ccoerBh[yM+1 h ] = ccoer 2 ah(yM+1 h ,yM+1 h ) ≤ ccoerψ1(cα) according to (85), together with (83) guarantees that ∥yM+1 h ∥H2 h(Ω) + ∥yM h ∥H2 h(Ω) + Cϕ,Φ ≤ ψ2(cα), 28 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES where ψ2(cα) is a positive increasing function of the argument cα which is indepen- dent of h,M, and the individual parameters (αi)3 i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Substituting back yields ∥δyM+1 h ∥2 H2 h(Ω) + τEh[yM+1 h ] − τEh[yM h ] ≤ τ∣log hmin∣ψ2(cα)∥δyM+1 h ∥2 H2 h(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Consequently, since τ ≤ (2α1∣log hmin∣)−1, it remains to choose (αi)3 i=1 so that ψ2(cα) ≤ α1, to derive the desired estimate (77) for N = M + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The validity of this estimate will be justified in Step (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step (iii): proof of (78) for N = M + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since δyn+1 h ∈ Fh(yn h), expanding I[yn+1 h ](xT ) = [(∇yn+1 h )T ∇yn+1 h ](xT ) and using the definition (72) of Fh(yn h) yields (86) Dh[yn+1 h ](xT ) = Dh[yN h ](xT ) + I[δyn+1 h ](xT ) ∀T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Applying Lemma 14 (discrete Sobolev inequality), followed by the discrete Friedrichs inequality (75) to estimate ∥∇hδyn+1 h ∥L2(Ω), implies (87) ∣I[δyn+1 h ](xT )∣ ≤ ∥∇hδyn+1 h ∥2 L∞(T) ≲ ∣log hmin∣∥δyn+1 h ∥2 H2 h(Ω), because ∣log(hmin)∣ ≥ 1 and δyn+1 h ∈ Vk h(0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Summing over 0 ≤ n ≤ M, and using telescopic cancellation along with y0 h ∈ Ah,0, yields (88) ∣Dh[yM+1 h ](xT )∣ ≲ ∣log hmin∣ M ∑ n=0 ∥δyn+1 h ∥2 H2 h(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Exploiting the energy decay (77), proved for N = M + 1 in Step (ii), gives (89) M ∑ n=0 ∥δyn+1 h ∥2 H2 h(Ω) ≤ 2τ(Eh[y0 h] − Eh[yM+1 h ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We now need a lower bound for the energy Eh[yM+1 h ], which is a consequence of (47) provided yM+1 h ∈ Ah,ϵ for some ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To this end, we resort again to (86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We first bound the second term on the right-hand side upon combining the intermediate estimate (85) for ∥δyM+1 h ∥H2 h(Ω) with (87) ∣I[δyM+1 h ](xT )∣ ≤ 2τ∣log hmin∣ψ1(cα) ≤ α−1 1 ψ1(cα), because τ ≤ (2α1∣log hmin∣)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Using this bound in (86), along with the fact that yM h ∈ Ah,δ for δ = c2 α − 1 according to the induction assumption (80), implies ∣Dh[yM+1 h ]](xT )∣ ≤ c2 α − 1 + α−1 1 ψ1(cα) =∶ ϵ, whence yM+1 h ∈ Ah,ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Inserting Eh[yM+1 h ] ≥ −˜ccoer(1 + ϵ)2 from (47) into (89) gives M ∑ n=0 ∥δyn+1 h ∥2 H2 h(Ω) ≤ 2τ(Eh[y0 h] + ˜ccoer(c2 α + α−1 1 ψ1(cα)) 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Returning to (88), we arrive at ∣Dh[yM+1 h ](xT )∣ ≤ cIτ∣log hmin∣(Eh[y0 h] + ˜ccoer(c2 α + α−1 1 ψ1(cα)) 2), LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 29 where cI is a constant independent of h, M and (αi)3 i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The desired control on the isometry defect (78) is thus guaranteed provided α3 ≥ cI, α2 ≥ ˜ccoer(c2 α + α−1 1 ψ1(cα)) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step (iv): choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' α = (αi)3 i=1 must satisfy α1 ≥ 1 2, ψ2(cα) ≤ α1, α3 ≥ cI, α2 ≥ ˜ccoer(c2 α + α−1 1 ψ1(cα)) 2 =∶ ψ3(α1,cα), where cα is defined in (81) and the functions ψ1,ψ2 are positive and increasing in their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' One admissible set of parameters is α3 = cI, α2 = α 1 2 1 , with α1 ≥ 1 2 sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In fact, we note that as α1 → ∞ cα ↓ 1, ψ1(cα) ↓ ψ1(1) ≥ 0, ψ2(cα) ↓ ψ2(1) ≥ 0, ψ3(α1,cα) ↓ ˜ccoer, and the condition α1 ≥ max{1 2,ψ2(cα),ψ3(α1,cα)2} admits a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This com- pletes the induction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ It is worth realizing that the ℓ∞-control of the isometry defect (78) implies that yh ∈ Ah,δ provided τ is so small that α3τ∣log hmin∣(Eh[y0 h] + α2) ≤ δ, where Ah,δ is defined in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This property is novel in the context of DG approx- imations [13, 14, 16, 17, 38], but is inspired by a similar one at element vertices shown by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bartels and Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Palus for Kirchhoff elements [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It is responsible for the explicit treatment of the cubic term in (73), which in turn converts (73) into a linear system to solve for δyn+1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The fact that H2(Ω) does not embed in W 1 ∞(Ω) in two dimensions, but is borderline instead, explains the critical nature of the estimates (77) and (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The discrete H2-metric of the gradient flow (73), combined with Lemma 14 (discrete Sobolev inequality), makes it possible to exploit this borderline structure discretely at the expense of a log term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' No weaker metric for the gradient flow than H2 would allow for ℓ∞-control of the isometry defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lagrange multipliers for the isometry constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We enforce tangen- tial variations δyn+1 h ∈ Fh(yn h) at each step of the gradient flow using Lagrange multipliers within the space of symmetric piecewise constant tensors Λh ∶= {λh ∶ Ω → R2×2 ∶ λT h = λh, λh ∈ [V0 h] 2×2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For any wh ∈ Vk h(ϕ,Φ), we define the bilinear form bh(wh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='⋅,⋅) on Vk h(0,0) × Λh as (90) bh(wh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='vh,µh) ∶= ∑ T∈Th ∣T∣L[vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='wh](xT ) ∶ µh, where the linearized isometry constraint L[vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='wh] is given in (71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Note that bh is continuous with a continuity constant uniform in h (91) ∣bh(wh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='vh,µh)∣ ≲ ∥wh∥H2 h(Ω)∥vh∥H2 h(Ω)∥µh∥L2(Ω), 30 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES thanks to the inverse inequality ∣L[vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='wh](xT )∣ ≲ h−1 T ∥L[vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='wh]∥L2(T) and the dis- crete Sobolev inequality ∥∇hwh∥L4(Ω) ≲ ∥wh∥H2 h(Ω) valid for all wh ∈ [Vk h]3, see [14, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='9)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We also observe that bh(wh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='vh,µh) = 0 for all µh ∈ Λh implies vh ∈ Fh(wh) according to (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, in each step of the gradient flow augmented with the linearized metric constraint, we seek (δyn+1 h ,λn+1 h ) ∈ Vk h(0,0) × Λh such that τ −1(δyn+1 h ,vh)H2 h(Ω)+ah(δyn+1 h ,vh)+bh(yn h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='vh,λn+1 h ) + bh(yn h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='δyn+1 h ,µh)=ℓ[yn h](vh)−ah(yn h,vh), (92) for all (vh,µh) ∈ Vk h(0,0) × Λh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The proposed strategy is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Algorithm 1: (discrete-H2 gradient flow with Lagrange multipliers) Given a pseudo-time step τ > 0 and a target tolerance tol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Choose an initial guess y0 h ∈ Ah,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' while τ −1∣Eh[yn+1 h ] − Eh[yn h]∣ >tol do Solve (92) for (δyn+1 h ,λn+1 h ) ∈ Vk h(0,0) × Λh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Update yn+1 h = yn h + δyn+1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' end It is worth pointing out that utilizing Lagrange multipliers is ubiquitous to enforce linearized metric constraints [4, 7, 8, 9, 13, 14, 17, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In particular, the system (92) is solved using the Schur complement approach, whose performance depends on the inf-sup stability of bh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' [38, 11] and refer to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='1 for additional details on the practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Unfortunately, there are no results available in the literature guaranteeing a uniform inf-sup not even for the continuous problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In this section, we make a first step towards a better understanding of the situation in that we derive a sub-optimal estimate of the inf-sup constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We start with a linear algebra lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Lemma 16 (solvability of a matrix equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Given a 2 × 2 symmetric matrix C and a full-rank 3 ×2 matrix B, there exists a 3 ×2 matrix A that solves the equation (93) (AT B + BT A) ∶ C = ∣C∣2 and satisfies ∣A∣ ≤ ∣C∣ 2σ2(B), where ∣ ⋅ ∣ denotes the Frobenius norm of matrices and σ2(B) > 0 is the smallest singular value of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Using the cyclic properties of the trace operator yields AT B ∶ C = tr(BT AC) = tr(CAT B) = tr(AT BC) = BT A ∶ C = A ∶ BC, whence (93) is equivalent to A ∶ BC = 1 2∣C∣2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let B = UΣV T be the singular value decomposition of B, where U ∈ R3×3 and V = R2×2 are orthogonal matrices, and Σ = [σ1(B),0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='0,σ2(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='0,0] ∈ R3×2 carries LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 31 the singular values σ1(B) ≥ σ2(B) ≥ 0 of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since B is full-rank, we deduce that σ2(B) > 0 and ∣BC∣2 = ∣UΣV T C∣2 = ∣ΣC∣2 ≥ σ2(B)2∣C∣2 = σ2(B)2∣C∣2, where C = V T C and thus ∣C∣ = ∣C∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We can now assume that C ≠ 0, for otherwise A = 0 solves (93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We then realize that ∣BC∣ > 0 and A = (BC)∣C∣2 2∣BC∣2 is clearly a solution to (93) as well as ∣A∣ = ∣C∣2 2∣BC∣ ≤ ∣C∣ 2σ2(B), which is the desired estimate ∣A∣ ≲ ∣C∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ The following sub-optimal estimate of the discrete inf-sup constant is a conse- quence of the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since only the gradient of vh ∈ Vk h(0,0) appears in (90), but the underlying norm of Vk h(0,0) is the discrete H2-norm, it seems natural to consider a negative Sobolev norm of order −1 for the space of Lagrange multipli- ers Λh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' However, the fact that ∇hvh is discontinuous makes it problematic to pair it with a distribution in a negative Sobolev space of order −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This leads to the embedding of Λh into [L2(Ω)]2×2, which is somehow responsible for suboptimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Theorem 17 (discrete inf-sup constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For any n ≥ 0 and yn h ∈ Ah,δ, there exists a constant β independent of n and h such that βh = βhmin > 0 satisfies (94) inf µh∈Λh sup vh∈Vk h(0,0) bh(yn h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='vh,µh) ∥vh∥H2 h(Ω)∥µh∥L2(Ω) ≥ βh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We proceed in two steps: we first construct a suitable vh and next show (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step 1: Construction of vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Given µh ∈ Λh, let µh,T = µh∣T be the constant symmet- ric 2 × 2 restriction of µh to any element T ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Thanks to Lemma 16 (solvability of a matrix equation), there exists a 3 × 2 constant matrix AT such that L[AT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='yn h](xT ) ∶ µh,T = (AT T ∇yn h(xT ) + ∇yn h(xT )T AT ) ∶ µh,T = ∣µh,T ∣2, and ∣AT ∣ ≤ ∣µh,T ∣ 2σmin(∇yn h(xT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Let λmin ∶ M2×2 → R be the smallest eigenvalue function defined over the space of symmetric matrices M2×2 into R, which turns out to be continuous with respect to any norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In particular, because yn h ∈ Ah,δ we have Dh[yn h](xT ) = ∣I[yn h](xT ) − I2∣ ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' and there is a constant c independent of h and n so that ∣λmin(I[yn h](xT )−I2)∣ ≤ cδ, or λmin(I[∇yn h](xT )) ≥ 1 − cδ, Consequently, for δ sufficiently small we deduce σmin((∇yn h(xT )) = (λmin(I[yn h](xT ))) 1 2 ≥ (1 − cδ) 1 2 , 32 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES is bounded away from 0 and we have ∣AT ∣ ≤ ∣µh,T ∣ 2(1−cδ) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We finally define vh(x)∣T ∶= AT (x − xT ) on each T ∈ Th, where xT is the barycenter of T, and observe that vh ∈ [Vk h]3 for k ≥ 2 and ∇vh∣T = AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Step 2: Discrete inf-sup property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We first compute bh(yn h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='vh,µh) = ∑ T∈Th ∣T∣L[vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='yn h](xT ) ∶ µh,T = ∑ T∈Th ∣µh,T ∣2∣T∣ = ∥µh∥2 L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' we don’tSince D2 hvh = 0 for vh piecewise linear, combining a trace inequality with the Poincar´e inequality on each element T gives ∥vh∥2 H2 h(Ω) = ∑ e∈Ea h ∥h−3/2[vh]∥2 L2(e) + ∥h−1/2[∇vh]∥2 L2(e) ≲ ∑ T∈Th h−4∥vh∥2 L2(T) + h−2∥∇vh∥2 L2(T) ≲ ∑ T∈Th h−2∥∇vh∥2 L2(T) due to the the fact that vh ∈ Vk h(0,0) has vanishing mean value on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Therefore, (95) ∥vh∥2 H2 h(Ω) ≲ ∑ T∈Th h−2 ∫T ∣AT ∣2 ≲ h−2 min(1 − cδ)−1∥µh∥2 L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In summary, we have shown that for every µh ∈ Λh, there exists vh ∈ Vk h(0,0) such that bh(yn h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='vh,µh) = ∥µh∥2 L2(Ω) and ∥vh∥H2 h(Ω) ≲ h−1 min∥µh∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This is the desired inf-sup condition in disguised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Numerical experiments In this section we present several numerical experiments, some motivated by com- putations [8, 7, 9, 16] and other by lab experiments [1, 30, 36, 32, 35, 37, 39, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We carry out simulations with several spontaneous curvature matrices Z and both Dirichlet and free boundary conditions, so as to capture a variety of insightful config- urations exhibiting large bending deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We consider the effect of different aspect ratios of rectangular domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We also explore properties of a novel model inspired by [6], which allows folding across curved creases (bilayer origami).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Our numerical simulations illustrate the computational performance of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We start with a few comments on the implementation of the gradient flow (92) and Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Saddle-point structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We resort to a Schur complement method to solve the dis- crete problem (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We refer to [13] for full implementation details of a similar linear algebra structure, but emphasize here how Theorem 17 (inf-sup stability) guarantees solvability and affects the solver efficiency in the spirit of [11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To make explicit the Schur complement matrix and deduce its condition number, we denote by {ϕi}N i=1 a basis for Vk h(0,0) and by {ψi}M i=1 an orthonormal basis for Λh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The matrix representations of the bilinear forms Ah(⋅,⋅) ∶= τ −1(⋅,⋅)H2 h(Ω)+ah(⋅,⋅) and bh(yn h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='⋅,⋅) used to define the gradient flow (92) are thus given by A ∶= (Ah(ϕj,ϕi)) N i,j=1, Bn ∶= (bh(yn h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='ϕj,ψi)) M,N i=1,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 33 With this notation, the Schur complement matrix reads Sn ∶= BnA−1BT n and satisfies (Snm,m) = (A−1/2BT nm,A−1/2BT nm) = sup w∈RN ((w,A−1/2BT nm) ∥w∥2 ) 2 = sup v∈RN ((v,BT nm) ∥A1/2v∥2 ) 2 = sup vh∈Vk h(0,0) bh(yn h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='vh,µh)2 Ah(vh,vh) , where µh ∶= ∑M i=1 miψi ∈ Λh,vh ∶= ∑N j=1 vjϕj ∈ Vk h(0,0) with m = (mi)M i=1,v = (vj)N j=1, and ∥ ⋅ ∥2 = (⋅,⋅)1/2 is the Euclidean norm in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' On one hand, the continuity (91) of bh and the coercivity estimate (46) for Bh[vh] = 1 2ah(vh,vh) yield (Snm,m) ≲ τ∥yn h∥2 H2 h(Ω)∥µh∥2 L2(Ω) ≲ τ∥µh∥2 L2(Ω) = τ∥m∥2 2, because ∥yn h∥H2 h(Ω) ≲ 1 in view of the energy stability (77) satisfied by yn h and the coercivity of total energy (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' On the other hand, the inf-sup stability (94) and the continuity estimate (46) for Bh[vh] = 1 2ah(vh,vh) imply τh2 min∥m∥2 2 = τh2 min∥µh∥2 L2(Ω) ≲ (Snm,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Combining these two inequalities yields an estimate for the condition number of Sn (96) κ(Sn) ∶= max m∈RM (Snm,m) ∥m∥2 2 ( min m∈RM (Snm,m) ∥m∥2 2 ) −1 ≲ h−2 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Estimate (96) guarantees that the saddle-point system is invertible but ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We use a conjugate gradient (CG) iterative solver for the numerical experiments be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Classical convergence theory for CG asserts that the number of iterations to achieve a desired accuracy is of order √ κ(Sn) [27, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Our numerical experiments reveal that the number of iterations needed in the CG solver roughly behaves like h−1 min, which is consistent with (96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We emphasize that solving the linear system (92) by the Schur complement method for several steps of the gradient flow remains the bottle neck in terms of computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We leave the design of suitable preconditioners open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Since the scalar product ⟨⋅,⋅⟩H2 h(Ω) and bilinear form ah do not change in the course of the gradient flow, we assemble them once for all before the main loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In contrast, we assemble the bilinear form bh(yn h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='⋅,⋅) and right hand side ℓ[yn h](.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=') at each step of the loop as they depend on the previous iterate yn h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Computing the discrete Hessian Hh[yh] is the most expensive part in the assembly process, as it requires solving the linear systems (25) and (26) for lifting operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In order to save computing time, we find the discrete Hessian of each basis function at the beginning of the simulation and store its values for later use;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' this pre-processing drastically decreases the assembly time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Software and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We implement our LDG method within the software platform deal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='ii [3] and visualize the outcome with paraview [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For all the simulations, we 34 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES fix the polynomial degree k of the deformation yh and the two liftings l1,l2 of the discrete Hessian Hh[yh], as well as the stabilization parameters γ1,γ2 to be k = l1 = l2 = 2, γ0 = γ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Recall that LDG is stable for any positive choice of parameters γ1,γ2, which con- trasts with IPDG that requires γ1,γ2 large for stability purposes [17, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In the following numerical simulations, we consider both clamped Dirichlet (ΓD ≠ ∅) and free boundary conditions (ΓD = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' For the latter, the discrete equation (73) is no longer well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To fix the system kernel, we add an L2-term to the metric (⋅,⋅)H2 h(Ω), while all other implementation aspects are similar to the case ΓD ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We refer to [13] for implementation details of free boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In either situation, a natural choice of initial deformation is that of a flat plate y0 h(x1,x2) = (x1,x2,0) ∀(x1,x2) ∈ Ω, and satisfies clamped boundary conditions and the isometry constraint I[y0 h] = I2 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This is much simpler than prestrained plates [13, 14], which require preprocessing of both boundary condition and metric constraint to construct suitable initial deformations for LDG to start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Clamped plate: Isotropic curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We consider a rectangular plate Ω = (−5,5) × (−2,2), clamped on the side {−5} × [−2,2], with isotropic spontaneous curvature Z = I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The deformation with minimal energy corresponds to a cylinder of radius 1 and energy 20 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This is confirmed by our simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2, which displays iterations of the discrete gradient flow with number of elements is 1024 (30720 dofs), τ = 5 × 10−3 and tol = 10−4 in Algorithm 1, as in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We notice that surface self-intersecting develop during the relaxation dynamics of Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' this is similar to [9] but different from [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, it takes fewer iterations for Algorithm 1 to reach the cylindrical equilibrium configuration, with the same or even smaller time step τ, than FEMs in [8, 9, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, we keep the time step τ = 5×10−3 fixed and consider two quasi-uniform meshes with 256 and 1024 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We obtain bending energies Eh = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='8627 and Eh = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='8038 (16% and 11% relative error) respectively, which exhibit smaller errors than the corresponding ones Eh = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='961 and Eh = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='544 with the Kirchhoff FEM of [8] for the same mesh-size and time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In addition, the energy error compares favorably with the new Kirchhoff FEM in [9], which computes with Z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='5I2 and produces a 36% relative error even with a finer mesh of 5120 triangular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Free plate: Anisotropic curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We now explore a cigar-type configu- ration motivated by experiments [30] and computations [8, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The plate is again the rectangle Ω = (−5,5) × (−2,2), but now we impose no boundary condition (free boundary) along with the anisotropic spontaneous curvature (97) Z = [ 3 −2 −2 3 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We observe that the eigenpairs of Z are (1,[1,1]T ) and (5,[1,−1]T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We thus expect that the plate deforms at −45 degrees with respect to the Cartesian axes in LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 35 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Isotropic curvature: Relaxation dynamics of Algorithm 1 to- wards the cylinder equilibrium shape of a clamped rectangular plate with the isotropic spontaneous curvature Z = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The bilayer plate is depicted at times 0,50,1000,9000,18000,36050,48100,56050,72100 of the gradient flow (counter-clockwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' a symmetric way and eventually reaches a cigar-like configuration, as in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We confirm this in Figure 3, that displays computations with 1024 elements (30720 dofs) and τ = 5 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The final energy is Eh = 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='3898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Remarkably, Algorithm 1 takes fewer iterations to reach the equilibrium configuration than [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Anisotropic curvature: Relaxation dynamics of Algorithm 1 towards the cigar-type equilibrium of a free rectangular plate with the anisotropic spontaneous curvature of (97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The bilayer plate is depicted at times 0,50,200,1000,10000,30000 of the gradient (counter-clockwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 36 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Anisotropic indefinite curvature: Relaxation dynamics of Al- gorithm 1 with spontaneous curvature (98) towards a DNA-like equilibrium configuration of a free rectangular strip with large aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The bilayer plate is depicted at times 0,100,200,1000,4000,12600 of the gradient flow (left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Free plate: Helix shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We present a helix-type shape motivated by a DNA-like configuration [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We consider a high aspect ratio plate Ω = (−8,8) × (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='5), with free boundary condition and anisotropic spontaneous curvature (98) Z = [ 1 −3/2 −3/2 1 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We point out that the eigenpairs of Z are (−1 2,[1,1]T ) and (5 2,[1,−1]T ), which again correspond to principal directions that form an angle of 45 degrees with the coordinate axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This, together with eigenvalues of opposite sign and high aspect ratio, leads to a deformation that resembles the twisting of DNA molecules, as in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We display several snapshots of the relaxation dynamics of Algorithm 1 in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The simulation is carried out with 1024 elements and τ = 10−2, and yields a final energy Eh = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Moreover, it again takes fewer iterations for LDG to reach the equilibrium configuration than the DG method of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Climate responsive architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bilayer devices can be used to control the temperature or moisture inside a room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The HygroSkin project [36, 37, 39, 32, 35] exploits this technology by designing visually appealing humidity responsive apertures to a pavilion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Heat and moisture are thus dynamically controlled without any high-tech equipment owing to the dominant orientation of fibers in plywood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To simulate this device with our bilayer model, we consider an equilateral triangle with side length 1 and vertices (0,0), (1,0) and (1 2, √ 3 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The actual climate respon- sive device consists of 6 of these triangular shapes suitably rotated and arranged LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 37 together as to form a flat regular hexagon, with the exterior edge of each triangle clamped;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' we refer to Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' To mimic the effect of different relative humidity values, we choose several anisotropic spontaneous curvatures (99) Z = (0 0 0 α), with α = 0,1,2,3,4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' for the triangle with exterior edge parallel to the x-axis and suitably rotated for the other triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This matrix favors bending along the y-axis exclusively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Climate responsive device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The undeformed plate is made of 6 equilateral triangles that together form a regular hexagon (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Finite element partition of each triangle into trapezoids (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Upon actuation, the climate device automatically opens as depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The matching of the computed (left) and actual (right) equilibrium shapes in Fig- ure 6 is quite remarkable for a model with just one parameter α within Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We run this simulation with time step τ = 1 and stopping tolerance tol = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Folding Model: Bilayer Origami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We finally explore computationally the combined effect of spontaneous curvature, as driving mechanism, and folding across a preassigned crease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The corresponding bilayer model and LDG method are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We consider below the setting from [6, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='2] and refer to [12] for additional numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The computational domain is a rectangle Ω = (0,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='6) × (0,15) and the folding crease is a quadratic curve C passing through the points (0,2), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='6,2), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content='8,6), which can be exactly represented by the isoparametric mesh Th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' In order to generate a configuration similar to the flapping mechanism in [6], which is obtained by compression of the lateral boundary, we set the spontaneous curvatures Z = (0 0 0 1), Z = (0 0 0 −1 2 ), below the folding arc and above of it, respectively, and do not impose any boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The resulting equilibrium shape is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We point out the crucial role played by the sign of principal curvatures λ = 1,−1 2 corresponding to the same coordinate eigendirection: bending of the lower and upper plates occurs in opposite directions which gives rise to folding across the crease and yields a rather large compatible deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 38 LDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Climate responsive device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (Left) Approximate deforma- tion for different values of spontaneous curvature (99) with parameter α = 0,1,2,3,4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (from left to right and top to bottom) (Right) Exper- imental deformations of a device made of plywood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Picture taken from [35] (courtesy of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Achim Menges);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' see also [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The matching is remark- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bilayer origami: Flapping mechanism generated by folding of a bilayer plate across a crease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (Left) Conforming subdivision of Ω with quadratic crease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' (Right) Different perspectives of the resulting very large deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Conclusions In this article, we present a new LDG method for large bending isometric defor- mations of bilayer plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We summarize our contributions in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It consists of replacing the Hessian D2y by a reconstructed Hessian Hh[yh] in the bending energy Bh[yh], and by a reduced (piecewise constant) 30%RH 36%RH 43% RH 49%·RH BS%RH 62%RH 69%·RH Z5%·RHLDG APPROXIMATION OF LARGE DEFORMATIONS OF BILAYER PLATES 39 discrete Hessian Hh[yh] in the cubic energy Ch[yh], which encodes the interaction with spontaneous curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We use the mid-point quadrature to integrate Ch[yh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Linearized isometry constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This allows for a slight violation of the isometry constraint I[yh] = I2 while providing control of the ℓ∞-norm of the isometry defect ∣I[yh] − I2∣ at element barycenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This turns out to be a significant improvement over previous DG methods that enforce such defect as sum of averages over elements [13, 14, 17, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Γ-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The key novelty of the Γ-convergence of discrete energies is the construction of the recovery sequence of any admissible deformation y ∈ [H2(Ω) ∩ W 1 ∞(Ω)]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' It hinges on a quadratic Taylor expansion at element barycenters of a suitable regularization of y, and exploits that both the reduced quadrature of Ch[yh] and the isometry defect are imposed at element barycenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Bilayer model with foldings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We extend the LDG method to deal with a piece- wise quadratic crease C and prove its convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The construction of a recovery sequence for one absolute minimizer y∗ ∈ [H2(Ω/C)∩W 1 ∞(Ω)]3 requires the slightly stronger assumption that y∗ is C1 in each subdomain created by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Fully linear solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We design a semi-implicit discrete gradient flow that treats Bh[yh] implicitly and Ch[yh] explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' This leads to linear problems at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' The scheme retains the key property of being energy diminishing and controls the isometry defect provided the fictitious time step τ satisfies a mild constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Sub-optimal discrete inf-sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' As in customary in the literature [4, 7, 8, 9, 13, 14, 17, 16], we rely on Lagrange multipliers to enforce the linearized isometry constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We prove a sub-optimal inf-sup condition for the resulting saddle-point system, which seems to be the first such result for these type of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' We present several insightful numerical experiments with large isometric deformations, including a climate responsive device and the folding of a plate across a quadratic crease that yields a bilayer origami as equilibrium shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' References [1] Alben, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=', Balakrisnan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=', and Smela, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Edge effects determine the direction of bilayer bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Nano letters 11, 6 (2011), 2280–2285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' [2] Ayachit, U.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=', Guignard, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=', Nochetto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=', and Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' LDG approximation of large deformations of prestrained plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQfZgSA/content/2301.03151v1.pdf'} +page_content=' Journal of Computational Physics 448 (2022), 110719.' metadata={'source': 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Griffin,4, 8, † and Adolfo G. Grushin2, ‡ +1Donostia International Physics Center, 20018 Donostia-San Sebastian, Spain +2Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France +3Department of Materials Science, University of California, Berkeley, California 94720, USA +4Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA +5Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden +6Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany +7Department of Physics, University of California, Berkeley, California 94720, USA +8Molecular Foundry Division, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA +(Dated: January 10, 2023) +While topological materials are not restricted to crystals, there is no efficient method to diagnose topology in +non-crystalline solids such as amorphous materials. Here we introduce the structural spillage, a new indicator +that predicts the unknown topological phase of a non-crystalline solid, which is compatible with first-principles +calculations. We illustrate its potential with tight-binding and first-principles calculations of amorphous bismuth, +predicting a bilayer to be a new topologically nontrivial material. Our work opens up the efficient prediction of +non-crystalline solids via first-principles and high-throughput searches. +Introduction-. +Predicting which solids host non-trivial +electronic topological phases is a central problem in con- +densed matter physics. For crystalline solids, first principles +methods take advantage of crystal symmetries to identify topo- +logical materials [1–5]. However, symmetry-based methods +cannot be applied to diagnose non-trivial topology in ma- +terials that lack translational invariance such as amorphous, +polycrystalline, and quasicrystalline materials. In fact, given +the far greater ubiquity of non-crystalline materials in con- +densed matter, solving this challenge would open up several +new material classes far more numerous than crystals, with +both fundamental interest for novel phenomena unique to non- +crystalline matter [6–36], and for their possible greater ease of +integration into devices [37, 38]. +Prior work on topology in non-crystalline materials used +convenient amorphous tight-binding models with average and +local symmetries [11, 14–16, 39], however these do not include +the full chemical and structural specificity found in real matter. +Similarly, real-space invariants [40–43], including Wannier- +based tight-binding formalism, require the system be treated +on a case-by-case basis and can be computationally costly. +To overcome this methodological problem, we introduce the +‘structural spillage’, which is inherently compatible with first- +principles approaches. +Since the characterization of topol- +ogy in general relies on the comparison with a known refer- +ence [44], we propose that in our case the appropriate compar- +ison is between the wavefunctions of the non-crystalline target +system and a crystalline reference state. A similar approach +was proposed to identify topological band inversions in crys- +tals by Liu and Vanderbilt [45] who compared the wavefunc- +tion overlap in crystals with and without spin-orbit coupling +(the ‘spin-orbit’ spillage). Inspired by this idea, we define the +structural spillage as a measure of the overlap between wave- +functions with different structural configurations. By com- +paring this structural spillage for crystals, whose topological +characterization can be efficiently calculated using standard +symmetry-based methods [1–5], with those of non-crystalline +solids, the topological characterization of the latter can be +determined (Fig. 1). +We first define the general formulation of structural spillage +and how it can be used to diagnose topology in non-crystalline +systems once a known reference phase is identified. We next +exemplify its potential by diagnosing topological phase transi- +tions in amorphous bismuth, a previously identified non-trivial +amorphous system, using both a tight-binding model and den- +sity functional theory (DFT). Our results indicate that the struc- +tural spillage can accurately identify amorphous bismuthene as +topologically non-trivial [13, 46], and predicts that amorphous +bilayer bismuth is a novel topological material. By definition, +the structural spillage is applicable to generic non-crystalline +materials. It is suitable to establish a high-throughput cata- +logue of potential non-crystalline topological materials, using +currently available DFT codes based on plane waves in our +current formalism. +Structural spillage-. +The total spillage γ measures the +mismatch between two projectors P and ˜P into occupied +states [45] +γ = 1 +2Tr +�� +P − ˜P +�2� += Tr +� +P(1 − ˜P) +� +, +(1) +where the trace acts on the entire Hilbert space, and the last +equality holds under the assumption that both systems have the +same total number of occupied states Nocc = Tr[P] = Tr[ ˜P]. +By definition, γ ≥ 0 and can be viewed as the variance between +two distributions with the same average. When P = ˜P the +spillage vanishes. However, when the overlap between the two +projectors is zero, it equals the total number of occupied states +Nocc. Therefore, γ acts as an indicator of band inversions +caused by the parameters that differ in P and ˜P [45]. +To predict topological band inversions in crystals, Liu and +arXiv:2301.02686v1 [cond-mat.mes-hall] 6 Jan 2023 + +2 +(a) +(b) +Topo. +Topo +Spillage +Trivial +Trivial +( ) +spillage +spin-orbit +structural +high +high +low +low +Figure 1. (a) The spillage γ is high or low depending on whether +a test wavefunction |ψ⟩ is in the same or different topological state +compared to a known reference wavefunction | ˜ψ⟩. (b) The spin-orbit +spillage [45] compares wavefunctions with and without SOC. The +structural spillage takes advantage of the knowledge of the topological +state of a crystalline solid to find the topological state of an amorphous +solid. +Vanderbilt [45] chose P and ˜P to be projectors onto the sub- +space of occupied states of crystalline insulators with and +without spin-orbit coupling (SOC), respectively. Lattice pe- +riodicity allows these to be written in Bloch momentum k as +P(k) = � +n∈occ |ψnk⟩⟨ψnk|, which defines a k-resolved spin- +orbit Bloch spillage, γB(k) = nocc − Tr[P(k) ˜P(k)], where +nocc = Nocc/Ncells is the number of occupied bands. The to- +tal spillage is recovered by summing over all momenta in the +Brillouin zone (BZ), γ = � +k γB(k). The spin-orbit Bloch +spillage γB(k) thus quantifies the band inversion caused by +SOC at each k; it is large at points in the BZ where the band +inversion is sizable. Ref. [45] showed that at certain points in +the BZ the spin-orbit Bloch spillage has to be larger than some +given value if the SOC induces a topologically non-trivial +phase from Wannier obstruction arguments. For instance, this +lower bound equals two for a time-reversal symmetric topo- +logical insulator. +From the above properties, γB(k) can be used to signal topo- +logical band inversions in crystals, and is straight-forward to +calculate using DFT [45]. Indeed, it has recently been applied +to high-throughput searches for topological crystals [47, 48]. +We note, however, that a large spillage is a necessary but not +sufficient condition for non-trivial topology: in certain cases, +e.g., when many bands close to the Fermi level are slightly +mixed by SOC, the spillage may be fooled by trivial insula- +tors [45]. Consequently, more recent searches for topological +crystals favor symmetry-based methods. +In most practical +cases, the spillage is expected to be an accurate indicator of +topology in crystals [45]. +In this work, we propose a spillage that compares an amor- +phous system with a crystalline counterpart. +In doing so, +we take advantage of the well-developed methods of sym- +metry indicators for the topological characterization of crys- +tals [2]. To this end, we now reformulate our spillage in a +plane-wave basis for incorporation into standard plane-wave +DFT codes. Moreover, it is also well defined for both crys- +talline and non-crystalline systems. We write the total spillage +γ in the plane wave basis |pα⟩, where p is the plane-wave +momentum (not necessarily restricted to the first BZ) and α +denotes spin. To calculate the spillage, we need the projector +onto occupied states of the amorphous and reference systems, +P = � +N∈occ |ψN⟩⟨ψN|, where |ψN⟩ are the eigenstates. By +projecting these onto plane waves, we then have access to +the projector matrix elements P αβ +p,p′ = ⟨pα| P |p′β⟩, which +are well-defined for crystalline and non-crystalline systems. +Any plane-wave momentum p can be uniquely decomposed as +p = k + G, the sum of a crystal momentum k in the first BZ +plus a reciprocal lattice vector G, both of the reference crystal. +Then, by substituting the plane-wave expansion into Eq. (1), +we can define the quasi-Bloch spillage as +γqB(k) = 1 +2 +� +k′ +� +GG′ +� +αβ +� +P αβ +k+G,k′+G′P βα +k′+G′,k+G − P αβ +k+G,k′+G′ ˜P βα +k′+G′,k+G +� ++ +� +P ↔ ˜P +� += +(2a) += 1 +2 +� +� +� +�� +Gα +P αα +k+G,k+G +� ++ ˜nocc(k) − +� +Gα +� +G′β +� +P αβ +k+G,k+G′ ˜P βα +k+G′,k+G + ˜P αβ +k+G,k+G′P βα +k+G′,k+G +� +� +� +� +(2b) +In Eq. (2b) we have used the fact that the reference projector +˜P corresponds to a crystal, which allows us to set k′ = k +in terms involving at least one ˜P, since there is no scattering +between different crystal momenta due to the discrete transla- +tional symmetry. Note that γqB(k) fulfills the same sum rule +as the Bloch spillage, γ = � +k γqB(k). Therefore, applied +to two insulating crystals, γqB(k) recovers the Bloch spillage. +Moreover, it can also be applied to semimetallic systems with +the advantage of it being bounded by zero, in contrast to recent +extensions to semimetallic materials [47, 48]. +Our key result is that the structural quasi-Bloch spillage, +defined by Eq. (2), can be used as an efficient topological +indicator in non-crystalline systems. Crucially, it can be effi- +ciently computed with plane-wave-based DFT methods, since +the projector matrix elements are an output of the calcula- +tion. Consequently, this method is suitable for high-throughput +identification of non-crystalline topological materials. + +3 +0 +1 +2 +λ/tσ +0.0 +0.2 +0.4 +0.6 +ρnon−hex +topological +trivial +γTB +qB (k = 0) +0.0 +0.5 +1.0 +1.5 +0 +1 +2 +λ/tσ +0.0 +0.2 +0.4 +0.6 +ρnon−hex +topological +trivial +conductance +� +e2 +h +� +0.0 +1.0 +2.0 +3.0 +4.0 +γTB +qB (k) +0.0 +0.5 +1.0 +1.5 +(a) +(b) +(c) +(d) +Figure 2. Structural spillage in the tight-binding approximation. (a) +Example of a real-space structure with a density of non-hexagonal +plaquettes ρnon-hex ≃ 0.53. +(b) Structural quasi-Bloch spillage +γTB +qB (k) in the BZ comparing topological amorphous bismuthene +with ρnon-hex ≃ 0.53 and λ = 0.22tσ with a trivial crystal with +λ/tσ = ∞. (c), (d) Phase diagrams as a function of SOC λ and the +density of non-hexagonal plaquettes ρnon-hex. (c) Conductance in the +“armchair” ribbon configuration (see SM [49] A 3). (d) Structural +quasi-Bloch spillage γTB +qB (k = 0) comparing the amorphous system +to a trivial crystal with λ/tσ = ∞. +Structural spillage in the tight-binding approximation-. +Defining a structural spillage that is useful in the tight-binding +approximation requires us to develop further Eq. (2). +The +reason is that two issues emerge as we define plane wave +states projected into the tight-binding Hilbert space of Nsites as +��pα⟩ = +1 +√Nsites +� +r eip·r��rα⟩, where r labels the position of +each site and α labels internal quantum numbers, such as spin +or the orbital type. First, because the tight-binding model’s +Hilbert space does not span the entire real space but only po- +sitions defined by the charge centers, our plane waves are non- +orthogonal. Therefore, their overlap depends on the atomic +positions, and therefore on the amount of structural disorder. +Since we expect continuous translational symmetry to be re- +covered after averaging over different disorder realizations, we +may solve this issue by neglecting the scattering between dif- +ferent momenta in Eq. (2), i.e. assuming that P αβ +p,p′ ∝ δp,p′. +This assumption has been successfully used to determine the +topology of non-crystalline systems using the effective Hamil- +tonian approach [14–16, 35]. +A second issue of the tight-binding approximation is that +the projected plane waves form an over-complete set. A well- +defined basis for a crystal with Ns/c sites per unit cell consist +of a subset with momenta in Ns/c Brillouin zones. However, +there are different types of Brillouin zones depending on the +phase factor eiG·t, where t are the relative positions of the sites +inside the unit cell [50]. For instance, in the honeycomb lattice +there are 3 types of BZ, since e−iG·t = eia2π/3, with a ∈ Z3 +(see Supplemental Material (SM) [49] C). This issue can be +handled by replacing the sum over reciprocal lattice vectors G +by an average over the different types of G, and multiplying +by Ns/c. +With these modifications, the structural spillage Eq. (2) can +be defined in the tight-binding approximation as +γTB +qB (k) = 1 +2 +Ns/c +NBZs +� +G∈BZs +tr +�� +Pk+G − ˜Pk+G +�2� +, +(3) +where the sum over G runs over one BZ of each of the NBZs +types, the trace acts over the internal degrees of freedom α, and +we have defined the single-momentum projector P αβ +p += P αβ +p,p. +Eqs. (3) and (2) define the structural spillage to be used in the +tight-binding approximation and first-principles calculations, +respectively. In the remainder of the paper, we demonstrate +how they capture topological phase transitions of amorphous +systems, using low-dimensional bismuth as an example. +Tight-binding benchmark: +bismuthene on a substrate-. +Crystalline bismuthene consists of a 2D honeycomb monolayer +of bismuth atoms. Experiments suggest it to be a quantum spin +Hall insulator with topological helical edge states when grown +on SiC(0001) [51] or Ag(111) [52] substrates. The effect of +the substrate is crucial: it filters the pz orbitals away from +the Fermi level leaving the px,y orbitals, resulting in a large +gap (∼ 0.67eV) and a non-zero strong Z2 topological index. +Moreover, amorphous bismuthene on a substrate is predicted +to remain topological via first-principles calculations [13, 46], +making it a convenient system to benchmark our proposed +structural spillage. +The low-energy physics of bismuthene is captured by a tight- +binding model with px,y orbitals in the honeycomb lattice, +coupled by nearest-neighbour hoppings tσ and tπ, a large on- +site SOC λ, and a substrate-induced Rashba SOC λR (which +we take proportional to λ) [51]. +To extend this model to +amorphous structures while preserving the short-range order +expected in amorphous systems [37], we use the voronization +of a pointset [8, 14] (see SM [49] A 1). When the pointset is tri- +angular, the voronization produces its dual honeycomb lattice. +By randomly displacing the triangular pointset according to a +characteristic length r, the voronization produces lattices with +threefold coordination, as the honeycomb lattice, but with a +finite density of non-hexagonal plaquettes (see Fig. 2(a)) [53]. +Therefore, r continuously controls how amorphous are our +lattices, allowing us to study the effect of structural disorder +on topological properties. In the following, we quantify how +amorphous our systems are by the (configuration-averaged) +density of non-hexagonal plaquettes ρnon-hex, which is in one- +to-one correspondence to the parameter r (see SM [49] A 1). +In Fig. 2 we present the topological phase diagram of amor- +phous bismuthene as a function of ρnon-hex and λ, benchmark- +ing γTB +qB (k) against the two-terminal conductance results. In +the crystalline limit (ρnon-hex = 0), the system starts as a Dirac +semimetal for vanishing λ, and a finite λ opens up a topological +gap, similarly to graphene [54]. Above a critical λ, where the + +4 +RDF +(c) +(a) Top view +Crystalline +Low disorder +High disorder +b +a +c +(b) Side view +c +b +a +Figure 3. Bismuth bilayer supercells used in DFT calculations. (a) +and (b) show in-plane and out of plane views of the supercell, re- +spectively. The colors indicate different degrees of disorder: crystal +(blue), low disorder (green) and high-disorder (orange). (c) Radial +distribution function (RDF) showing the statistics of the bond lengths +in the disordered bismuth bilayer and their deviations from the per- +fect crystal (vertical dashed lines). The disorder is sampled from a +Gaussian distribution with a standard deviation of 0.15 Å for the low +disorder and 0.30 Å for the high disorder. +gap closes at the Γ point, the system becomes a topologically +trivial insulator, adiabatically connected to the atomic limit in +which only the onsite SOC is non-zero. +Both the conductance (Fig. 2(c)) and the structural quasi- +Bloch spillage (Fig. 2(d)) capture the topological transition, +even at finite structural disorder (ρnon-hex ̸= 0). The conduc- +tance in the topological insulator phase is equal to 2e2/h, +originating from the helical edge states, while it reduces to +zero after the phase transition to the trivial insulator. Con- +comitantly, γTB +qB (k = 0) is large in the topological phase and +small in the trivial phase because we choose the reference sys- +tem to be a trivial crystal, only with non-zero onsite λ. Had +we chosen the topological state as reference, the magnitude +of the spillage in each phase would be inverted; see SM [49] +A 1. The critical λ at the transition for the crystal is correctly +predicted by γTB +qB (k = 0). In agreement with Refs. [13, 46], +we find that increasing disorder decreases the topological gap +and hence the critical λ. Nevertheless, the realistic value of +λ ≃ 0.22tσ [51] lies in the topological phase also in the amor- +phous case. +Lastly, Fig. 2(b) shows γTB +qB (k) for fixed λ = 0.22tσ and +ρnon-hex = 0.53. +γTB +qB (k) is peaked around k = 0 with a +value ∼ 1.5, reminiscent of the crystalline topological band +inversion occurring at the Γ point. +Structural spillage in DFT: free-standing Bi bilayer-. +To +show that Eq. (2) is well suited for high-throughput screening +of amorphous topological materials, we calculate the struc- +tural spillage from the output wavefunctions of first-principles +γqb +0 +2 +Low +disorder +High +disorder +Tight binding +DFT +a-SOC +x-SOC +x-noSOC +a-SOC +vs +vs +Figure 4. Structural quasi-Bloch spillage γqB(k) for the bismuth +bilayer. First row: comparison between an amorphous system with +SOC (a-SOC) and a crystalline system without SOC (x-noSOC). +Comparing an amorphous system without SOC with a crystalline +sample with SOC leads to similar results. Second row: comparison +between the amorphous and crystalline systems with SOC (a-SOC and +x-SOC, respectively). γqB(k) is high at k = 0 for the first row while +small for the second row, indicating that amorphous bismuth bilayer +is a topological insulator. The last column shows a comparison with +the tight-binding quasi-Bloch spillage γTB +qB (k) (see SM [49] A 2). +calculations (see full details in SM). We choose previously- +studied free-standing bismuth (111) bilayer as an example. +This 2D bismuth allotrope, whose crystalline phase consists of +a buckled honeycomb lattice with lattice constant a = 4.33 Å, +is also predicted to be a strong topological insulator crystal +with Z2 = 1 [55–58]. However, no prediction exists for its +amorphous counterpart. +To represent amorphous structures given the periodic +boundary conditions of the calculations, we create 5 × 5 × 1 +supercells comprising of 50 Bi atoms per bilayer. Their elec- +tronic structure is calculated for a single supercell momentum, +the center of the supercell BZ. Starting from a crystalline su- +percell, the structure is disordered by adding random displace- +ments in the x, y, and z directions, sampled from a Gaussian +distribution. +The structures and their corresponding radial +distribution functions are shown in Fig. 3. +To predict the topological phase of amorphous Bi bilayer +with SOC we compute Eq. (2) with plane-wave-based DFT +(see SM [49] B) to compare it with its crystalline counterpart +without and with SOC. When SOC is not included, and hence +when it is topologically trivial (Fig. 4, first row), γqB(k) is +peaked at k = 0, with γqB(k = 0) > 2. Increasing disorder +smooths γqB(k), yet it remains peaked at Γ with a value greater +than 2. In contrast, when we include SOC in calculations of +both the disordered Bi bilayer and the pristine crystal (Fig. 4, +second row) the spillage is always small. Both rows together +show that amorphous bismuth bilayer with SOC is in the same +topological state as the crystal with SOC, a strong topological +insulator crystal with Z2 = 1. +We have performed a similar analysis using a tight-binding + +5 +model for the amorphous Bi (111) bilayer (introduced in +SM [49] A 2). The results, displayed in the last column of +Fig. 4, show that for comparable disorder strengths γTB +qB (k) is +broader and its maximum value is smaller than γqB(k) in DFT. +It is thus apparent that, due to the approximations in the tight- +binding calculation of the spillage, which lacks information of +the real space extension of the orbitals, the spillage method is +more suitable for DFT, an advantageous feature compared to +other topological indicators available for non-crystalline sys- +tems. +Discussion-. +We have introduced the structural spillage +as an efficient method to signal non-crystalline topological +phases, compatible with tight-binding and ab-initio simula- +tions. We have used it to predict amorphous Bi bilayer as a +novel topological insulator. +As was the case for spin-orbit spillage in crystals, we expect +the structural spillage to signal a large fraction of promising +materials, but not to be infallible: if multiple band inversions +are introduced upon amorphization, the spillage might also be +artificially large. However, unlike for crystals, the spillage is +currently the only systematic, model-independent method that +is compatible with ab-initio calculations. Additionally, we ob- +serve that, for different disorder realizations, its fluctuations +are smaller compared to scattering methods like calculating +the conductance. It can also be applied to systems without a +spectral gap, where the effective Hamiltonian approach [35] +can fail [14]. Lastly, while Eq. (2) is general, the definition +of the spillage is relatively versatile and can accommodate +less standard cases. For example, when no crystalline coun- +terpart exists, one may define a plane-wave-resolved spillage +(see SM [49] D) by using Eq. (2a) without the sum over G, a +modification worth studying in the future. +The structural spillage establishes a clear road-map to con- +struct a high-throughput catalogue of non-crystalline (amor- +phous, polycrystalline, quasicrystalline) topological materials +by screening existing amorphous databases, or by scrutinizing +realistic structures obtained using existing ab-initio molecu- +lar dynamics packages [59]. This methodology may enable +for the first time the systematic prediction and discovery of a +potentially large number of amorphous materials that are cur- +rently inaccessible, suitable to develop affordable and scalable +topological devices. +Acknowledgements-. +We are grateful to S. Franca, F. de +Juan, J. Hannukainen, D. López-Cano, R. Queiroz, Q. Marsal, +A. Soluyanov, R. M. Martin, and J. Vinson for fruitful dis- +cussions and related collaborations. This work was partially +funded by the U.S. Department of Energy, Office of Science, +Office of Basic Energy Sciences, Materials Sciences and Engi- +neering Division under Contract No. DE-AC02-05-CH11231 +within the Nonequilibrium Magnetic Materials Program (MS- +MAG), specifically the work by P.C., F.H., and S.M.G. D.M.S. +is supported by an FPU predoctoral contract from Spanish +MCIU No. FPU19/03195. A.G.G. acknowledges financial +support from the European Research Council (ERC) Consol- +idator grant under grant agreement No. 101042707 (TOPO- +MORPH). D.V. was supported by the Swedish Research Coun- +cil (VR) and the Knut and Alice Wallenberg Foundation. +Computational resources were provided by the National En- +ergy Research Scientific Computing Center and the Molecular +Foundry, DOE Office of Science User Facilities supported by +the Office of Science, U.S. Department of Energy under Con- +tract No. DEAC02-05CH11231. 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Defining the structural spillage in the tight-binding approximation +16 +1. General remarks and motivation +16 +2. System with a single site per unit cell +17 +a. Setting the stage: crystalline system +17 +b. Spillage comparing two crystals +17 +c. Structural spillage comparing an amorphous system to a crystal +18 +3. System with several sites per unit cell +18 +a. Crystal: definitions and types of Brillouin zones +18 +b. Crystal: recovering the exact results using plane waves +19 +c. Comparing an amorphous system to a crystal using the structural spillage: no-scattering approximation +20 +d. Taking into account different types of Brillouin zones +21 +e. Structural spillage without scattering in the tight-binding approximation +21 +4. Phase transition criterion in the tight-binding approximation +23 +D. Absence of a corresponding crystal: spin-orbit plane-wave spillage +24 +Appendix A: Tight-binding models +This Appendix describes the method for generating the amorphous tight-binding models used in the maint text. We include as +well further calculation details and some additional discussion regarding the phase diagrams that one can obtain using different +reference systems of the structural spillage. +1. +Model for bismuthene on a substrate +This section describes how to generate the amorphous bismuthene structure and tight-binding Hamiltonian that we have used +to benchmark the structural spillage method in Fig. 2. +a. +Tight-binding Hamiltonian +Crystalline bismuthene consists of a 2D honeycomb monolayer of bismuth atoms [51]. An effective tight-binding of crystalline +bismuthene on a substrate was proposed by Ref. [51]. It consists of px and py orbitals in the honeycomb lattice, coupled by nearest- +neighbour hoppings, a large onsite SOC, and a substrate-induced Rashba SOC. In real space and in the basis {px↑, px↓, py↑, py↓}, + +8 +the Hamiltonian reads: +H = − 1 +2 +� +⟨ij⟩ +� +(tσ − tπ) τ0 + (tσ + tπ) +� +c(2) +ij τz + s(2) +ij τx +�� +σ0 + +� +i +[λτyσz] + ++ +� +⟨ij⟩ +i +� +λA +Rτ0 [sijσx − cijσy] + λE +R [(cijτx − sijτz) σx − (cijτz + sijτx) σy] +� +, +(A1) +where we have defined cij = cos(θij), sij = sin(θij), c(2) +ij += cos(2θij), and s(2) +ij += sin(2θij), with θij the angle between the +bond joining site i to site j and the x axis. τµ and σµ are the Pauli matrices acting on the orbital {px, py} and spin {↑, ↓} +degrees of freedom, respectively. tσ and tπ are the sigma and pi nearest-neighbour hoppings, λ is the onsite SOC, and λA +R and +λE +R are the orbital-independent and orbital-dependent Rashba SOC, respectively. As in Ref. [51], in this work we will assume +that λA +R = λE +R = λR. The values used in Ref. [51] are tσ ≃ 2.0eV, tπ ≃ 0.21eV ≃ 0.11tσ, λ ≃ 0.44eV ≃ 0.22tσ, and +λR ≃ 0.032eV ≃ 0.074λ. In our calculations, we will take tσ as the unit of energy, we will use the same value for tπ = 0.11tσ, +and we will vary both the onsite SOC λ as well as the Rashba SOC proportionally to the former, λR = 0.074λ. +The Hamiltonian (A1) can readily be applied to an amorphous lattice once we define which sites are nearest neighbours of each +other. In principle, it could be generalized to include a dependence on the distance in the hoppings, such as the Harrison law [60]. +However, we will consider fixed values for the hoppings, which can be a good approximation for covalently-bonded amorphous +solids, which usually display a rather narrow distribution of bond distances [37]. Moreover, this approximation enables us to +isolate the effect of structural disorder. +b. +Construction of amorphous structures +Covalently-bonded amorphous materials usually preserve local environments similar to the ones in the corresponding crystals, +since they are set by the strong covalent bonds. Therefore, most amorphous materials have average coordination numbers, bond +distances, bond angles, etc., which are centered around those of the crystal [37]. With this in mind, our amorphous models +preserve, for every site, the threefold coordination of the honeycomb lattice. This is achieved by applying the Voronoi method +similar to Ref. [14], but with a modification that enables us to control the degree of amorphization. +In particular, we first construct a pointset forming a triangular lattice with lattice constant a, whose points will be called seeds. +We then randomly displace the seeds from their initial positions following an exponential distribution with characteristic distance +r · a in the radial direction, and a uniform distribution in the angular direction. We thereafter compute their corresponding +Voronoi diagram, which is defined by the Voronoi cells, i.e., the regions consisting of all points closer to one seed point than to +any other. The vertices of such cells, called Voronoi vertices, form a threefold coordinated lattice with the edges of the Voronoi +cells corresponding to the nearest-neighbour bonds (only the vertices at the boundaries of the system have fewer than three +neighbours). +The lattices obtained in this way have large variances in the bond angle and bond length distributions, which might not be very +realistic. In order to reduce this artifact, we apply a simple iterative relaxation procedure. We select the threefold coordinated +sites one by one and displace them to the barycenter formed by their three nearest neighbours. We iterate this process until +convergence is reached, i.e., until the displacements are smaller than some small cutoff. This relaxation procedure tends to set +the bond angles as close as possible to the crystalline angle, 120◦. Finally, once the lattice is relaxed, we rescale the distances +so that the average nearest-neighbour distance is a/ +√ +3, which is the corresponding value in the crystalline honeycomb lattice. +Fig. S1(a) shows the resulting histograms of the relative positions of atoms for two amorphous structures with different disorder +strengths, r = 0.3 (top) and r = 0.5 (bottom). Both structures are isotropic at long distances, although for small disorder +the nanocrystalline domains (see for example Fig. 2(a) in the main text) give rise to broad nearest neighbour peaks around the +crystalline positions. For high disorder, the correlation hole for distances under a/ +√ +3 and an annular peak are visible. +The parameter r, characterizing the exponential distribution by which the seeds are displaced from the regular triangular lattice, +continuously controls the amorphousness of the resulting Voronoi lattice. Indeed, since the Voronoi diagram of a triangular +lattice is a honeycomb lattice, we recover the crystal in the r → 0 limit. Increasing r introduces non-hexagonal plaquettes in +the Voronoi lattice, at least until r ≳ 1, when the seed becomes completely random (since all the information from the initial +triangular seed is lost). This can be observed in Fig. S1(b), which shows that the configuration-averaged standard deviations of +the distributions of bond angles, bond distances, and plaquettes start to saturate at about r ≳ 0.6. +Structural disorder can be quantified by several properties. These include the standard deviations of the distributions of +nearest-neighbour distances, angles and plaquettes (normalized by the corresponding average values), as well as the density +of non-crystalline plaquettes (in our models, where the crystalline limit consists of a honeycomb lattice, the non-crystalline +plaquettes correspond to the non-hexagonal ones). In order to take into account the finite-size effects, for each parameter r, we +consider the configuration-average of these quantities over 100 realizations. + +9 +(a) +(b) +(c) +(d) +Figure S1. (a) Histograms of the relative positions of atoms for two amorphous structures with different disorder strengths, r = 0.3 (top) and +r = 0.5 (bottom). (b) Configuration-averaged structural quantities as a function of the parameter r controlling the amorphousness: standard +deviations (std) of the distributions of nearest neighbour bond angles, bond distances (both for the planar bismuthene as well as for the buckled +Bi bilayer), and plaquettes, as well as density of non-hexagonal plaquettes. For each disorder intensity r, the results have been averaged over +100 different realizations. (c) Distribution of the ratios of non-hexagonal plaquettes ρnon-hex obtained with 100 disorder realizations with fixed +disorder r = 0.3. (d) Distribution of plaquettes for a given disorder realization with r = 0.3 (corresponding to ρnon-hex ≃ 0.55). +As shown in Fig. S1(b), all these configuration-averaged quantities have the same qualitative dependence with the parameter r. +In particular, there exists a one-to-one correspondence between our control parameter r and any of these configuration-averaged +quantities. However, for particular disorder realizations in a finite system, there are fluctuations that make their relation to r +not one-to-one before performing the configuration average. This is illustrated by the distribution of ratios of non-hexagonal +plaquettes ρnon-hex shown in Fig. S1(c) for different realizations with fixed r = 0.3. Therefore, we have chosen to physically +characterize the amorphousness of a system by the configuration-averaged density of non-hexagonal plaquettes formed by the +nearest neighbour sites ρnon-hex. This measure could be generalized to other models whose crystalline limit consisted of lattices +other than the honeycomb. Finally, Fig. S1(d) shows an example distribution of plaquettes obtained for a particular disorder +realization with r = 0.3, which corresponds to ρnon-hex ≃ 0.55, while the configuration-average for this r corresponds to +ρnon-hex ≃ 0.53. +The above procedure generates structures with open boundary conditions, which is useful to compute e.g. the local density of +states at the edges or the longitudinal conductance once some leads have been attached. However, for spectral quantities such as +the spillage, we can reduce the possible finite-size effects by imposing periodic boundary conditions, or equivalently by putting +the system on a torus. An amorphous system might have a different number of atoms at opposite edges, so the periodic boundary +conditions cannot be imposed directly, but rather before computing the Voronoi tessellation, as described below. +Before explaining the procedure to impose the periodic boundary conditions, let us note that our periodic systems consist of +a rectangular supercell with sides Lx and Ly. In order for the periodic boundary conditions to be applicable to systems with +an arbitrary amount of structural disorder, including the crystalline limit, Lx and Ly are restricted to the values such that the +supercell is commensurate with the initial crystalline unit cell. In our models, where the crystalline limit is a honeycomb lattice, +the previous condition imposes that Lx = nxa and Ly = ny +√ +3a, where a is the lattice constant, and nx, ny are integer numbers. +Taking this into account, let us now describe the procedure to impose periodic boundary conditions on a system with an +arbitrary amount of disorder. First, we generate a triangular seed within the supercell x ∈ [0, Lx), y ∈ [0, Ly), and we disorder +choosing a finite value of r. Then, we repeat this initial seed in the eight nearest-neighbour supercells, i.e., we copy the seed +points displaced from their initial positions x to x + L = x + (nxLx, nyLy), with nx, ny ∈ {1, 0, −1}. Then, the Voronoi +tessellation of the whole system (composed by the nine supercells) is determined. This gives rise to a threefold coordinated +lattice with the following convenient feature: the supercell defined by the sites inside the region x ∈ [0, Lx), y ∈ [0, Ly) has the +same number of sites in opposite sides. Therefore, the periodic boundary conditions can be now applied to this supercell (all +the sites outside this supercell are discarded). Finally, we carry out the relaxation procedure of this supercell, being careful to +preserve the periodic boundary conditions. +To conclude this section, let us mention that we generate the systems with open boundary conditions starting from a system +with periodic boundary conditions, by first removing the bonds at the edges of the supercell and then removing the dangling sites. +This way, the bulk of the periodic structure where the spillage is computed is the same as the bulk of the open system where the +conductance is determined, which allows us to safely compare their predictions of the topological phase. + +2 +a +0 +-2 +2 +0 +2 +α/aaverage relative std +0.6 +plaquettes std +0.2 +2D angles std +0.4 +3D angles std +2D distances std +0.1 +0.2 +3D distances std +Pnon-hex +0.0 +0.0 +0.0 +0.2 +0.4 +0.6 +raverage = 0.53 +std = 0.02 +30 +probability density +20 +10 +0 +0.50 +0.52 +0.54 +0.56 +Pnon-hexaverage = 6.0 +std = 1.0 +0.4 +probability density +0.3 +0.2 +0.1 +0.0 +3 +4 +5 +6 +7 +8 +9 +10 +# sides of plaquettes2 +a +0 +9 +-2 +0 +210 +(a) +(b) +(c) +(d) +Figure S2. Phase diagrams of different quantities as a function of SOC λ and amorphousness ρnon-hex for the bismuthene model. (a) Density of +states at the Fermi level of the system with periodic boundary conditions. (b) Two-terminal longitudinal conductance in the “zigzag” ribbon. +(c) Structural quasi-Bloch spillage γTB +qB (k = 0) comparing the amorphous system with a topological bismuthene crystal with λ = 0.1tσ. (d) +Structural quasi-Bloch spillage γTB +qB (k = 0) comparing the amorphous system with SOC λ to the corresponding crystal with the same SOC λ. +c. +Additional results: density of states and structural spillage for different reference systems +In this section we discuss further different phase diagrams that may be obtained for the bismuthene tight-binding model and its +spillage in the tight-binding approximation. Fig. S2 shows phase diagrams for the density of states, conductance and structural +spillage corresponding to the same bismuthene structures as the ones presented in the main text in Fig. 2. In particular, Fig. S2(a) +shows that the density of states at the Fermi level increases with ρnon-hex when the SOC is such that the crystal is in the topological +phase (λ ≲ 1.3tσ). This is due to the band broadening due to the disorder, and also from the appearance of low-energy states +induced by a sublattice imbalance in a bipartite lattice [61]. At high disorder, this induces the band inversion that drives the +system from topological to trivial at a smaller SOC than in the crystal. +In order to show that the quantized conductance does not arise from disorder-robust trivial edge states present in one particular +crystalline direction, we display in Fig. S2(b) the longitudinal two-terminal conductance along the direction perpendicular to the +one displayed in the main Fig. 2 (the edges here would correspond to a zigzag ribbon in the crystalline case). As expected, both +conductances coincide, which is a signature of the topological helical edge states, which live at all the boundaries of the system. +Let us now explore how the structural spillage changes when we choose a topological reference system, as opposed to a trivial +reference system used in the main text, Fig. 2. Fig. S2(c) shows the structural quasi-Bloch spillage when the reference system is +a topological crystal with SOC λ = 0.1tσ. Contrary to the trivial reference case shown in the main in Fig. 2, now the spillage +is small in the topological phase and large in the trivial one, as expected from Fig. 1. Importantly, the transition is predicted at +approximately the same SOC irrespective of the reference system, which shows the robustness of the spillage. +Finally, in order to isolate the effect of the structural disorder on the topological band inversion from the effect of SOC, we +have also computed the structural quasi-Bloch spillage comparing each amorphous system with amorphousness ρnon-hex and SOC +λ to a reference crystal with the same SOC λ, shown in Fig. S2(d). This choice highlights the regions where disorder induces a +topological band inversion. For example, if the reference crystal is topological for a given λ, this spillage will have a large value +if the disorder induces a trivial state. Therefore, interpreting Fig. S2(d) requires knowledge of the topological phase of the crystal +at each λ. For λ ≲ 1.3tσ, the reference crystal is topological. Since the spillage is small for λ ≲ 1.1tσ, the amorphous system +is topological for λ ≲ 1.1tσ. However, at high disorder, the spillage becomes large between λ ≃ 1.1tσ and λ ≃ 1.3tσ, which +indicates that the disorder induces a trivial phase. Lastly, for λ ≳ 1.3tσ, the reference crystal is trivial, and the spillage is low, +indicating that the amorphous system is also trivial. +In conclusion, all phase diagrams Fig. S2 (b-d) agree qualitatively. The spillage is able to predict the topological phase +transition independent of the reference system. +2. +Model for free-standing bismuth (111) bilayer +In this section, we introduce a tight-binding model for the amorphous bismuth bilayer, for which we study the structural +spillage. After introducing the model and describing the method to generate the amorphous structures, we analyze its topological +phase diagram to further benchmark the structural spillage. Finally, we compare the tight-binding results and DFT calculations, +as shown in Fig. 4. We conclude that, while both qualitatively agree, the structural spillage method works better in DFT. + +DOS(EF)[a.u.] +0.6 +1000 +Pnon-hex +750 +0.4 +500 +0.2 +250 +0.0 +0 +0 +1 +2 +入/toconductance +h +4.0 +0.6 +3.0 +0.4 +2.0 +0.2 +1.0 +topological +trivial +0.0 +0.0 +0 +1 +2 +入/t。B(k = 0) +1.5 +0.6 +1.0 +Pnon-hex +0.4 +0.5 +0.2 +topological +trivial +0.0 +0.0 +0 +1 +2 +入/t。B(k = 0) +1.5 +0.6 +1.0 +Pnon-hex +0.4 +0.5 +0.2 +topological +trivial +0.0 +0.0 +0 +1 +2 +入/t。11 +a. +Tight-binding Hamiltonian +Crystalline bismuth (111) bilayer consists of a buckled honeycomb lattice of bismuth atoms, where each sublattice has a +different height [57]. An effective tight-binding of crystalline Bi bilayer was introduced by Ref. [62], where the three p orbitals +are relevant due to the absence of the substrate in this case. Their model consists of spinful px, py and pz orbitals in the buckled +honeycomb lattice with up to third nearest-neighbour hoppings. For simplicity, we will restrict ourselves to nearest-neighbour +hoppings and onsite SOC. In real space and in the basis {px↑, px↓, py↑, py↓, pz↑, pz↓}, the Hamiltonian reads: +H = +� +⟨ij⟩ +� +�tπτ0σ0 − (tσ + tπ) +� +� +(dij · ux)2 +(dij · ux) (dij · uy) (dij · ux) (dij · uz) +(dij · uy) (dij · ux) +(dij · uy)2 +(dij · uy) (dij · uz) +(dij · uz) (dij · ux) (dij · uz) (dij · uy) +(dij · uz)2 +� +� σ0 +� +� + ++ +� +i +� +�E0z +� +� +0 0 0 +0 0 0 +0 0 1 +� +� σ0 + λL · σ +� +� , +(A2) +where E0z is the difference between the onsite energy of the pz and px,y orbitals, dij is the unit vector along the bond from site +i to site j, and ua, a = x, y, z, are the unit vectors along the three cartesian axes. We have also defined the angular momentum +matrices La, which act on the orbital subspace {px, py, pz}: +Lx = +� +� +0 0 +0 +0 0 −i +0 i +0 +� +� +; +Ly = +� +� +0 +0 i +0 +0 0 +−i 0 0 +� +� +; +Lz = +� +� +0 −i 0 +i +0 +0 +0 +0 +0 +� +� . +(A3) +In our calculations, we will take tσ as the unit of energy, and fix the value of tπ = 0.25tσ and E0z = −0.4tσ. We vary the +onsite SOC λ. From the DFT-derived tight-binding model of Ref. [62], we can estimate that the actual SOC for the Bi bilayer is +λ ∼ 0.7tσ. The height of the bilayer enters via the vectors dij. Different DFT calculations have predicted heights ranging from +dz = 0.35a to dz = 0.40a [57, 58, 62, 63]. In this work, we will use dz = 0.9a/ +√ +6 ≃ 0.37a. +b. +Construction of amorphous structures +Our structures of amorphous Bi bilayers are constructed in a similar way to monolayer bismuthene. Indeed, the first step is +generating an amorphous bismuthene lattice following the procedure outlined in Appendix A 1 b. We then have to assign different +heights to the sites. In the crystalline limit, each sublattice has a different fixed height because of the buckling. Sublattices are +no longer well-defined in an amorphous lattice, but we can still define some effective sublattices. One differentiating property +between the two sublattices in a crystalline honeycomb lattice is the direction of their nearest-neighbour bonds: if the bonds +from sublattice A point at polar angles θA +1 = π/2, θA +2 = −11π/12 and θA +3 = −π/12, then the ones from sublattice B point +at θB +1 = −π/2, θB +2 = π/12 and θB +3 = 11π/12. Therefore, η(S) = sign +��� +l θS +l mod 2π +� +− π +� +is equal to +1 for sublattice +S = A and −1 for S = B. Using η(S) = ±1 to define the effective sublattices in the amorphous structures, we then assign +a height ±dz/2. Finally, we add some random disorder to the height of each site sampled from a Gaussian distribution with +standard deviation rz ·a. In particular, we choose the height disorder rz proportional to r, the parameter that controls the in-plane +amorphousness. In the calculations presented in this work, we take rz = rdz/(4a) ≃ 0.09r. Fig. S3(a) shows the top and side +views of a representative structure. +c. +Topological phase diagrams +In this section, we study the topological phase diagram of the amorphous Bi bilayer tight-binding model (A2), and show that, +as for Bimsuthene, the structural spillage correctly predicts the topological band inversion in this model. +Before analyzing the results, let us briefly review the current status regarding the topological characterization of crystalline Bi +(111) bilayer. In the crystalline case with SOC, the Bi bilayer has been predicted to be a strong topological insulator [55–58]. +Our model can also describe other materials with the same lattice, such as the antimony (111) bilayer. Due to the smaller SOC, +the Sb bilayer becomes a strong topological insulator only when strained [64]. Therefore, our model in the crystalline case starts +as a Z2 = 0 insulator for vanishing λ. A band inversion occurs at a finite value of λ, driving the system to a Z2 = 1 topological +insulating phase. For the parameters used in this work (see Appendix A 2 a), this band inversion in the crystal occurs at Γ for +λ ≃ 0.27tσ. + +12 +(c) +(e) +(d) +(b) +(a) +Figure S3. Bi bilayer tight-binding model structure and phase diagrams as a function of SOC λ and amorphousness ρnon-hex. (a) Top and side +views of an example structure for amorphousness ρnon-hex = 0.53 (r = 0.3). Sites are colored according to their out-of-plane positions: red/blue +indicates the effective sublattice, and the color intensity scales with the actual out-of-plane position. The positions in the out-of-plane direction +have been rescaled by a factor 10 for visualization purposes. (b) Momentum resolved tight-binding quasi-Bloch spillage for ρnon-hex = 0.53 +(r = 0.3) and SOC λ = 0.7tσ. These parameters are equal to those in Fig. 4, with a change in color to match that of (e). (c) Phase diagram +of the density of states at the Fermi level of the system with periodic boundary conditions. (d) Phase diagram of the two-terminal longitudinal +conductance in the “armchair” ribbon configuration. (e) Phase diagram of the structural quasi-Bloch spillage γTB +qB (k = 0) comparing the +amorphous system with SOC λ to a topological crystal with λ = tσ. +As shown in Fig. S3(b), the structural quasi-Bloch spillage γTB +qB (k) of the amorphous system with amorphousness ρnon-hex = +0.53 (r = 0.3) and SOC λ = 0.7tσ is maximum at k = 0, with a value > 0.75, when the reference system is a trivial crystal +with λ = 0. Per our topological criterion, explained in detailed in Appendix C 4, this indicates that there is still a band inversion +at k = 0 in the presence of disorder. +Let us now analyze the topological phase diagram of the amorphous Bi bilayer tight-binding model. Figs. S3(d) and (e) show +the conductance and the structural quasi-Bloch spillage, computed for a reference topological crystal with λ = tσ, respectively, +as a function of amorphousness, ρnon-hex, and SOC, λ. +Both phase diagrams show a transition from a trivial insulator at +λ ∼ 0.2 − 0.3tσ. +First, note that the conductance shows a metallic region around the transition, also in the crystalline case. This is an artifact +of the finite precision in computing the Fermi level with the kernel polynomial method, compounded with finite-size effects (see +Appendix A 3). These effects also broaden the otherwise sharp transition in the structural spillage at low disorder. We have +checked that this transition region is reduced upon increasing the kernel polynomial method precision and the system size. Note +that these issues only appear as one approaches the transition, where the gap is increasingly small. For further related details, +see also the discussion of Fig. S8 in Appendix C. +Let us now focus on the phases away from the transition. The trivial insulator phase at small λ, characterized by a vanishing +conductance and a large spillage (since the reference crystal is topological), survives with amorphousness up to slightly higher +λ than in the crystalline case. On the other hand, the topological insulator phase, indicated by a quantized 2e2/h conductance +and a small spillage, only survives for small disorder, and the system seems to become slightly metallic for higher disorder. This +metallic phase is further signaled by the finite density of states at the Fermi level shown in Fig. S3(c). Notice that, despite the +absence of Rashba SOC in this model, the onsite λ is already spin-non-conserving, and therefore a metallic phase can be the +ground state. Nevertheless, we cannot discard the possibility that the metallic conductance is arising from finite-size effects with +an Anderson localized bulk but with a localization length longer than the system sizes considered. A scaling study would be +needed to discern the nature of this metallic conductance, but this lies beyond the scope of this work. In any case, the spillage is +not specifically designed to capture such metallic feature, and it just indicates that the topological band inversion still (partially) + +B(k = 0) +1.5 +0.6 +1.0 +Pnon-hex +0.4 +0.5 +0.2 +trivial +topological +0.0 +0.0 +0.0 +0.5 +1.0 +入/t。B(k) +1.5 +1.0 +0.5 +0.0DOS(EF)[a.u.] +0.6 +4000 +Pnon-hex +0.4 +3000 +2000 +0.2 +1000 +0.0 +0 +0.0 +0.5 +1.0 +入/tconductance +h +4.0 +0.6 +3.0 +Pnon-hex +0.4 +2.0 +0.2 +1.0 +trivial +topological +0.0 +0.0 +0.0 +0.5 +1.0 +入/t。13 +occurs for high disorder. Nevertheless, the larger spillage at high disorder, where the disorder induces this potential metallic +phase starting from a topological state, provides a signature for the partial loss of this band inversion. This partial melting of the +band inversion is also compatible with the increasing density of states at the Fermi level shown in Fig. S3(c). +In summary, both conductance and spillage phase diagrams agree qualitatively and predict the topological phase transition. +Quantitative differences only arise in the metallic regions, where the band inversion is just partial. As for bismuthene, we have +also checked that the conductance with leads in the perpendicular direction and the spillages with other reference systems give +similar results. +d. +Comparison with DFT +In this section, we comment on the comparison of the results of the previous section with the DFT results presented in the +main text. In particular, let us compare the latter to the tight-binding results for the realistic SOC λ ≃ 0.7tσ. As shown in Fig. 4, +the structural spillage predicts a topological band inversion in the amorphous Bi bilayer in both DFT and tight binding. Both +methods also agree on the fact that, above a certain disorder, the spectral gap closes (see Figs. S3 and S5). Crucially, because +we are forced to neglect the momentum scattering in the tight-binding approximation (see Appendix C), the structural spillage +in DFT takes higher values and it is also less broad. Consequently, the structural spillage not only is a topological indicator +compatible with DFT, but it works better in DFT than in tight-binding modeling. +3. +Calculation details +This section describes in detail the methods used to solve the tight-binding models, and some related subtleties. +We use the Kwant software package [65] to generate the tight-binding Hamiltonians and perform the calculations. To be able +to treat larger system sizes, we apply the kernel polynomial method (KPM) [66] to estimate the density of states (DOS) and the +projector onto the occupied states. The projector is computed following the procedure of Ref. [67] and using plane waves as +initial KPM vectors, which allows us to calculate the projector matrix elements ⟨pα|P|pβ⟩. We use a KPM energy resolution of +0.01tσ (645 moments) for the bismuthene structures, and of 0.005tσ (887 moments) for the bilayer ones. The DOS is computed +by performing a KPM stochastic trace with 50 and 100 random vectors in the cases of bismuthene and bilayer, respectively. The +system sizes considered are 21a × 12 +√ +3a for the bismuthene case and 41a × 24 +√ +3a for the Bi bilayer one. Both the resolution +and the size of the Bi bilayer system are taken to be larger than those of bismuthene since the gap in the former case is smaller, +and therefore finite-size effects are larger. Additionally, our model for the Bi bilayer displays some trivial edge states that affect +the calculation of the Fermi level considerably. +The structural quasi-Bloch spillage is computed in the systems with periodic boundary conditions using Eq. (3), which reduces +to Eq. (C24) in our models, since the crystalline phase has a honeycomb lattice. On the other hand, the conductance is determined +with the Kwant software in the systems with open boundary conditions. In order to avoid possible artifacts arising from trivial +edge states in some particular termination, the conductance is calculated using leads in both x and y directions, such that in the +crystalline case the edges are zigzag and armchair, respectively. Since the aim of the conductance is to identify the insulating and +topological insulating regions, which have a quantized conductance of 0 and 2e2/h, respectively, regardless of the shape of the +leads, we use leads consisting of a 2D planar square lattice with nearest-neighbour hoppings such that their bandwidth is larger +than that of the system. These leads are attached to all the atoms on the corresponding edge of the system. Fig. S4 shows two +example configurations with the leads in the y (armchair) and x (zigzag) directions. +Our Bi bilayer models, display at low disorder some trivial edge states close to the Fermi level over a wide range of values +of SOC, which appear in both zigzag and armchair edges. These change the Fermi level of a finite system with open boundary +conditions Eopen +F +with respect to the one computed with periodic boundary conditions Eperiodic +F +. For the system sizes we are able +to treat numerically the change in the Fermi level Eopen +F +is enough for it to lie outside of the bulk gap, since the thermodynamic +gap in the crystal is rather small (∼ 0.1tσ). Therefore, the conductance computed at Eopen +F +in the crystal would show metallic +regions even in the insulating and topological insulating phases due to this artifact. In order to avoid this issue, in the Bi bilayer +systems we compute the conductance at Eperiodic +F +determined with periodic boundary conditions. We note that this problem does +not appear in the bismuthene models. It is also worth highlighting that the metallic phase observed at large SOC and disorder is +not an artifact (see Appendix A 2), since we observe that the trivial edge states merge into bulk states in this region and therefore +Eperiodic +F +≃ Eopen +F +. +Lastly, to compute the phase diagrams we only need a single disorder realization for each r. The reason is twofold. First, we +noticed that for sufficiently large systems sizes, as the ones considered in this work, the fluctuations of the structural spillage +for different disorder realizations are rather small. Indeed, they are smaller than the fluctuations in the conductance, which is +another convenient feature for the use of the structural spillage in high-throughput searches for topological amorphous materials. + +14 +(a) +(b) +Figure S4. Examples of Bi bilayer systems with leads where conductance is calculated. (a) Top and side views of a system with leads in the +x axis, which would correspond to a zigzag ribbon in the crystalline case. (b) Top and side views of a system with leads in the y axis, which +would correspond to an armchair ribbon in the crystalline case. +Second, while extracting a precise topological phase diagram from the conductance would require a configuration average, it is +not strictly necessary if we just aim to use it as a benchmark for the structural spillage. +Appendix B: DFT calculation details +We performed Density Functional Theory (DFT) calculations using the projector augmented wave (PAW) formalism in the +Vienna ab-initio Simulation Package (VASP) [68, 69]. The exchange-correlation potentials were treated within the generalized +gradient approximation (GGA) of Perdew-Burke-Ernzerbof (PBE) [70]. The wavefunctions were expanded in plane waves to an +energy cutoff of 700 eV. SOC was added self-consistently for all calculations in which it was used. For supercell calculations, we +performed Gamma point only calculations. For self-consistent calculations of the unit cell, we used a k-point grid of 21x21x1 +with Gamma for the BZ sampling. We then sampled the 25 k-points ( n1 +N1 b1 + n2 +N2 b2) that would backfold to Gamma in the +5x5x1 supercell. To compare the same momenta between the unit cell and the supercell, the two must be commensurate and the +supercell lattice vectors must be multiples of the unit cell lattice vectors. If this were not the case, one could linearly interpolate +the coefficients of the supercell wavefunctions at the appropriate momenta from the closest supercell reciprocal lattice vectors. +Unlike in the tight-binding approximation, the structural spillage of Eq. (2) can be directly implemented in DFT. Here, the +overlap between two systems is well-defined irrespective of them having atoms at different positions. However, strictly speaking, +the continuous set of plane waves is always overcomplete in any numerical scheme. Nevertheless, the structural spillage of Eq. (2) +is still well-defined in DFT implemented with both a plane-wave or a localized basis. On the one hand, plane-wave-based DFT +codes feature discretized momenta (imposed by the periodic boundary conditions of the supercell) and a high-momentum cutoff. +These features do not constitute any fundamental problem for comparing two systems with different atomic structures, as long as +one has access to (or can interpolate) the information at the same momenta in both systems. On the other hand, implementations +of DFT with a localized basis, such as Gaussian or hydrogenic orbitals, do not directly output the information in plane-wave +momentum space. However, knowing the shape of the orbitals, a Fourier transform gives access to it, and no problem appears +regardless of the atomic structure. +To calculate the structural spillage in DFT using Eq. (2), we extract the projector matrix elements on an orthonormal plane wave +basis. The pseudo-wavefunctions generated with VASP are orthonormal with respect to an overlap operator [71]. Therefore, by +using the PAW approach, we perform a transformation to an orthonormal basis that spans the same space as the full wavefunctions. +Future improvements could use norm-conserving pseudopotentials, reconstructed full wavefunctions, or all-electron approaches. +Besides imposing this orthonormality, we rearrange the wavefunction coefficient arrays of the amorphous supercell so that we +compare the same momenta between both the amorphous supercell and the crystalline unit cell. +To corroborate that the spillage Eq. (2) is correctly implemented, we compared a crystalline supercell to a crystalline unit cell, +which should recover the exact Bloch spillage. In particular, we considered crystalline Bi2Se3 as well as crystalline BiTeI, and + +15 +SOC +No SOC +Crystal +Low-disorder +High-disorder +Figure S5. Orbital-resolved density of states (DOS) of the Bi (111) bilayer calculated with DFT, showing the contributions of the Bi p +orbitals near the Fermi level (indicated by a vertical dashed line). First row: DOS without SOC. Second row: DOS with SOC. Each column +corresponds to a different structure: crystal in the the first column, low-disorder structure (standard deviation of 0.15Å) in the second column, +and high-disorder system (standard deviation of 0.30Å) in the third column (see Fig. 3 in the main text for a real space view of these lattice +structures). SOC drives a band inversion that occupies the pz orbital and empties the px,y orbitals. +our method accurately diagnosed the band inversion in both systems. In crystalline Bi2Se3 a band inversion at Gamma leads to +a topological insulator phase which results in a spillage value of 2.12 [45]. When comparing the crystalline Bi2Se3 supercell to +the unit cell we obtain a spillage of 2.09 which exactly matches the result given by pymatgen [72]. For the case of disordered +BiTeI, previous work showed that small amounts of disorder in the atomic positions cause the system to undergo a topological +phase transitions from a trivial insulator (crystal) to a topological insulator (disordered) as a result of an induced band inversion +[24]. This is caused by the modified crystal field of the orbitals near the Fermi level which pushes these states closer together +when disordered. In the latter case, all point group symmetries are broken but translational symmetry is still present. In this +case, we find a spillage value of 5.17 at the A point where the band inversion occurss, and values of 3.03 at other BZ points +indicating there is a larger orbital spillage throughout the BZ. The method still captures the topological band inversion in this +case and exactly matches the results given by pymatgen. +Finally, let us comment further on the results obtained for the Bi (111) bilayer. The disordered structures, shown in Fig. 3, +are obtained by randomly displacing the atoms from their high-symmetry crystal positions following a Gaussian distribution. +We choose the standard deviations to be 0.15Å and 0.30Å for the low and high disorder systems, respectively. For standard +deviations of 0.15Å the deviation from equilibrium position is small which preserves the bulk electronic gap while demonstrating +our method works in the presence of disorder. Standard deviations of 0.30Å lead to an average atomic displacement of 0.41Å +which is similar to atomic displacements seen in topological materials in the presence of disorder [24]. The structural spillage, +shown in Figs. 3 and S6, demonstrate that SOC drives a band inversion at the Gamma point with the result that all the crystalline +and the disordered structures are topologically non-trivial. This band inversion is confirmed by the density of states of Fig. S5, +which further illustrates that the band inversion occurs between the pz and the px,y orbitals. Indeed, the crystal and the amorphous +systems display an increased occupation of the pz orbital after SOC is included. Additionally, Fig. S5 illustrates that the Bi bilayer +becomes metallic for sufficiently high structural disorder, in agreement to the tight-binding model (see section A 2). However, +studying whether the amorphous system is extended or localized for strong disorder lies beyond the scope of this work. + +16 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Figure S6. Calculated structural spillage of the crystalline Bi bilayer from DFT. The value of 2 at the Gamma point indicates that the crystalline +Bi bilayer with SOC is topological. +Appendix C: Defining the structural spillage in the tight-binding approximation +1. +General remarks and motivation +In the main text we use the tight-binding spillage as a benchmark, and argue that the structural spillage is most useful within +DFT calculations. For completeness, in this appendix we give a pedagogical justification of Eq. (3) for computing the structural +quasi-Bloch spillage in the tight-binding approximation. It is aimed to aid future studies in understanding the approximations +that go into applying the structural spillage to tight-binding models, as alternative to topological markers. Thus it can be skipped +by readers only interested in applying Eq. 2. +Let us first highlight the problem of applying the general formulation of the structural spillage of Eq. (2) in the tight- +binding approximation. By tight-binding approximation we refer to the phenomenological tight-binding models where the only +information about the wavefunctions is the position of their Wannier charge centers (and possibly their transformation properties +under symmetries), but their spatial structure is unknown and therefore considered to be a Dirac delta. An implicit assumption +of Eq. (2) is that the Hilbert space of the system is the whole real space (in addition to the spin space), in which the plane +waves constitute an orthonormal basis. While this is applicable in DFT (see Appendix B), it is not true in the tight-binding +approximation, where the Hilbert space is just spanned by the positions of the Wannier charge centers (with the internal degrees +of freedom of spin and orbital type). The fundamental problem for comparing two tight-binding systems with different lattice +structures, as done by the structural spillage, stems from the fact that their Hilbert spaces are different, and therefore their overlap +is ill-defined. When projected to the tight-binding Hilbert space, the plane waves constitute a non-orthogonal and overcomplete +set. The overlap between these projected plane waves depends on the lattice structure, and therefore the usual formalism of +non-orthogonal bases (see e.g. [73]) cannot be applied. +However, by using the plane waves and the approximations described in this Appendix, one can derive a physically motivated +expression for the structural spillage in the tight-binding approximation, Eq. (3). The line of the argument for solving this +problem works as follows. The structural spillage (2) contains the matrix elements of the products of two projectors in the plane +wave basis. By neglecting the momentum scattering, i.e., by assuming that these operators are diagonal in momentum space, +the fundamental problem of the disorder-dependent plane-wave overlaps is circumvented. However, this introduces some new +issues. To bypass these, we choose the solution which, in the crystalline limit, gives results closer to the exact ones. Our solution +gives the exact results for the quantities containing matrix elements of just one projector. In the case of the structural spillage, +which contains matrix elements of the product of two projectors, our results in the crystalline limit are not exact. However, we +argue and numerically show for selected models that the results are similar in absolute value, and more importantly that the sharp +changes in the spillage that signal topological transitions still show up. +In order to separately understand the different issues that appear in the tight-binding, let us first consider the simple case of +a system whose corresponding crystalline limit has a single site per unit cell, where the majority of problems suffered by the +structural spillage in the tight binding do not appear. Then, we will analyze the general multi-site case. + +17 +2. +System with a single site per unit cell +a. +Setting the stage: crystalline system +Consider a crystalline tight-binding system with Ncells unit cells and one site per unit cell, i.e., only one Wyckoff position +with multiplicity one is occupied by an atom, Ns/c = 1. Therefore, the number of sites is the same as the number of cells, +Nsites = Ncells. The number of internal degrees of freedom (orbitals and spins) at each site does not influence the discussion +below, so we omit this internal index for simplicity in the notation. In the tight-binding approximation, Wannier functions are +unknown in real space, and therefore considered to be Dirac delta distributions, i.e., the Wannier function |φR⟩ at the lattice site +R has wavefunction: +φR(r) = ⟨r|φR⟩ = δ(r − R). +(C1) +We will always assume that the Wannier functions are orthonormal: +⟨φR′|φR⟩ = δR,R′. +(C2) +The plane wave with momentum p projected to the tight-binding Hilbert space is a state with a phase p · R at the site R, and +normalized in the total volume of the system. Then, the Wannier functions in the plane wave basis read: +φR(p) = ⟨p|φR⟩ = +1 +√Nsites +e−ip·R. +(C3) +Moreover, the Bloch states defined at crystal momentum k in the first BZ are: +|φk⟩ = +1 +√Ncells +� +R +eik·R|φR⟩, +(C4) +The overlap between the Bloch states and the plane waves is thus: +⟨p|φk⟩ = +1 +Nsites +� +R +ei(k−p)·R = +� +G +δp,k+G, +(C5) +where G are the reciprocal lattice vectors, i.e., G · R/2π ∈ Z. Therefore, all the BZs are exactly equivalent in a crystalline +one-atom tight-binding, since +⟨k + G|φk⟩ = 1 +(C6) +does not depend on G. In other words, ⟨p|p + G⟩ = 1 for the crystal, i.e., both plane waves are projected to the same state, +which is exactly the Bloch state at k too. +Finally, as a side remark, it is worth mentioning that even if there is a single site per unit cell, the BZs of a crystal are no longer +equivalent if the orbitals have a finite spread in real space. Indeed, in this case, the overlap between the Bloch state and the plane +waves is: +⟨k + G|φk⟩ = +1 +Ncells +� +R +eik·R⟨k + G|φR⟩ = +1 +Ncells +� +R +e−iG·R⟨k + G|φ0⟩ = φ0(k + G), +(C7) +where φ0(k + G) is the Fourier transform of the orbital located at the origin, which is generically not constant. +b. +Spillage comparing two crystals +Let us remember that plane waves are an overcomplete set in the tight-binding Hilbert space. In this single-site case, the +Hilbert space dimension is Nsites, which is the number of linearly independent plane waves needed for a basis. One possible +choice is selecting all the Ncells = Nsites momenta in one BZ (e.g. the first BZ). These are linearly independent and orthogonal +in the crystalline case (and also for an amorphous structure in the infinite size limit). Therefore, this choice constitutes an +orthonormal basis. Therefore, in this basis we can directly apply Eq. (2b) for the spillage, choosing to compare two crystals, +with the particularity that the sums over reciprocal lattice vectors G disappear since there is only one in the basis. The key +difference from the general multi-site case is that observables are the same irrespective of the BZ where the momenta for the +basis are chosen, i.e., irrespective of the G chosen in the basis. Moreover, thanks to the equivalence between plane waves and +Bloch states in this single-site case, observables projected to a plane wave p are equal to the crystalline quantities computed at +Bloch momentum k = p mod G. In particular, the quasi-Bloch spillage (2), which is equal to the Bloch spillage because we are +comparing two crystals, is also equal to the quasi-Bloch spillage without scattering (3) in this crystalline one-site case. + +18 +c. +Structural spillage comparing an amorphous system to a crystal +The previous basis choice is also orthonormal for an amorphous system in the infinite-size limit. Consequently, unlike in +the multi-site case that will be analyzed in the next section, the issue of the overlap between plane waves being different for +the amorphous and crystalline systems does not appear. Therefore, the structural quasi-Bloch spillage including scattering of +Eq. (2) can also be applied for comparing the amorphous structure with a crystalline one in this single-site tight-binding case +(again the sums over reciprocal lattice vectors G drop out in this single-site case). As mentioned in the previous section, when +comparing two crystals with a single site per unit cell, the quasi-Bloch spillage including scattering of Eq. (2) coincides with the +one without scattering of Eq. (3). This is no longer true when comparing an amorphous structure to a crystal, since the scattering +resummation over k′ in the amorphous projector, which is carried out in Eq. (2), is neglected in Eq. (3). +Now, although the structural quasi-Bloch spillage including scattering of Eq. (2) could in principle be applied, this would entail +a high computational cost. Indeed, other methods to indicate the topology in the tight-binding would be equally efficient (such +as the local topological markers [40–43]), questioning the usefulness of the structural spillage applied to a tight-binding model. +Therefore, to implement efficiently the structural spillage, we assume the no-scattering approximation of Eq. (3). Because we +neglect the scattering resummation over k′, the structural spillage of Eq. (3) becomes much more computationally efficient. +However, an important inconvenience arising from neglecting the scattering is that the spillage depends on the BZ where the +momenta for the plane wave basis are chosen. This is because momenta from different crystalline BZs will no longer lead to +equivalent results in the amorphous system, unlike in the single-site crystal. In fact, |p + G⟩ and |p⟩ no longer project to the +same state (⟨p|p + G⟩ = 0 for the amorphous case in the infinite size limit), and the quantities projected in |p + G⟩ differ from +those projected onto |p⟩. +This problem raises the question of how to compute correctly the structural spillage in the no-scattering approximation +between an amorphous material and a crystal, even in this single-site case. Although there is no unique answer, we now provide +a justification for using momenta just in the first BZ. The tight binding has no information about the spatial extent of the orbitals, +although we know that they are exponentially localized around the atom. Therefore, the tight-binding approximation captures +well long-distance physics, but there is a short-distance-cutoff below which the tight-binding results are no longer reliable. It is +reasonable to assume that this cutoff is of the order of the nearest-neighbour distance rnn, which coincides with the lattice constant +a in the crystalline single-site tight-binding. Therefore, only plane-wave momenta below ∼ 2π/a are reliable. Consequently, the +quasi-Bloch spillage computed just with plane-wave momenta in the first BZ is a sensible option (optionally, one could average +over the first BZ and second BZs). Considering just the first BZ, the structural quasi-Bloch spillage without scattering reads +γsingle-site-TB +qB +(k) = 1 +2tr +�� +Pk − ˜Pk +�2� +, +(C8) +which is just Eq. (3) in the single-site case because, as mentioned before, all BZs are equivalent in the crystal, and therefore there +is a single type of BZ, NBZs = 1. +3. +System with several sites per unit cell +In this section, we will show that if there are more than one site in the unit cell, then a phase factor depending on the relative +positions of the sites appears in the observables. Unlike in the single-site case, this leads to some BZs being inequivalent in the +crystal, requiring us to upgrade the single-site structural spillage Eq. (C8). +a. +Crystal: definitions and types of Brillouin zones +Consider a crystal with Ncells unit cells at positions R and Ns/c sites per unit cell at positions tA with respect to the center of +the cell R, so that the total number of sites is Nsites = Ncells · Ns/c. The Bloch states with a definite sublattice are, therefore: +|φA +k ⟩ = +1 +√Ncells +� +R +eik·(R+tA)|φA +R⟩. +(C9) +The projection of the Wannier functions onto plane-waves reads: +φA +R(p) = ⟨p|φA +R⟩ = +1 +√Nsites +e−ip·(R+tA). +(C10) + +19 +1 +e +i2 /3 +ei2 /3 +e +i2 /3 +ei2 /3 +e +i2 /3 +ei2 /3 +Figure S7. BZ types for the honeycomb lattice. Colors are different for each type. Red corresponds to a = 0 mod 3, and therefore a phase +e−iG·tAB = eia2π/3 = 1. Blue represents a = 1 mod 3, i.e., a phase ei2π/3. Finally, green refers to a = 2 mod 3, i.e., a phase e−i2π/3. +Therefore, the overlap between the Bloch states and the plane waves is: +⟨k + G|φA +k ⟩ = +1 +√Ns/c +e−iG·tA. +(C11) +However, the band eigenvectors are combinations of these Bloch states in different sublattices: +|ψn +k⟩ = +� +A +cnA +k |φA +k ⟩, +(C12) +and, therefore, their overlap with the plane waves reads: +⟨k + G|ψn +k⟩ = +1 +√Ns/c +� +A +cnA +k e−iG·tA, +(C13) +Let us now show that observables projected to a plane wave with momentum p = k+G depend on the phase factors e−iG·tAB, +where tAB = tA −tB are the relative positions of the different sublattices. For concreteness, let us start considering the simplest +observable, that will be a building block for e.g. the spillage: the projector onto band n at crystal momentum k, P n +k = +��ψn +k⟩⟨ψn +k +��: +⟨k + G +��P n +k +��k + G⟩ = +��⟨k + G +��ψn +k⟩ +��2 = +1 +Ns/c +� +A,B +cnA +k +� +cnB +tk +�∗ e−iG·tAB = +1 +Ns/c +� +�1 + +� +A̸=B +cnA +k +� +cnB +k +�∗ e−iG·tAB +� +� , (C14) +which is different from tr [P n +k ] = 1 in general. These phase factors, which depend on G, lead to at least some BZs being +inequivalent even if the orbitals are still Dirac deltas. Therefore, the types of BZs in the multi-site crystal can be classified by +the set of phase factors +� +e−iG·tAB� +. In general, some BZs become inequivalent whenever there is structure inside the unit cell, +irrespective of whether it comes from spatially-extended orbitals or from several sites. +As an example, consider the honeycomb lattice, where there are Ns/c = 2 sublattices A and B such that tAB = −a +� +0, 1/ +√ +3 +� +. +The reciprocal lattice basis vectors are G1 = 4π/ +√ +3a +�√ +3/2, 1/2 +� +, and G2 = 4π/ +√ +3a [0, 1]. A general reciprocal lattice +vector G = n1G1 + n2G2, with n1, n2 ∈ Z, satisfies G · tAB = −4π/3(2n2 + n1) = 2π/3 · 2(2n2 + n1). Therefore, +e−iG·tAB = eia2π/3, with a ∈ Z3, so there are NBZs = 3 different types of BZs depending on the value of this phase factor. If +we consider all possible momenta, from zero to infinity, then the multiplicity in momentum space of each type of BZ is the same. +On the other hand, if we only consider momenta up to a cutoff pmax, then the multiplicity in momentum space of each type of +BZ can be different. Fig. S7 shows the type of the first BZ and the six nearest-neighbour second BZs. Note that the first BZ has +G = 0, and therefore it is always characterized by a = 0, i.e., by a phase e−iG·tAB = eia2π/3 = 1. +b. +Crystal: recovering the exact results using plane waves +We now ask the question of how to recover the exact values of the observables in the crystalline tight binding, this time using +the plane waves. We also keep in mind that we want to later extend our definitions to the amorphous case. +First, we have to choose a basis of plane waves for this crystalline multi-site case. The tight-binding Hilbert space has +dimension Nsites = Ns/c · Ncells. Therefore, a possibility is to select Ncells plane waves in Ns/c inequivalent BZs. Decomposing + +20 +the plane-wave momenta as p = k + G, we find that plane waves with different k are orthogonal. However, in contrast to the +single-site case, plane waves with the same k but differing in a reciprocal lattice vector G are generically neither orthogonal nor +equivalent in the crystalline case. It is only when the differing reciprocal lattice vector G verify +� +e−iG·tAB� += {1}, i.e., when +the BZs are equivalent, that the projected plane waves are equivalent states. +For instance, in the honeycomb lattice, where Ns/c = 2, we can choose the basis in the first BZ (G0 = 0) and in the +G1 = 4π/ +√ +3a(0, 1) BZ. In this example, the overlap between plane waves is |⟨k + G0|k + G1⟩| = |⟨k|k + G1⟩| = 0.5. +Therefore, we have to use the formalism of non-orthogonal bases (see, e.g., Ref. [73]) and properly modify the quasi-Bloch +spillage of Eq. (2a). Within this formalism, the closure relation reads: +1 = +� +k +� +GG′ +��k + G⟩ +� +S−1� +G,G′ ⟨k + G′��, +(C15) +where the overlap matrix is defined as SG,G′ = ⟨k + G +��k + G′⟩, which depends only on the difference G′ − G. Also, the +sums over the reciprocal lattice vectors G run over the Ns/c BZs chosen in the basis. In the previous example of the honeycomb +lattice, they would run over G0 = 0 and G1 = 4π/ +√ +3a(0, 1). Using this expression for the closure relation, we can derive the +expressions for the observables in this non-orthogonal plane-wave basis. For example, the trace of the projector onto band n at +crystal momentum k, tr [P n +k ], becomes +tr [P n +k ]non-orth = +� +GG′ +⟨k + G +��P n +k +��k + G′⟩ +� +S−1� +G′,G , +(C16) +Importantly, Eq. (C16) recovers the expected crystalline value tr [P n +k ] = 1, irrespective of the chosen plane-wave basis. +Furthermore, in this non-orthogonal basis, the quasi-Bloch spillage is given by the appropriate generalization of Eq. (2a): +γnon-orth +qB +(k) = 1 +2 +� +k′ +� +G1G2G3G4 +� +αβ +� +P αβ +k+G1,k′+G2 +� +S−1� +G2,G3 P βα +k′+G3,k+G4 +� +S−1� +G4,G1 − +−P αβ +k+G1,k′+G2 +� +S−1� +G2,G3 ˜P βα +k′+G3,k+G4 +� +S−1� +G4,G1 +� ++ +� +P ↔ ˜P +� +. +(C17) +Crucially, when comparing two crystals, Eq. (C17) exactly recovers the Bloch spillage, regardless of the plane wave basis chosen. +c. +Comparing an amorphous system to a crystal using the structural spillage: no-scattering approximation +Let us now try to compute the structural spillage between a crystalline and an amorphous structure. Aside from the issues +already discussed for the single-site case, here is where comparing two tight bindings with sites at different positions becomes +problematic. The reason is that overlap between the plane waves is different in the crystal and in the amorphous cases. In the +crystal, as discussed in section C 3 a, some plane waves +��p + G⟩ are different states from +��p⟩, yet their overlap is non-zero, +⟨p +��p + G⟩ ̸= 0. In the amorphous system, in the limit of infinite size, all plane waves are inequivalent (as in the single-site +case), and more significantly, they are orthogonal. In the structural spillage of Eq. (C17), the crystalline and the amorphous +projector appear sandwiched between the overlap matrices, but this overlap depends on the system. Therefore, we cannot apply +the previous non-orthogonal formalism. +As explained in the main text, this issue can be avoided by neglecting the momentum scattering, i.e., by setting k′ = k and +G′ = G in Eq. (2a). Such approximation has been used previously to determine the topology of an amorphous system using +other methods such as the effective Hamiltonian approach [14, 35]. It is also inspired by the fact that continuous translational +symmetry is recovered after averaging over different disorder realizations. +Let us now write the expressions for the projector and the spillage within this approximation. On the one hand, the trace of the +projector into band n at crystal momentum k simplifies to: +tr [P n +k ]no scatt = +� +G +⟨k + G +��P n +k +��k + G⟩, +(C18) +where the sums over the reciprocal lattice vectors G again run over the Ns/c BZs chosen in the plane wave basis. On the other +hand, the corresponding expression for the structural quasi-Bloch spillage without scattering, which is obtained by setting k′ = k +and G′ = G in Eq. (2a), reads: +γno scatt +qB +(k) = 1 +2 +� +G +tr +�� +Pk+G − ˜Pk+G +�2� +, +(C19) + +21 +where the trace acts over the internal degrees of freedom α, and, as in the main text, P αβ +p += ⟨p|P|p⟩. Eq. (C19) is not yet the +definite expression of Eq. (3) for the structural spillage in the tight-binding approximation, since it still suffers from a problem +that we detail below. +d. +Taking into account different types of Brillouin zones +In contrast to the single-site case, the values of the observables computed within this no-scattering approximation depend on +the BZs chosen in the basis even in the crystal. The reason is the presence of different types of BZs (see Appendix C 3 a). In this +section, we will provide a method to circumvent this issue based on the condition that, when applied to crystals, it leads to values +as close as possible to the exact crystalline values, where rigorous proofs exist [45]. +In short, our solution consists of computing a observable without scattering, performing an average over the NBZs different +types of BZs, and then multiplying by the number of sites per unit cell Ns/c in the crystal. First, let us show that our proposal +recovers the correct crystalline result for the observables that depend only on one projector. Indeed, the BZ-averaged Eq. (C18) +representing the trace of the projector into the band n at crystal momentum k becomes: +tr [P n +k ]BZ av +no scatt = Ns/c +NBZs +� +a∈BZs +⟨k + Ga +��P n +k +��k + Ga⟩ = 1 + +� +A̸=B +cnA +k +� +cnB +k +�∗ +� +1 +NBZs +� +a∈BZs +e−iGa·tAB +� += 1, +(C20) +where the sum over a runs over a representative BZ of each type, and we have used Eq. (C14) and the fact that the term inside the +square brackets vanishes identically for A ̸= B. If there is a finite number NBZs of BZ types, this term vanishes because the NBZs +phases e−iGa·tAB are the 1/NBZs roots of unity. If there are infinite BZ types, which might occur, e.g., if the sites are located +at a generic nonsymmetric Wyckoff position incommensurate with the reciprocal lattice vectors, then this term vanishes due to +the infinite sum of a continuum of phases. In the example of the honeycomb lattice, where NBZs = 3 and e−iGa·tAB = eia2π/3 +with a ∈ Z3 if A ̸= B, and e−iGa·tAB = 1 if A = B, we obtain, as expected: +1 +3 +� +a=0,1,2 +e−iGa·tAB = δAB. +(C21) +We have also verified that the correct crystalline results are obtained numerically in our bismuthene and Bi bilayer tight-binding +models. Indeed, Fig. S8 shows the number of occupied states per unit cell � +n∈occ tr [P n +k ]BZ av +no scatt at k = 0 as a function of the +onsite SOC for crystalline bismuthene and Bi bilayer. In both models, this number of occupied states (or filling) is constant and +equal to 4 and 6, as expected, since they correspond to half-filling in bismuthene and Bi bilayer, respectively. Note that the filling +artificially deviates from these values close to the topological transition. However, this is an artifact stemming from the finite +KPM resolution. Indeed, this artifact only appears close to the transition, which is where the bulk gap is smaller, and therefore +is where the required precision to obtain the correct results is higher. We have checked that the deviations from the exact filling +shrink when increasing the KPM precision and the system size. +In summary, we have shown that, by averaging over the BZ types and multiplying by Ns/c, we recover the correct values in +the crystal for the quantities that involve the trace of one projector. This exact result is recovered despite neglecting both the +scattering by different reciprocal lattice vectors and the non-orthogonality of the plane waves. This means that the scattering +does not play a crucial role in the quantities that involve the trace of only one projector. +e. +Structural spillage without scattering in the tight-binding approximation +Now, let us consider quantities that involve the trace of two projectors, such as the spillage. Unlike in the quantities involving +just one projector, here scattering plays an important role. Indeed, we will show that scattering should be included to obtain the +exact result in the crystalline limit (see, e.g., Eq. (2b), where the sum over G′ represents the scattering). However, as explained +in Appendix C 3 c, the scattering has to be neglected in order to be able to use the structural spillage to compare amorphous and +crystalline systems. Nevertheless, we will also show that, even if the crystalline results are not exactly recovered, our method +gives reasonably good results, which allows the structural spillage to work as a topological indicator also in the tight-binding +approximation. +Consider, the trace of (P n +k )2, which should be equal to one if P n +k is a projector. If we include scattering and average over + +22 +(a) +(b) +Figure S8. Sum over occupied bands of the trace of one and two projectors, � +n∈occ tr [P n +k ]BZ av +no scatt and � +n∈occ tr +� +(P n +k )2�BZ av +no scatt, as a function +of onsite SOC, computed using the formalism of Eqs. (C20) and (C23) at k = 0. (a) Bismuthene crystal. (b) Bi bilayer crystal. On the one +hand, the filling � +n∈occ tr [P n +k ]BZ av +no scatt recovers the exact crystalline result, except close to the transition due to finite precision effects. On the +other hand, the trace of the projector square � +n∈occ tr +� +(P n +k )2�BZ av +no scatt, which should be equal to the filling, is just slightly (∼ 8 − 25%) smaller +due to neglecting the momentum scattering. +Brillouin zones this exact condition is fulfilled for the crystal, as can be checked explicitly: +tr +� +(P n +k )2�BZ av +scatt = Ns/c +NBZs +� +a∈BZs +Ns/c +NBZs +� +a′∈BZs +� +⟨k + Ga +��P n +k +��k + Ga + Ga′⟩⟨k + Ga + Ga′��P n +k +��k + Ga⟩ +� += += +� +A,B,C,D +cnA +k +� +cnB +k +�∗ cnC +k +� +cnD +k +�∗ +1 +NBZs +� +a∈BZs +e−iGa·(tAB+tCD) +1 +NBZs +� +a′∈BZs +e−iGa′·tCB = += +� +A,B,D +cnA +k +��cnB +k +��2 � +cnD +k +�∗ +1 +NBZs +� +a∈BZs +e−iGa·tAD = +� +A,B +��cnA +k +��2��cnB +k +��2 = 1. +(C22) +However, including scattering is not possible in general, unlike BZ averaging. As explained above, the scattering cannot be +taken into account when the two projectors belong to systems with a different lattice structure. Therefore, when computing +two-projector quantities we still perform the BZ average on the external sum over Ga, but are forced to neglect the scattering +resummation over Ga′: +tr +� +(P n +k )2�BZ av +no scatt = Ns/c +NBZs +� +a∈BZs +� +⟨k + Ga +��P n +k +��k + Ga⟩⟨k + Ga +��P n +k +��k + Ga⟩ +� += += +1 +Ns/c +� +A,B,C,D +cnA +k +� +cnB +k +�∗ cnC +k +� +cnD +k +�∗ +1 +NBZs +� +a∈BZs +e−iGa·(tAB+tCD) = += +1 +Ns/c +� +A,B,C,D +cnA +k +� +cnB +k +�∗ cnC +k +� +cnD +k +�∗ δtAB+tCD,0. +(C23) +Although this equation does not exactly recover the crystalline value, we have numerically verified that the sum over occupied +bands of this Eq. (C23), � +n∈occ tr[(P n +k )2]BZ av +no scatt, gives values just ∼ 8 − 25% smaller than � +n∈occ tr[P n +k ]BZ av +no scatt in the crystal, +as shown in Fig. S8. Therefore, we take this as a reasonable approximation, especially taking into account that this quantity can +also be computed when one of the projectors corresponds to an amorphous structure. Applying this method to the structural +quasi-Bloch spillage, we arrive at Eq. (3). +In order to implement the tight-binding spillage of Eq. (3) we need to account for a final detail: the choice of a representative +BZ of each type. This is a requirement because we introduced the average over BZ types in Eqs. (C20)-(C23). To perform this +average, one has to select one representative for each type of BZ. To this end, let us consider the example of the honeycomb +lattice relevant to our Bi models, which has NBZs = 3 types of BZ, as sketched in Fig. S7. Due to the argument which lead +us to Eq. (C8) in Appendix C 2 c, the optimal criterium for choosing the BZ representatives is to consider the ones whose +reciprocal lattice vector is smaller in modulus. For example, the first BZ will always be chosen as the representative of the BZs +characterized by a phase eiG·tAB = 1. There can still be several options, such as the three possibilities for the BZs with phases +eiG·tAB = e±i2π/3. In this case, one can choose any of them. A better choice however is to perform an angular average over + +Zn +tr[Pn] +neoco +8 +cell +6 +: states +4 +# +2 +1 +2 +入/t。neocc +12 + cell +9 +: states +6 +# +3 +0.0 +0.5 +1.0 +入/t。23 +them. Indeed, while the crystal is anisotropic, the amorphous structure is effectively isotropic. In particular, although the total +traces in the crystal are exactly the same in all equivalent BZs, some orbital-resolved quantities might vary. For instance, in +the honeycomb lattice, if the occupied eigenstate at G = 4π/ +√ +3(0, 1) is of py character, the eigenstate at the threefold rotated +ˆC3G = 4π/ +√ +3(− +√ +3/2, −1/2) is of the threefold rotated −( +√ +3/2)px − (1/2)py character. On the other hand, for sufficiently +large samples, amorphous structures are expected to be isotropic in momentum space. Therefore, one would ideally perform an +angular average over the G corresponding to equivalent BZs with the same modulus, but pointing in a different direction. In +the honeycomb lattice, the quantity corresponding to the BZs with phase eiG·tAB = e+i2π/3 would be an average over the three +BZs shown in blue in Fig. S7. Consequently, when the corresponding crystal displays a honeycomb lattice, the angle-averaged +Eq. (3) for the structural quasi-Bloch spillage in the tight-binding approximation reads: +γTB +qB (k) = 2 +3 +� +� +� +1 +2tr +�� +Pk+G0 − ˜Pk+G0 +�2� ++ 1 +3 +� +Gm +1 +1 +2tr +�� +Pk+Gm +1 − ˜Pk+Gm +1 +�2� ++ 1 +3 +� +Gm +2 +1 +2tr +�� +Pk+Gm +2 − ˜Pk+Gm +2 +�2�� +� +� , +(C24) +where: +G0 = 0 +⇒ e−iG0·tAB = 1, +(C25) +� +� +� +G0 +1 = 4π/ +√ +3(0, 1) +G1 +1 = ˆC3G0 +1 = 4π/ +√ +3(− +√ +3/2, −1/2) +G2 +1 = ( ˆC3)2G0 +1 = 4π/ +√ +3( +√ +3/2, −1/2) +� +� +� ⇒ e−iGm +1 ·tAB = ei2π/3, +(C26) +� +� +� +G0 +2 = 4π/ +√ +3(0, −1) +G1 +2 = ˆC3G0 +2 = 4π/ +√ +3( +√ +3/2, 1/2) +G2 +2 = ( ˆC3)2G0 +2 = 4π/ +√ +3(− +√ +3/2, 1/2) +� +� +� ⇒ e−iGm +2 ·tAB = e−i2π/3. +(C27) +Eq. (C24) is a specific instance of the general Eq. (3) that we used for computing the spillage in our bismuthene and Bi bilayer +tight-binding models. However, we have also checked that in these models, for the system sizes considered, performing the +angular average or not does not noticeably change the results. +In summary, our proposed method for computing two-projector quantities, such as the structural spillage, consists of neglecting +the momentum scattering, performing an average over the different types of BZs, and multiplying by the number of sites per unit +cell in the corresponding crystal. Applying this method to the structural quasi-Bloch spillage, we arrive at the final expression for +the structural spillage in the tight-binding approximation, Eq. (3) of the main text. To conclude, we highlight that, in the specific +case when the number of types of BZs is infinite or very large, (3) would involve reciprocal lattice vectors |G| ≫ 2π/a, with a +the crystalline lattice constant. In this case, as in the single-site case, we may introduce a momentum cutoff and consider only +the reciprocal lattice vectors G smaller than this cutoff. +4. +Phase transition criterion in the tight-binding approximation +In this section we define our criterion to choose the topological transition. To this end it is important to note first that, as +mentioned above, Eq. (3) does not exactly recover the values of the Bloch spillage when applied to two crystals with and without +SOC, because we neglected scattering. However, we have numerically verified that it results in similar values. In particular, +the maximum spillage without scattering is max +� +γTB +qB (k = 0) +� += 1.5 in the two models, which is a factor of 4/3 smaller than +the exact spillage max [γqB(k = 0)] = 2 that would be recovered after considering the scattering. This is related to the fact +that � +n∈occ tr +� +(P n +k=0)2�BZ av +no scatt is a factor of 4/3 smaller than � +n∈occ tr [P n +k=0]BZ av +no scatt in the topological and trivial phases for +the bismuthene and Bi bilayer tight-binding models, respectively (see Fig. S8). There is no reason to believe that this factor is +universal, and thus we consider it model dependent. +With this in mind, in order to identify the topological phases in a tight-binding phase diagram, we take the criterion that the +topological transition occurs when the quasi-Bloch spillage of Eq. (3) equals to half the maximum value of the spillage between +two topologically different crystals when scattering is neglected. In both our models, this critical value equals 0.75. However, +in general, this critical value of the tight-binding structural spillage will be model-dependent, and must be determined in a +case-to-case basis. + +24 +Appendix D: Absence of a corresponding crystal: spin-orbit plane-wave spillage +One of our assumptions for applying the structural quasi-Bloch spillage of Eqs. (2)-(3) is that there exists a crystalline structure +with similar local environments to the non-crystalline one. While this is a quite generic feature [37], there are also some +amorphous and quasicrystalline structures whose local environment is different to any crystalline phase of the same material. +In this case, while the structural quasi-Bloch spillage could still be calculated, it would probably not be very indicative of the +topology, since many possibly trivial band inversions could occur. +In this case, one could again resort to computing the spin-orbit Bloch spillage comparing an amorphous supercell with and +without SOC, as proposed for crystals by Liu and Vanderbilt [45]. However, as mentioned in the main text, this would always +be a large quantity due to the big size of the supercell. Liu and Vanderbilt proposed to fix this issue by analyzing valence- and +conduction-band-resolved spillages. However, these are not gauge-invariant, and a careful analysis is required to discern the +topological character using this method. These solutions are not practical from the point of view of a performing high-throughput +screening of amorphous materials, where it is desirable to define a quantity that is easily implemented and analyzed using +ab-initio codes. +For such cases without a crystalline counterpart, we propose instead a plane-wave-resolved spin-orbit spillage comparing an +amorphous system with and without SOC. This spin-orbit plane-wave spillage γpw(p) is defined as in Eq. (2a) but without the +sum over crystalline reciprocal lattice vectors G: +γpw(p) = 1 +2 +� +p′ +� +αβ +� +P αβ +p,p′P βα +p′,p − P αβ +p,p′ ˜P βα +p′,p +� ++ +� +P ↔ ˜P +� +(D1) +where p and p′ are plane-wave momenta. For a supercell Gamma calculation in DFT, p and p′ would be the supercell reciprocal +lattice vectors. Now, since both systems that are being compared have the same structure, Eq. (D1) can also be applied within a +tight-binding approximation. However, for the latter approximation, one could first compute the much more efficient plane-wave +spillage without scattering, which would read: +γno scatt +pw +(p) = 1 +2tr +�� +Pp − ˜Pp +�2� +, +(D2) +We however leave the benchmarking of the plane-wave spillage for future work. + diff --git a/NtE0T4oBgHgl3EQf0gJJ/content/tmp_files/load_file.txt b/NtE0T4oBgHgl3EQf0gJJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..69a1f2b01eb6c912e978f0de10d4e7392107961e --- /dev/null +++ b/NtE0T4oBgHgl3EQf0gJJ/content/tmp_files/load_file.txt @@ -0,0 +1,1723 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf,len=1722 +page_content='Structural spillage: an efficient method to identify non-crystalline topological materials Daniel Muñoz-Segovia*,1, 2, ∗ Paul Corbae*,3, 4 Dániel Varjas,5, 6 Frances Hellman,7, 4 Sinéad M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Griffin,4, 8, † and Adolfo G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Grushin2, ‡ 1Donostia International Physics Center, 20018 Donostia-San Sebastian, Spain 2Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Grenoble Alpes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Grenoble INP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Institut Néel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 38000 Grenoble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' France 3Department of Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' California 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' USA 4Materials Sciences Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' USA 5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Stockholm University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' AlbaNova University Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 106 91 Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Sweden 6Max Planck Institute for the Physics of Complex Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Nöthnitzer Strasse 38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 01187 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Germany 7Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' California 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' USA 8Molecular Foundry Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' USA (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2023) While topological materials are not restricted to crystals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' there is no efficient method to diagnose topology in non-crystalline solids such as amorphous materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Here we introduce the structural spillage, a new indicator that predicts the unknown topological phase of a non-crystalline solid, which is compatible with first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We illustrate its potential with tight-binding and first-principles calculations of amorphous bismuth, predicting a bilayer to be a new topologically nontrivial material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Our work opens up the efficient prediction of non-crystalline solids via first-principles and high-throughput searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Introduction-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Predicting which solids host non-trivial electronic topological phases is a central problem in con- densed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For crystalline solids, first principles methods take advantage of crystal symmetries to identify topo- logical materials [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, symmetry-based methods cannot be applied to diagnose non-trivial topology in ma- terials that lack translational invariance such as amorphous, polycrystalline, and quasicrystalline materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In fact, given the far greater ubiquity of non-crystalline materials in con- densed matter, solving this challenge would open up several new material classes far more numerous than crystals, with both fundamental interest for novel phenomena unique to non- crystalline matter [6–36], and for their possible greater ease of integration into devices [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Prior work on topology in non-crystalline materials used convenient amorphous tight-binding models with average and local symmetries [11, 14–16, 39], however these do not include the full chemical and structural specificity found in real matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Similarly, real-space invariants [40–43], including Wannier- based tight-binding formalism, require the system be treated on a case-by-case basis and can be computationally costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To overcome this methodological problem, we introduce the ‘structural spillage’, which is inherently compatible with first- principles approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Since the characterization of topol- ogy in general relies on the comparison with a known refer- ence [44], we propose that in our case the appropriate compar- ison is between the wavefunctions of the non-crystalline target system and a crystalline reference state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A similar approach was proposed to identify topological band inversions in crys- tals by Liu and Vanderbilt [45] who compared the wavefunc- tion overlap in crystals with and without spin-orbit coupling (the ‘spin-orbit’ spillage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Inspired by this idea, we define the structural spillage as a measure of the overlap between wave- functions with different structural configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' By com- paring this structural spillage for crystals, whose topological characterization can be efficiently calculated using standard symmetry-based methods [1–5], with those of non-crystalline solids, the topological characterization of the latter can be determined (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We first define the general formulation of structural spillage and how it can be used to diagnose topology in non-crystalline systems once a known reference phase is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We next exemplify its potential by diagnosing topological phase transi- tions in amorphous bismuth, a previously identified non-trivial amorphous system, using both a tight-binding model and den- sity functional theory (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Our results indicate that the struc- tural spillage can accurately identify amorphous bismuthene as topologically non-trivial [13, 46], and predicts that amorphous bilayer bismuth is a novel topological material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' By definition, the structural spillage is applicable to generic non-crystalline materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' It is suitable to establish a high-throughput cata- logue of potential non-crystalline topological materials, using currently available DFT codes based on plane waves in our current formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural spillage-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The total spillage γ measures the mismatch between two projectors P and ˜P into occupied states [45] γ = 1 2Tr �� P − ˜P �2� = Tr � P(1 − ˜P) � , (1) where the trace acts on the entire Hilbert space, and the last equality holds under the assumption that both systems have the same total number of occupied states Nocc = Tr[P] = Tr[ ˜P].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' By definition, γ ≥ 0 and can be viewed as the variance between two distributions with the same average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' When P = ˜P the spillage vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, when the overlap between the two projectors is zero, it equals the total number of occupied states Nocc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, γ acts as an indicator of band inversions caused by the parameters that differ in P and ˜P [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To predict topological band inversions in crystals, Liu and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='02686v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='mes-hall] 6 Jan 2023 2 (a) (b) Topo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Topo Spillage Trivial Trivial ( ) spillage spin-orbit structural high high low low Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (a) The spillage γ is high or low depending on whether a test wavefunction |ψ⟩ is in the same or different topological state compared to a known reference wavefunction | ˜ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (b) The spin-orbit spillage [45] compares wavefunctions with and without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The structural spillage takes advantage of the knowledge of the topological state of a crystalline solid to find the topological state of an amorphous solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Vanderbilt [45] chose P and ˜P to be projectors onto the sub- space of occupied states of crystalline insulators with and without spin-orbit coupling (SOC), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Lattice pe- riodicity allows these to be written in Bloch momentum k as P(k) = � n∈occ |ψnk⟩⟨ψnk|, which defines a k-resolved spin- orbit Bloch spillage, γB(k) = nocc − Tr[P(k) ˜P(k)], where nocc = Nocc/Ncells is the number of occupied bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The to- tal spillage is recovered by summing over all momenta in the Brillouin zone (BZ), γ = � k γB(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The spin-orbit Bloch spillage γB(k) thus quantifies the band inversion caused by SOC at each k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' it is large at points in the BZ where the band inversion is sizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [45] showed that at certain points in the BZ the spin-orbit Bloch spillage has to be larger than some given value if the SOC induces a topologically non-trivial phase from Wannier obstruction arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For instance, this lower bound equals two for a time-reversal symmetric topo- logical insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' From the above properties, γB(k) can be used to signal topo- logical band inversions in crystals, and is straight-forward to calculate using DFT [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, it has recently been applied to high-throughput searches for topological crystals [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We note, however, that a large spillage is a necessary but not sufficient condition for non-trivial topology: in certain cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', when many bands close to the Fermi level are slightly mixed by SOC, the spillage may be fooled by trivial insula- tors [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Consequently, more recent searches for topological crystals favor symmetry-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In most practical cases, the spillage is expected to be an accurate indicator of topology in crystals [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this work, we propose a spillage that compares an amor- phous system with a crystalline counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In doing so, we take advantage of the well-developed methods of sym- metry indicators for the topological characterization of crys- tals [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To this end, we now reformulate our spillage in a plane-wave basis for incorporation into standard plane-wave DFT codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Moreover, it is also well defined for both crys- talline and non-crystalline systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We write the total spillage γ in the plane wave basis |pα⟩, where p is the plane-wave momentum (not necessarily restricted to the first BZ) and α denotes spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To calculate the spillage, we need the projector onto occupied states of the amorphous and reference systems, P = � N∈occ |ψN⟩⟨ψN|, where |ψN⟩ are the eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' By projecting these onto plane waves, we then have access to the projector matrix elements P αβ p,p′ = ⟨pα| P |p′β⟩, which are well-defined for crystalline and non-crystalline systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Any plane-wave momentum p can be uniquely decomposed as p = k + G, the sum of a crystal momentum k in the first BZ plus a reciprocal lattice vector G, both of the reference crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Then, by substituting the plane-wave expansion into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (1), we can define the quasi-Bloch spillage as γqB(k) = 1 2 � k′ � GG′ � αβ � P αβ k+G,k′+G′P βα k′+G′,k+G − P αβ k+G,k′+G′ ˜P βα k′+G′,k+G � + � P ↔ ˜P � = (2a) = 1 2 � � � �� Gα P αα k+G,k+G � + ˜nocc(k) − � Gα � G′β � P αβ k+G,k+G′ ˜P βα k+G′,k+G + ˜P αβ k+G,k+G′P βα k+G′,k+G � � � � (2b) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2b) we have used the fact that the reference projector ˜P corresponds to a crystal, which allows us to set k′ = k in terms involving at least one ˜P, since there is no scattering between different crystal momenta due to the discrete transla- tional symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Note that γqB(k) fulfills the same sum rule as the Bloch spillage, γ = � k γqB(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, applied to two insulating crystals, γqB(k) recovers the Bloch spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Moreover, it can also be applied to semimetallic systems with the advantage of it being bounded by zero, in contrast to recent extensions to semimetallic materials [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Our key result is that the structural quasi-Bloch spillage, defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2), can be used as an efficient topological indicator in non-crystalline systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Crucially, it can be effi- ciently computed with plane-wave-based DFT methods, since the projector matrix elements are an output of the calcula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Consequently, this method is suitable for high-throughput identification of non-crystalline topological materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 3 0 1 2 λ/tσ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='6 ρnon−hex topological trivial γTB qB (k = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 0 1 2 λ/tσ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='6 ρnon−hex topological trivial conductance � e2 h � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 γTB qB (k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 (a) (b) (c) (d) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural spillage in the tight-binding approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (a) Example of a real-space structure with a density of non-hexagonal plaquettes ρnon-hex ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (b) Structural quasi-Bloch spillage γTB qB (k) in the BZ comparing topological amorphous bismuthene with ρnon-hex ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='53 and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='22tσ with a trivial crystal with λ/tσ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (c), (d) Phase diagrams as a function of SOC λ and the density of non-hexagonal plaquettes ρnon-hex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (c) Conductance in the “armchair” ribbon configuration (see SM [49] A 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (d) Structural quasi-Bloch spillage γTB qB (k = 0) comparing the amorphous system to a trivial crystal with λ/tσ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural spillage in the tight-binding approximation-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Defining a structural spillage that is useful in the tight-binding approximation requires us to develop further Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The reason is that two issues emerge as we define plane wave states projected into the tight-binding Hilbert space of Nsites as ��pα⟩ = 1 √Nsites � r eip·r��rα⟩, where r labels the position of each site and α labels internal quantum numbers, such as spin or the orbital type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' First, because the tight-binding model’s Hilbert space does not span the entire real space but only po- sitions defined by the charge centers, our plane waves are non- orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, their overlap depends on the atomic positions, and therefore on the amount of structural disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Since we expect continuous translational symmetry to be re- covered after averaging over different disorder realizations, we may solve this issue by neglecting the scattering between dif- ferent momenta in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' assuming that P αβ p,p′ ∝ δp,p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This assumption has been successfully used to determine the topology of non-crystalline systems using the effective Hamil- tonian approach [14–16, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A second issue of the tight-binding approximation is that the projected plane waves form an over-complete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A well- defined basis for a crystal with Ns/c sites per unit cell consist of a subset with momenta in Ns/c Brillouin zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, there are different types of Brillouin zones depending on the phase factor eiG·t, where t are the relative positions of the sites inside the unit cell [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For instance, in the honeycomb lattice there are 3 types of BZ, since e−iG·t = eia2π/3, with a ∈ Z3 (see Supplemental Material (SM) [49] C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This issue can be handled by replacing the sum over reciprocal lattice vectors G by an average over the different types of G, and multiplying by Ns/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' With these modifications, the structural spillage Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) can be defined in the tight-binding approximation as γTB qB (k) = 1 2 Ns/c NBZs � G∈BZs tr �� Pk+G − ˜Pk+G �2� , (3) where the sum over G runs over one BZ of each of the NBZs types, the trace acts over the internal degrees of freedom α, and we have defined the single-momentum projector P αβ p = P αβ p,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) and (2) define the structural spillage to be used in the tight-binding approximation and first-principles calculations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the remainder of the paper, we demonstrate how they capture topological phase transitions of amorphous systems, using low-dimensional bismuth as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Tight-binding benchmark: bismuthene on a substrate-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Crystalline bismuthene consists of a 2D honeycomb monolayer of bismuth atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Experiments suggest it to be a quantum spin Hall insulator with topological helical edge states when grown on SiC(0001) [51] or Ag(111) [52] substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The effect of the substrate is crucial: it filters the pz orbitals away from the Fermi level leaving the px,y orbitals, resulting in a large gap (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='67eV) and a non-zero strong Z2 topological index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Moreover, amorphous bismuthene on a substrate is predicted to remain topological via first-principles calculations [13, 46], making it a convenient system to benchmark our proposed structural spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The low-energy physics of bismuthene is captured by a tight- binding model with px,y orbitals in the honeycomb lattice, coupled by nearest-neighbour hoppings tσ and tπ, a large on- site SOC λ, and a substrate-induced Rashba SOC λR (which we take proportional to λ) [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To extend this model to amorphous structures while preserving the short-range order expected in amorphous systems [37], we use the voronization of a pointset [8, 14] (see SM [49] A 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' When the pointset is tri- angular, the voronization produces its dual honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' By randomly displacing the triangular pointset according to a characteristic length r, the voronization produces lattices with threefold coordination, as the honeycomb lattice, but with a finite density of non-hexagonal plaquettes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2(a)) [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, r continuously controls how amorphous are our lattices, allowing us to study the effect of structural disorder on topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the following, we quantify how amorphous our systems are by the (configuration-averaged) density of non-hexagonal plaquettes ρnon-hex, which is in one- to-one correspondence to the parameter r (see SM [49] A 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2 we present the topological phase diagram of amor- phous bismuthene as a function of ρnon-hex and λ, benchmark- ing γTB qB (k) against the two-terminal conductance results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the crystalline limit (ρnon-hex = 0), the system starts as a Dirac semimetal for vanishing λ, and a finite λ opens up a topological gap, similarly to graphene [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Above a critical λ, where the 4 RDF (c) (a) Top view Crystalline Low disorder High disorder b a c (b) Side view c b a Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Bismuth bilayer supercells used in DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (a) and (b) show in-plane and out of plane views of the supercell, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The colors indicate different degrees of disorder: crystal (blue), low disorder (green) and high-disorder (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (c) Radial distribution function (RDF) showing the statistics of the bond lengths in the disordered bismuth bilayer and their deviations from the per- fect crystal (vertical dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The disorder is sampled from a Gaussian distribution with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='15 Å for the low disorder and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='30 Å for the high disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' gap closes at the Γ point, the system becomes a topologically trivial insulator, adiabatically connected to the atomic limit in which only the onsite SOC is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Both the conductance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2(c)) and the structural quasi- Bloch spillage (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2(d)) capture the topological transition, even at finite structural disorder (ρnon-hex ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The conduc- tance in the topological insulator phase is equal to 2e2/h, originating from the helical edge states, while it reduces to zero after the phase transition to the trivial insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Con- comitantly, γTB qB (k = 0) is large in the topological phase and small in the trivial phase because we choose the reference sys- tem to be a trivial crystal, only with non-zero onsite λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Had we chosen the topological state as reference, the magnitude of the spillage in each phase would be inverted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' see SM [49] A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The critical λ at the transition for the crystal is correctly predicted by γTB qB (k = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In agreement with Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [13, 46], we find that increasing disorder decreases the topological gap and hence the critical λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Nevertheless, the realistic value of λ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='22tσ [51] lies in the topological phase also in the amor- phous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Lastly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2(b) shows γTB qB (k) for fixed λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='22tσ and ρnon-hex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' γTB qB (k) is peaked around k = 0 with a value ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5, reminiscent of the crystalline topological band inversion occurring at the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural spillage in DFT: free-standing Bi bilayer-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To show that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) is well suited for high-throughput screening of amorphous topological materials, we calculate the struc- tural spillage from the output wavefunctions of first-principles γqb 0 2 Low disorder High disorder Tight binding DFT a-SOC x-SOC x-noSOC a-SOC vs vs Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural quasi-Bloch spillage γqB(k) for the bismuth bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' First row: comparison between an amorphous system with SOC (a-SOC) and a crystalline system without SOC (x-noSOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Comparing an amorphous system without SOC with a crystalline sample with SOC leads to similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Second row: comparison between the amorphous and crystalline systems with SOC (a-SOC and x-SOC, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' γqB(k) is high at k = 0 for the first row while small for the second row, indicating that amorphous bismuth bilayer is a topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The last column shows a comparison with the tight-binding quasi-Bloch spillage γTB qB (k) (see SM [49] A 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' calculations (see full details in SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We choose previously- studied free-standing bismuth (111) bilayer as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This 2D bismuth allotrope, whose crystalline phase consists of a buckled honeycomb lattice with lattice constant a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='33 Å, is also predicted to be a strong topological insulator crystal with Z2 = 1 [55–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, no prediction exists for its amorphous counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To represent amorphous structures given the periodic boundary conditions of the calculations, we create 5 × 5 × 1 supercells comprising of 50 Bi atoms per bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Their elec- tronic structure is calculated for a single supercell momentum, the center of the supercell BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Starting from a crystalline su- percell, the structure is disordered by adding random displace- ments in the x, y, and z directions, sampled from a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The structures and their corresponding radial distribution functions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To predict the topological phase of amorphous Bi bilayer with SOC we compute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) with plane-wave-based DFT (see SM [49] B) to compare it with its crystalline counterpart without and with SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' When SOC is not included, and hence when it is topologically trivial (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 4, first row), γqB(k) is peaked at k = 0, with γqB(k = 0) > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Increasing disorder smooths γqB(k), yet it remains peaked at Γ with a value greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In contrast, when we include SOC in calculations of both the disordered Bi bilayer and the pristine crystal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 4, second row) the spillage is always small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Both rows together show that amorphous bismuth bilayer with SOC is in the same topological state as the crystal with SOC, a strong topological insulator crystal with Z2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We have performed a similar analysis using a tight-binding 5 model for the amorphous Bi (111) bilayer (introduced in SM [49] A 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The results, displayed in the last column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 4, show that for comparable disorder strengths γTB qB (k) is broader and its maximum value is smaller than γqB(k) in DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' It is thus apparent that, due to the approximations in the tight- binding calculation of the spillage, which lacks information of the real space extension of the orbitals, the spillage method is more suitable for DFT, an advantageous feature compared to other topological indicators available for non-crystalline sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Discussion-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We have introduced the structural spillage as an efficient method to signal non-crystalline topological phases, compatible with tight-binding and ab-initio simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We have used it to predict amorphous Bi bilayer as a novel topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As was the case for spin-orbit spillage in crystals, we expect the structural spillage to signal a large fraction of promising materials, but not to be infallible: if multiple band inversions are introduced upon amorphization, the spillage might also be artificially large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, unlike for crystals, the spillage is currently the only systematic, model-independent method that is compatible with ab-initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Additionally, we ob- serve that, for different disorder realizations, its fluctuations are smaller compared to scattering methods like calculating the conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' It can also be applied to systems without a spectral gap, where the effective Hamiltonian approach [35] can fail [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Lastly, while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) is general, the definition of the spillage is relatively versatile and can accommodate less standard cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For example, when no crystalline coun- terpart exists, one may define a plane-wave-resolved spillage (see SM [49] D) by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2a) without the sum over G, a modification worth studying in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The structural spillage establishes a clear road-map to con- struct a high-throughput catalogue of non-crystalline (amor- phous, polycrystalline, quasicrystalline) topological materials by screening existing amorphous databases, or by scrutinizing realistic structures obtained using existing ab-initio molecu- lar dynamics packages [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This methodology may enable for the first time the systematic prediction and discovery of a potentially large number of amorphous materials that are cur- rently inaccessible, suitable to develop affordable and scalable topological devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Acknowledgements-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We are grateful to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Franca, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' de Juan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Hannukainen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' López-Cano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Queiroz, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Marsal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Soluyanov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Martin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Vinson for fruitful dis- cussions and related collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This work was partially funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engi- neering Division under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' DE-AC02-05-CH11231 within the Nonequilibrium Magnetic Materials Program (MS- MAG), specifically the work by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' is supported by an FPU predoctoral contract from Spanish MCIU No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' FPU19/03195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' acknowledges financial support from the European Research Council (ERC) Consol- idator grant under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 101042707 (TOPO- MORPH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' was supported by the Swedish Research Coun- cil (VR) and the Knut and Alice Wallenberg Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Computational resources were provided by the National En- ergy Research Scientific Computing Center and the Molecular Foundry, DOE Office of Science User Facilities supported by the Office of Science, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Department of Energy under Con- tract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' DEAC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The work performed at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Department of Energy under the same contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Author contributions-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The original idea was conceived by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' derived the expressions for the quasi- Bloch spillage and performed tight-binding calculations as- sisted by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' performed the DFT calculations and de- veloped the spillage code for plane-waves assisted by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' All authors contributed to the interpretation of results and writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' ∗ daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='munozsegovia@dipc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='org † sgriffin@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='gov ‡ adolfo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='grushin@neel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='fr [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Kruthoff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' de Boer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} 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+page_content=' Cholia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Gunter, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Chevrier, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Persson, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Ceder, Computational Materials Science 68, 314 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [73] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Soriano and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Palacios, Physical Review B 90, 075128 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 7 CONTENTS References 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Tight-binding models 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Model for bismuthene on a substrate 7 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Tight-binding Hamiltonian 7 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Construction of amorphous structures 8 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Additional results: density of states and structural spillage for different reference systems 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Model for free-standing bismuth (111) bilayer 10 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Tight-binding Hamiltonian 11 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Construction of amorphous structures 11 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Topological phase diagrams 11 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Comparison with DFT 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Calculation details 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' DFT calculation details 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Defining the structural spillage in the tight-binding approximation 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' General remarks and motivation 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' System with a single site per unit cell 17 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Setting the stage: crystalline system 17 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Spillage comparing two crystals 17 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural spillage comparing an amorphous system to a crystal 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' System with several sites per unit cell 18 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Crystal: definitions and types of Brillouin zones 18 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Crystal: recovering the exact results using plane waves 19 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Comparing an amorphous system to a crystal using the structural spillage: no-scattering approximation 20 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Taking into account different types of Brillouin zones 21 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural spillage without scattering in the tight-binding approximation 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Phase transition criterion in the tight-binding approximation 23 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Absence of a corresponding crystal: spin-orbit plane-wave spillage 24 Appendix A: Tight-binding models This Appendix describes the method for generating the amorphous tight-binding models used in the maint text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We include as well further calculation details and some additional discussion regarding the phase diagrams that one can obtain using different reference systems of the structural spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Model for bismuthene on a substrate This section describes how to generate the amorphous bismuthene structure and tight-binding Hamiltonian that we have used to benchmark the structural spillage method in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Tight-binding Hamiltonian Crystalline bismuthene consists of a 2D honeycomb monolayer of bismuth atoms [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' An effective tight-binding of crystalline bismuthene on a substrate was proposed by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' It consists of px and py orbitals in the honeycomb lattice, coupled by nearest- neighbour hoppings, a large onsite SOC, and a substrate-induced Rashba SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In real space and in the basis {px↑, px↓, py↑, py↓}, 8 the Hamiltonian reads: H = − 1 2 � ⟨ij⟩ � (tσ − tπ) τ0 + (tσ + tπ) � c(2) ij τz + s(2) ij τx �� σ0 + � i [λτyσz] + + � ⟨ij⟩ i � λA Rτ0 [sijσx − cijσy] + λE R [(cijτx − sijτz) σx − (cijτz + sijτx) σy] � , (A1) where we have defined cij = cos(θij), sij = sin(θij), c(2) ij = cos(2θij), and s(2) ij = sin(2θij), with θij the angle between the bond joining site i to site j and the x axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' τµ and σµ are the Pauli matrices acting on the orbital {px, py} and spin {↑, ↓} degrees of freedom, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' tσ and tπ are the sigma and pi nearest-neighbour hoppings, λ is the onsite SOC, and λA R and λE R are the orbital-independent and orbital-dependent Rashba SOC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [51], in this work we will assume that λA R = λE R = λR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The values used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [51] are tσ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0eV, tπ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='21eV ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='11tσ, λ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='44eV ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='22tσ, and λR ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='032eV ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='074λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In our calculations, we will take tσ as the unit of energy, we will use the same value for tπ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='11tσ, and we will vary both the onsite SOC λ as well as the Rashba SOC proportionally to the former, λR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='074λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The Hamiltonian (A1) can readily be applied to an amorphous lattice once we define which sites are nearest neighbours of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In principle, it could be generalized to include a dependence on the distance in the hoppings, such as the Harrison law [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, we will consider fixed values for the hoppings, which can be a good approximation for covalently-bonded amorphous solids, which usually display a rather narrow distribution of bond distances [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Moreover, this approximation enables us to isolate the effect of structural disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Construction of amorphous structures Covalently-bonded amorphous materials usually preserve local environments similar to the ones in the corresponding crystals, since they are set by the strong covalent bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, most amorphous materials have average coordination numbers, bond distances, bond angles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', which are centered around those of the crystal [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' With this in mind, our amorphous models preserve, for every site, the threefold coordination of the honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This is achieved by applying the Voronoi method similar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [14], but with a modification that enables us to control the degree of amorphization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In particular, we first construct a pointset forming a triangular lattice with lattice constant a, whose points will be called seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We then randomly displace the seeds from their initial positions following an exponential distribution with characteristic distance r · a in the radial direction, and a uniform distribution in the angular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We thereafter compute their corresponding Voronoi diagram, which is defined by the Voronoi cells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', the regions consisting of all points closer to one seed point than to any other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The vertices of such cells, called Voronoi vertices, form a threefold coordinated lattice with the edges of the Voronoi cells corresponding to the nearest-neighbour bonds (only the vertices at the boundaries of the system have fewer than three neighbours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The lattices obtained in this way have large variances in the bond angle and bond length distributions, which might not be very realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In order to reduce this artifact, we apply a simple iterative relaxation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We select the threefold coordinated sites one by one and displace them to the barycenter formed by their three nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We iterate this process until convergence is reached, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', until the displacements are smaller than some small cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This relaxation procedure tends to set the bond angles as close as possible to the crystalline angle, 120◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Finally, once the lattice is relaxed, we rescale the distances so that the average nearest-neighbour distance is a/ √ 3, which is the corresponding value in the crystalline honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S1(a) shows the resulting histograms of the relative positions of atoms for two amorphous structures with different disorder strengths, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3 (top) and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Both structures are isotropic at long distances, although for small disorder the nanocrystalline domains (see for example Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2(a) in the main text) give rise to broad nearest neighbour peaks around the crystalline positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For high disorder, the correlation hole for distances under a/ √ 3 and an annular peak are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The parameter r, characterizing the exponential distribution by which the seeds are displaced from the regular triangular lattice, continuously controls the amorphousness of the resulting Voronoi lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, since the Voronoi diagram of a triangular lattice is a honeycomb lattice, we recover the crystal in the r → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Increasing r introduces non-hexagonal plaquettes in the Voronoi lattice, at least until r ≳ 1, when the seed becomes completely random (since all the information from the initial triangular seed is lost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S1(b), which shows that the configuration-averaged standard deviations of the distributions of bond angles, bond distances, and plaquettes start to saturate at about r ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural disorder can be quantified by several properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' These include the standard deviations of the distributions of nearest-neighbour distances, angles and plaquettes (normalized by the corresponding average values), as well as the density of non-crystalline plaquettes (in our models, where the crystalline limit consists of a honeycomb lattice, the non-crystalline plaquettes correspond to the non-hexagonal ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In order to take into account the finite-size effects, for each parameter r, we consider the configuration-average of these quantities over 100 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 9 (a) (b) (c) (d) Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (a) Histograms of the relative positions of atoms for two amorphous structures with different disorder strengths, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3 (top) and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (b) Configuration-averaged structural quantities as a function of the parameter r controlling the amorphousness: standard deviations (std) of the distributions of nearest neighbour bond angles, bond distances (both for the planar bismuthene as well as for the buckled Bi bilayer), and plaquettes, as well as density of non-hexagonal plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For each disorder intensity r, the results have been averaged over 100 different realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (c) Distribution of the ratios of non-hexagonal plaquettes ρnon-hex obtained with 100 disorder realizations with fixed disorder r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (d) Distribution of plaquettes for a given disorder realization with r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3 (corresponding to ρnon-hex ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S1(b), all these configuration-averaged quantities have the same qualitative dependence with the parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In particular, there exists a one-to-one correspondence between our control parameter r and any of these configuration-averaged quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, for particular disorder realizations in a finite system, there are fluctuations that make their relation to r not one-to-one before performing the configuration average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This is illustrated by the distribution of ratios of non-hexagonal plaquettes ρnon-hex shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S1(c) for different realizations with fixed r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, we have chosen to physically characterize the amorphousness of a system by the configuration-averaged density of non-hexagonal plaquettes formed by the nearest neighbour sites ρnon-hex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This measure could be generalized to other models whose crystalline limit consisted of lattices other than the honeycomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S1(d) shows an example distribution of plaquettes obtained for a particular disorder realization with r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3, which corresponds to ρnon-hex ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='55, while the configuration-average for this r corresponds to ρnon-hex ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The above procedure generates structures with open boundary conditions, which is useful to compute e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' the local density of states at the edges or the longitudinal conductance once some leads have been attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, for spectral quantities such as the spillage, we can reduce the possible finite-size effects by imposing periodic boundary conditions, or equivalently by putting the system on a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' An amorphous system might have a different number of atoms at opposite edges, so the periodic boundary conditions cannot be imposed directly, but rather before computing the Voronoi tessellation, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Before explaining the procedure to impose the periodic boundary conditions, let us note that our periodic systems consist of a rectangular supercell with sides Lx and Ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In order for the periodic boundary conditions to be applicable to systems with an arbitrary amount of structural disorder, including the crystalline limit, Lx and Ly are restricted to the values such that the supercell is commensurate with the initial crystalline unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In our models, where the crystalline limit is a honeycomb lattice, the previous condition imposes that Lx = nxa and Ly = ny √ 3a, where a is the lattice constant, and nx, ny are integer numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Taking this into account, let us now describe the procedure to impose periodic boundary conditions on a system with an arbitrary amount of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' First, we generate a triangular seed within the supercell x ∈ [0, Lx), y ∈ [0, Ly), and we disorder choosing a finite value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Then, we repeat this initial seed in the eight nearest-neighbour supercells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', we copy the seed points displaced from their initial positions x to x + L = x + (nxLx, nyLy), with nx, ny ∈ {1, 0, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Then, the Voronoi tessellation of the whole system (composed by the nine supercells) is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This gives rise to a threefold coordinated lattice with the following convenient feature: the supercell defined by the sites inside the region x ∈ [0, Lx), y ∈ [0, Ly) has the same number of sites in opposite sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, the periodic boundary conditions can be now applied to this supercell (all the sites outside this supercell are discarded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Finally, we carry out the relaxation procedure of this supercell, being careful to preserve the periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To conclude this section, let us mention that we generate the systems with open boundary conditions starting from a system with periodic boundary conditions, by first removing the bonds at the edges of the supercell and then removing the dangling sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This way, the bulk of the periodic structure where the spillage is computed is the same as the bulk of the open system where the conductance is determined, which allows us to safely compare their predictions of the topological phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2 a 0 2 2 0 2 α/aaverage relative std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='6 plaquettes std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='2 2D angles std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='4 3D angles std 2D distances std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='2 3D distances std Pnon-hex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='6 raverage = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='53 std = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='02 30 probability density 20 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='56 Pnon-hexaverage = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 std = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='4 probability density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 3 4 5 6 7 8 9 10 # sides of plaquettes2 a 0 9 2 0 210 (a) (b) (c) (d) Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Phase diagrams of different quantities as a function of SOC λ and amorphousness ρnon-hex for the bismuthene model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (a) Density of states at the Fermi level of the system with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (b) Two-terminal longitudinal conductance in the “zigzag” ribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (c) Structural quasi-Bloch spillage γTB qB (k = 0) comparing the amorphous system with a topological bismuthene crystal with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='1tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (d) Structural quasi-Bloch spillage γTB qB (k = 0) comparing the amorphous system with SOC λ to the corresponding crystal with the same SOC λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Additional results: density of states and structural spillage for different reference systems In this section we discuss further different phase diagrams that may be obtained for the bismuthene tight-binding model and its spillage in the tight-binding approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S2 shows phase diagrams for the density of states, conductance and structural spillage corresponding to the same bismuthene structures as the ones presented in the main text in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In particular, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S2(a) shows that the density of states at the Fermi level increases with ρnon-hex when the SOC is such that the crystal is in the topological phase (λ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3tσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This is due to the band broadening due to the disorder, and also from the appearance of low-energy states induced by a sublattice imbalance in a bipartite lattice [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' At high disorder, this induces the band inversion that drives the system from topological to trivial at a smaller SOC than in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In order to show that the quantized conductance does not arise from disorder-robust trivial edge states present in one particular crystalline direction, we display in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S2(b) the longitudinal two-terminal conductance along the direction perpendicular to the one displayed in the main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2 (the edges here would correspond to a zigzag ribbon in the crystalline case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As expected, both conductances coincide, which is a signature of the topological helical edge states, which live at all the boundaries of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Let us now explore how the structural spillage changes when we choose a topological reference system, as opposed to a trivial reference system used in the main text, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S2(c) shows the structural quasi-Bloch spillage when the reference system is a topological crystal with SOC λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='1tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Contrary to the trivial reference case shown in the main in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2, now the spillage is small in the topological phase and large in the trivial one, as expected from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Importantly, the transition is predicted at approximately the same SOC irrespective of the reference system, which shows the robustness of the spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Finally, in order to isolate the effect of the structural disorder on the topological band inversion from the effect of SOC, we have also computed the structural quasi-Bloch spillage comparing each amorphous system with amorphousness ρnon-hex and SOC λ to a reference crystal with the same SOC λ, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This choice highlights the regions where disorder induces a topological band inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For example, if the reference crystal is topological for a given λ, this spillage will have a large value if the disorder induces a trivial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, interpreting Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S2(d) requires knowledge of the topological phase of the crystal at each λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For λ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3tσ, the reference crystal is topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Since the spillage is small for λ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='1tσ, the amorphous system is topological for λ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='1tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, at high disorder, the spillage becomes large between λ ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='1tσ and λ ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3tσ, which indicates that the disorder induces a trivial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Lastly, for λ ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3tσ, the reference crystal is trivial, and the spillage is low, indicating that the amorphous system is also trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In conclusion, all phase diagrams Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S2 (b-d) agree qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The spillage is able to predict the topological phase transition independent of the reference system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Model for free-standing bismuth (111) bilayer In this section, we introduce a tight-binding model for the amorphous bismuth bilayer, for which we study the structural spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' After introducing the model and describing the method to generate the amorphous structures, we analyze its topological phase diagram to further benchmark the structural spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Finally, we compare the tight-binding results and DFT calculations, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We conclude that, while both qualitatively agree, the structural spillage method works better in DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' DOS(EF)[a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='6 1000 Pnon-hex 750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='4 500 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 Pnon-hex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='2 topological trivial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0 1 2 入/t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='11 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Tight-binding Hamiltonian Crystalline bismuth (111) bilayer consists of a buckled honeycomb lattice of bismuth atoms, where each sublattice has a different height [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' An effective tight-binding of crystalline Bi bilayer was introduced by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [62], where the three p orbitals are relevant due to the absence of the substrate in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Their model consists of spinful px, py and pz orbitals in the buckled honeycomb lattice with up to third nearest-neighbour hoppings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For simplicity, we will restrict ourselves to nearest-neighbour hoppings and onsite SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In real space and in the basis {px↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' px↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' py↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' py↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' pz↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' pz↓},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' the Hamiltonian reads: H = � ⟨ij⟩ � �tπτ0σ0 − (tσ + tπ) � � (dij · ux)2 (dij · ux) (dij · uy) (dij · ux) (dij · uz) (dij · uy) (dij · ux) (dij · uy)2 (dij · uy) (dij · uz) (dij · uz) (dij · ux) (dij · uz) (dij · uy) (dij · uz)2 � � σ0 � � + + � i � �E0z � � 0 0 0 0 0 0 0 0 1 � � σ0 + λL · σ � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (A2) where E0z is the difference between the onsite energy of the pz and px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='y orbitals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' dij is the unit vector along the bond from site i to site j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' and ua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' a = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' are the unit vectors along the three cartesian axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We have also defined the angular momentum matrices La, which act on the orbital subspace {px, py, pz}: Lx = � � 0 0 0 0 0 −i 0 i 0 � � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Ly = � � 0 0 i 0 0 0 −i 0 0 � � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Lz = � � 0 −i 0 i 0 0 0 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (A3) In our calculations, we will take tσ as the unit of energy, and fix the value of tπ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='25tσ and E0z = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='4tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We vary the onsite SOC λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' From the DFT-derived tight-binding model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [62], we can estimate that the actual SOC for the Bi bilayer is λ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='7tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The height of the bilayer enters via the vectors dij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Different DFT calculations have predicted heights ranging from dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='35a to dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='40a [57, 58, 62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this work, we will use dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='9a/ √ 6 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='37a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Construction of amorphous structures Our structures of amorphous Bi bilayers are constructed in a similar way to monolayer bismuthene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, the first step is generating an amorphous bismuthene lattice following the procedure outlined in Appendix A 1 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We then have to assign different heights to the sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the crystalline limit, each sublattice has a different fixed height because of the buckling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Sublattices are no longer well-defined in an amorphous lattice, but we can still define some effective sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' One differentiating property between the two sublattices in a crystalline honeycomb lattice is the direction of their nearest-neighbour bonds: if the bonds from sublattice A point at polar angles θA 1 = π/2, θA 2 = −11π/12 and θA 3 = −π/12, then the ones from sublattice B point at θB 1 = −π/2, θB 2 = π/12 and θB 3 = 11π/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, η(S) = sign ��� l θS l mod 2π � − π � is equal to +1 for sublattice S = A and −1 for S = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Using η(S) = ±1 to define the effective sublattices in the amorphous structures, we then assign a height ±dz/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Finally, we add some random disorder to the height of each site sampled from a Gaussian distribution with standard deviation rz ·a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In particular, we choose the height disorder rz proportional to r, the parameter that controls the in-plane amorphousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the calculations presented in this work, we take rz = rdz/(4a) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='09r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S3(a) shows the top and side views of a representative structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Topological phase diagrams In this section, we study the topological phase diagram of the amorphous Bi bilayer tight-binding model (A2), and show that, as for Bimsuthene, the structural spillage correctly predicts the topological band inversion in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Before analyzing the results, let us briefly review the current status regarding the topological characterization of crystalline Bi (111) bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the crystalline case with SOC, the Bi bilayer has been predicted to be a strong topological insulator [55–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Our model can also describe other materials with the same lattice, such as the antimony (111) bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Due to the smaller SOC, the Sb bilayer becomes a strong topological insulator only when strained [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, our model in the crystalline case starts as a Z2 = 0 insulator for vanishing λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A band inversion occurs at a finite value of λ, driving the system to a Z2 = 1 topological insulating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For the parameters used in this work (see Appendix A 2 a), this band inversion in the crystal occurs at Γ for λ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='27tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 12 (c) (e) (d) (b) (a) Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Bi bilayer tight-binding model structure and phase diagrams as a function of SOC λ and amorphousness ρnon-hex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (a) Top and side views of an example structure for amorphousness ρnon-hex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='53 (r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Sites are colored according to their out-of-plane positions: red/blue indicates the effective sublattice, and the color intensity scales with the actual out-of-plane position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The positions in the out-of-plane direction have been rescaled by a factor 10 for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (b) Momentum resolved tight-binding quasi-Bloch spillage for ρnon-hex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='53 (r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3) and SOC λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='7tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' These parameters are equal to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 4, with a change in color to match that of (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (c) Phase diagram of the density of states at the Fermi level of the system with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (d) Phase diagram of the two-terminal longitudinal conductance in the “armchair” ribbon configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (e) Phase diagram of the structural quasi-Bloch spillage γTB qB (k = 0) comparing the amorphous system with SOC λ to a topological crystal with λ = tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S3(b), the structural quasi-Bloch spillage γTB qB (k) of the amorphous system with amorphousness ρnon-hex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='53 (r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3) and SOC λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='7tσ is maximum at k = 0, with a value > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='75, when the reference system is a trivial crystal with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Per our topological criterion, explained in detailed in Appendix C 4, this indicates that there is still a band inversion at k = 0 in the presence of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Let us now analyze the topological phase diagram of the amorphous Bi bilayer tight-binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S3(d) and (e) show the conductance and the structural quasi-Bloch spillage, computed for a reference topological crystal with λ = tσ, respectively, as a function of amorphousness, ρnon-hex, and SOC, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Both phase diagrams show a transition from a trivial insulator at λ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='3tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' First, note that the conductance shows a metallic region around the transition, also in the crystalline case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This is an artifact of the finite precision in computing the Fermi level with the kernel polynomial method, compounded with finite-size effects (see Appendix A 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' These effects also broaden the otherwise sharp transition in the structural spillage at low disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We have checked that this transition region is reduced upon increasing the kernel polynomial method precision and the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Note that these issues only appear as one approaches the transition, where the gap is increasingly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For further related details, see also the discussion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S8 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Let us now focus on the phases away from the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The trivial insulator phase at small λ, characterized by a vanishing conductance and a large spillage (since the reference crystal is topological), survives with amorphousness up to slightly higher λ than in the crystalline case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the other hand, the topological insulator phase, indicated by a quantized 2e2/h conductance and a small spillage, only survives for small disorder, and the system seems to become slightly metallic for higher disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This metallic phase is further signaled by the finite density of states at the Fermi level shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Notice that, despite the absence of Rashba SOC in this model, the onsite λ is already spin-non-conserving, and therefore a metallic phase can be the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Nevertheless, we cannot discard the possibility that the metallic conductance is arising from finite-size effects with an Anderson localized bulk but with a localization length longer than the system sizes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A scaling study would be needed to discern the nature of this metallic conductance, but this lies beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In any case, the spillage is not specifically designed to capture such metallic feature, and it just indicates that the topological band inversion still (partially) B(k = 0) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 入/tconductance h 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 Pnon-hex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 trivial topological 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 入/t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='13 occurs for high disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Nevertheless, the larger spillage at high disorder, where the disorder induces this potential metallic phase starting from a topological state, provides a signature for the partial loss of this band inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This partial melting of the band inversion is also compatible with the increasing density of states at the Fermi level shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In summary, both conductance and spillage phase diagrams agree qualitatively and predict the topological phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Quantitative differences only arise in the metallic regions, where the band inversion is just partial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As for bismuthene, we have also checked that the conductance with leads in the perpendicular direction and the spillages with other reference systems give similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Comparison with DFT In this section, we comment on the comparison of the results of the previous section with the DFT results presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In particular, let us compare the latter to the tight-binding results for the realistic SOC λ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='7tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 4, the structural spillage predicts a topological band inversion in the amorphous Bi bilayer in both DFT and tight binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Both methods also agree on the fact that, above a certain disorder, the spectral gap closes (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S3 and S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Crucially, because we are forced to neglect the momentum scattering in the tight-binding approximation (see Appendix C), the structural spillage in DFT takes higher values and it is also less broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Consequently, the structural spillage not only is a topological indicator compatible with DFT, but it works better in DFT than in tight-binding modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Calculation details This section describes in detail the methods used to solve the tight-binding models, and some related subtleties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We use the Kwant software package [65] to generate the tight-binding Hamiltonians and perform the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To be able to treat larger system sizes, we apply the kernel polynomial method (KPM) [66] to estimate the density of states (DOS) and the projector onto the occupied states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The projector is computed following the procedure of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [67] and using plane waves as initial KPM vectors, which allows us to calculate the projector matrix elements ⟨pα|P|pβ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We use a KPM energy resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='01tσ (645 moments) for the bismuthene structures, and of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='005tσ (887 moments) for the bilayer ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The DOS is computed by performing a KPM stochastic trace with 50 and 100 random vectors in the cases of bismuthene and bilayer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The system sizes considered are 21a × 12 √ 3a for the bismuthene case and 41a × 24 √ 3a for the Bi bilayer one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Both the resolution and the size of the Bi bilayer system are taken to be larger than those of bismuthene since the gap in the former case is smaller, and therefore finite-size effects are larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Additionally, our model for the Bi bilayer displays some trivial edge states that affect the calculation of the Fermi level considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The structural quasi-Bloch spillage is computed in the systems with periodic boundary conditions using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3), which reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C24) in our models, since the crystalline phase has a honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the other hand, the conductance is determined with the Kwant software in the systems with open boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In order to avoid possible artifacts arising from trivial edge states in some particular termination, the conductance is calculated using leads in both x and y directions, such that in the crystalline case the edges are zigzag and armchair, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Since the aim of the conductance is to identify the insulating and topological insulating regions, which have a quantized conductance of 0 and 2e2/h, respectively, regardless of the shape of the leads, we use leads consisting of a 2D planar square lattice with nearest-neighbour hoppings such that their bandwidth is larger than that of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' These leads are attached to all the atoms on the corresponding edge of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S4 shows two example configurations with the leads in the y (armchair) and x (zigzag) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Our Bi bilayer models, display at low disorder some trivial edge states close to the Fermi level over a wide range of values of SOC, which appear in both zigzag and armchair edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' These change the Fermi level of a finite system with open boundary conditions Eopen F with respect to the one computed with periodic boundary conditions Eperiodic F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For the system sizes we are able to treat numerically the change in the Fermi level Eopen F is enough for it to lie outside of the bulk gap, since the thermodynamic gap in the crystal is rather small (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='1tσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, the conductance computed at Eopen F in the crystal would show metallic regions even in the insulating and topological insulating phases due to this artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In order to avoid this issue, in the Bi bilayer systems we compute the conductance at Eperiodic F determined with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We note that this problem does not appear in the bismuthene models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' It is also worth highlighting that the metallic phase observed at large SOC and disorder is not an artifact (see Appendix A 2), since we observe that the trivial edge states merge into bulk states in this region and therefore Eperiodic F ≃ Eopen F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Lastly, to compute the phase diagrams we only need a single disorder realization for each r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The reason is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' First, we noticed that for sufficiently large systems sizes, as the ones considered in this work, the fluctuations of the structural spillage for different disorder realizations are rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, they are smaller than the fluctuations in the conductance, which is another convenient feature for the use of the structural spillage in high-throughput searches for topological amorphous materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 14 (a) (b) Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Examples of Bi bilayer systems with leads where conductance is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (a) Top and side views of a system with leads in the x axis, which would correspond to a zigzag ribbon in the crystalline case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (b) Top and side views of a system with leads in the y axis, which would correspond to an armchair ribbon in the crystalline case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Second, while extracting a precise topological phase diagram from the conductance would require a configuration average, it is not strictly necessary if we just aim to use it as a benchmark for the structural spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Appendix B: DFT calculation details We performed Density Functional Theory (DFT) calculations using the projector augmented wave (PAW) formalism in the Vienna ab-initio Simulation Package (VASP) [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The exchange-correlation potentials were treated within the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerbof (PBE) [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The wavefunctions were expanded in plane waves to an energy cutoff of 700 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' SOC was added self-consistently for all calculations in which it was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For supercell calculations, we performed Gamma point only calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For self-consistent calculations of the unit cell, we used a k-point grid of 21x21x1 with Gamma for the BZ sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We then sampled the 25 k-points ( n1 N1 b1 + n2 N2 b2) that would backfold to Gamma in the 5x5x1 supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To compare the same momenta between the unit cell and the supercell, the two must be commensurate and the supercell lattice vectors must be multiples of the unit cell lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' If this were not the case, one could linearly interpolate the coefficients of the supercell wavefunctions at the appropriate momenta from the closest supercell reciprocal lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Unlike in the tight-binding approximation, the structural spillage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) can be directly implemented in DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Here, the overlap between two systems is well-defined irrespective of them having atoms at different positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, strictly speaking, the continuous set of plane waves is always overcomplete in any numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Nevertheless, the structural spillage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) is still well-defined in DFT implemented with both a plane-wave or a localized basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the one hand, plane-wave-based DFT codes feature discretized momenta (imposed by the periodic boundary conditions of the supercell) and a high-momentum cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' These features do not constitute any fundamental problem for comparing two systems with different atomic structures, as long as one has access to (or can interpolate) the information at the same momenta in both systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the other hand, implementations of DFT with a localized basis, such as Gaussian or hydrogenic orbitals, do not directly output the information in plane-wave momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, knowing the shape of the orbitals, a Fourier transform gives access to it, and no problem appears regardless of the atomic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To calculate the structural spillage in DFT using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2), we extract the projector matrix elements on an orthonormal plane wave basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The pseudo-wavefunctions generated with VASP are orthonormal with respect to an overlap operator [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, by using the PAW approach, we perform a transformation to an orthonormal basis that spans the same space as the full wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Future improvements could use norm-conserving pseudopotentials, reconstructed full wavefunctions, or all-electron approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Besides imposing this orthonormality, we rearrange the wavefunction coefficient arrays of the amorphous supercell so that we compare the same momenta between both the amorphous supercell and the crystalline unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To corroborate that the spillage Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) is correctly implemented, we compared a crystalline supercell to a crystalline unit cell, which should recover the exact Bloch spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In particular, we considered crystalline Bi2Se3 as well as crystalline BiTeI, and 15 SOC No SOC Crystal Low-disorder High-disorder Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Orbital-resolved density of states (DOS) of the Bi (111) bilayer calculated with DFT, showing the contributions of the Bi p orbitals near the Fermi level (indicated by a vertical dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' First row: DOS without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Second row: DOS with SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Each column corresponds to a different structure: crystal in the the first column, low-disorder structure (standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='15Å) in the second column, and high-disorder system (standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='30Å) in the third column (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 3 in the main text for a real space view of these lattice structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' SOC drives a band inversion that occupies the pz orbital and empties the px,y orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' our method accurately diagnosed the band inversion in both systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In crystalline Bi2Se3 a band inversion at Gamma leads to a topological insulator phase which results in a spillage value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='12 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' When comparing the crystalline Bi2Se3 supercell to the unit cell we obtain a spillage of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='09 which exactly matches the result given by pymatgen [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For the case of disordered BiTeI, previous work showed that small amounts of disorder in the atomic positions cause the system to undergo a topological phase transitions from a trivial insulator (crystal) to a topological insulator (disordered) as a result of an induced band inversion [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This is caused by the modified crystal field of the orbitals near the Fermi level which pushes these states closer together when disordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the latter case, all point group symmetries are broken but translational symmetry is still present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this case, we find a spillage value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='17 at the A point where the band inversion occurss, and values of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='03 at other BZ points indicating there is a larger orbital spillage throughout the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The method still captures the topological band inversion in this case and exactly matches the results given by pymatgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Finally, let us comment further on the results obtained for the Bi (111) bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The disordered structures, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 3, are obtained by randomly displacing the atoms from their high-symmetry crystal positions following a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We choose the standard deviations to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='15Å and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='30Å for the low and high disorder systems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For standard deviations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='15Å the deviation from equilibrium position is small which preserves the bulk electronic gap while demonstrating our method works in the presence of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Standard deviations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='30Å lead to an average atomic displacement of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='41Å which is similar to atomic displacements seen in topological materials in the presence of disorder [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The structural spillage, shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 3 and S6, demonstrate that SOC drives a band inversion at the Gamma point with the result that all the crystalline and the disordered structures are topologically non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This band inversion is confirmed by the density of states of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S5, which further illustrates that the band inversion occurs between the pz and the px,y orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, the crystal and the amorphous systems display an increased occupation of the pz orbital after SOC is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Additionally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S5 illustrates that the Bi bilayer becomes metallic for sufficiently high structural disorder, in agreement to the tight-binding model (see section A 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, studying whether the amorphous system is extended or localized for strong disorder lies beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='00 Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Calculated structural spillage of the crystalline Bi bilayer from DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The value of 2 at the Gamma point indicates that the crystalline Bi bilayer with SOC is topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Appendix C: Defining the structural spillage in the tight-binding approximation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' General remarks and motivation In the main text we use the tight-binding spillage as a benchmark, and argue that the structural spillage is most useful within DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For completeness, in this appendix we give a pedagogical justification of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) for computing the structural quasi-Bloch spillage in the tight-binding approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' It is aimed to aid future studies in understanding the approximations that go into applying the structural spillage to tight-binding models, as alternative to topological markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Thus it can be skipped by readers only interested in applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Let us first highlight the problem of applying the general formulation of the structural spillage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) in the tight- binding approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' By tight-binding approximation we refer to the phenomenological tight-binding models where the only information about the wavefunctions is the position of their Wannier charge centers (and possibly their transformation properties under symmetries), but their spatial structure is unknown and therefore considered to be a Dirac delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' An implicit assumption of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) is that the Hilbert space of the system is the whole real space (in addition to the spin space), in which the plane waves constitute an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' While this is applicable in DFT (see Appendix B), it is not true in the tight-binding approximation, where the Hilbert space is just spanned by the positions of the Wannier charge centers (with the internal degrees of freedom of spin and orbital type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The fundamental problem for comparing two tight-binding systems with different lattice structures, as done by the structural spillage, stems from the fact that their Hilbert spaces are different, and therefore their overlap is ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' When projected to the tight-binding Hilbert space, the plane waves constitute a non-orthogonal and overcomplete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The overlap between these projected plane waves depends on the lattice structure, and therefore the usual formalism of non-orthogonal bases (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [73]) cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, by using the plane waves and the approximations described in this Appendix, one can derive a physically motivated expression for the structural spillage in the tight-binding approximation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The line of the argument for solving this problem works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The structural spillage (2) contains the matrix elements of the products of two projectors in the plane wave basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' By neglecting the momentum scattering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', by assuming that these operators are diagonal in momentum space, the fundamental problem of the disorder-dependent plane-wave overlaps is circumvented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, this introduces some new issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To bypass these, we choose the solution which, in the crystalline limit, gives results closer to the exact ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Our solution gives the exact results for the quantities containing matrix elements of just one projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the case of the structural spillage, which contains matrix elements of the product of two projectors, our results in the crystalline limit are not exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, we argue and numerically show for selected models that the results are similar in absolute value, and more importantly that the sharp changes in the spillage that signal topological transitions still show up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In order to separately understand the different issues that appear in the tight-binding, let us first consider the simple case of a system whose corresponding crystalline limit has a single site per unit cell, where the majority of problems suffered by the structural spillage in the tight binding do not appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Then, we will analyze the general multi-site case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' System with a single site per unit cell a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Setting the stage: crystalline system Consider a crystalline tight-binding system with Ncells unit cells and one site per unit cell, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', only one Wyckoff position with multiplicity one is occupied by an atom, Ns/c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, the number of sites is the same as the number of cells, Nsites = Ncells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The number of internal degrees of freedom (orbitals and spins) at each site does not influence the discussion below, so we omit this internal index for simplicity in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the tight-binding approximation, Wannier functions are unknown in real space, and therefore considered to be Dirac delta distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', the Wannier function |φR⟩ at the lattice site R has wavefunction: φR(r) = ⟨r|φR⟩ = δ(r − R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C1) We will always assume that the Wannier functions are orthonormal: ⟨φR′|φR⟩ = δR,R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C2) The plane wave with momentum p projected to the tight-binding Hilbert space is a state with a phase p · R at the site R, and normalized in the total volume of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Then, the Wannier functions in the plane wave basis read: φR(p) = ⟨p|φR⟩ = 1 √Nsites e−ip·R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C3) Moreover, the Bloch states defined at crystal momentum k in the first BZ are: |φk⟩ = 1 √Ncells � R eik·R|φR⟩, (C4) The overlap between the Bloch states and the plane waves is thus: ⟨p|φk⟩ = 1 Nsites � R ei(k−p)·R = � G δp,k+G, (C5) where G are the reciprocal lattice vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', G · R/2π ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, all the BZs are exactly equivalent in a crystalline one-atom tight-binding, since ⟨k + G|φk⟩ = 1 (C6) does not depend on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In other words, ⟨p|p + G⟩ = 1 for the crystal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', both plane waves are projected to the same state, which is exactly the Bloch state at k too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Finally, as a side remark, it is worth mentioning that even if there is a single site per unit cell, the BZs of a crystal are no longer equivalent if the orbitals have a finite spread in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, in this case, the overlap between the Bloch state and the plane waves is: ⟨k + G|φk⟩ = 1 Ncells � R eik·R⟨k + G|φR⟩ = 1 Ncells � R e−iG·R⟨k + G|φ0⟩ = φ0(k + G), (C7) where φ0(k + G) is the Fourier transform of the orbital located at the origin, which is generically not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Spillage comparing two crystals Let us remember that plane waves are an overcomplete set in the tight-binding Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this single-site case, the Hilbert space dimension is Nsites, which is the number of linearly independent plane waves needed for a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' One possible choice is selecting all the Ncells = Nsites momenta in one BZ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' the first BZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' These are linearly independent and orthogonal in the crystalline case (and also for an amorphous structure in the infinite size limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, this choice constitutes an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, in this basis we can directly apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2b) for the spillage, choosing to compare two crystals, with the particularity that the sums over reciprocal lattice vectors G disappear since there is only one in the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The key difference from the general multi-site case is that observables are the same irrespective of the BZ where the momenta for the basis are chosen, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', irrespective of the G chosen in the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Moreover, thanks to the equivalence between plane waves and Bloch states in this single-site case, observables projected to a plane wave p are equal to the crystalline quantities computed at Bloch momentum k = p mod G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In particular, the quasi-Bloch spillage (2), which is equal to the Bloch spillage because we are comparing two crystals, is also equal to the quasi-Bloch spillage without scattering (3) in this crystalline one-site case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 18 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural spillage comparing an amorphous system to a crystal The previous basis choice is also orthonormal for an amorphous system in the infinite-size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Consequently, unlike in the multi-site case that will be analyzed in the next section, the issue of the overlap between plane waves being different for the amorphous and crystalline systems does not appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, the structural quasi-Bloch spillage including scattering of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) can also be applied for comparing the amorphous structure with a crystalline one in this single-site tight-binding case (again the sums over reciprocal lattice vectors G drop out in this single-site case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As mentioned in the previous section, when comparing two crystals with a single site per unit cell, the quasi-Bloch spillage including scattering of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) coincides with the one without scattering of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This is no longer true when comparing an amorphous structure to a crystal, since the scattering resummation over k′ in the amorphous projector, which is carried out in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2), is neglected in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Now, although the structural quasi-Bloch spillage including scattering of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2) could in principle be applied, this would entail a high computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, other methods to indicate the topology in the tight-binding would be equally efficient (such as the local topological markers [40–43]), questioning the usefulness of the structural spillage applied to a tight-binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, to implement efficiently the structural spillage, we assume the no-scattering approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Because we neglect the scattering resummation over k′, the structural spillage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) becomes much more computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, an important inconvenience arising from neglecting the scattering is that the spillage depends on the BZ where the momenta for the plane wave basis are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This is because momenta from different crystalline BZs will no longer lead to equivalent results in the amorphous system, unlike in the single-site crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In fact, |p + G⟩ and |p⟩ no longer project to the same state (⟨p|p + G⟩ = 0 for the amorphous case in the infinite size limit), and the quantities projected in |p + G⟩ differ from those projected onto |p⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This problem raises the question of how to compute correctly the structural spillage in the no-scattering approximation between an amorphous material and a crystal, even in this single-site case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Although there is no unique answer, we now provide a justification for using momenta just in the first BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The tight binding has no information about the spatial extent of the orbitals, although we know that they are exponentially localized around the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, the tight-binding approximation captures well long-distance physics, but there is a short-distance-cutoff below which the tight-binding results are no longer reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' It is reasonable to assume that this cutoff is of the order of the nearest-neighbour distance rnn, which coincides with the lattice constant a in the crystalline single-site tight-binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, only plane-wave momenta below ∼ 2π/a are reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Consequently, the quasi-Bloch spillage computed just with plane-wave momenta in the first BZ is a sensible option (optionally, one could average over the first BZ and second BZs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Considering just the first BZ, the structural quasi-Bloch spillage without scattering reads γsingle-site-TB qB (k) = 1 2tr �� Pk − ˜Pk �2� , (C8) which is just Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) in the single-site case because, as mentioned before, all BZs are equivalent in the crystal, and therefore there is a single type of BZ, NBZs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' System with several sites per unit cell In this section, we will show that if there are more than one site in the unit cell, then a phase factor depending on the relative positions of the sites appears in the observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Unlike in the single-site case, this leads to some BZs being inequivalent in the crystal, requiring us to upgrade the single-site structural spillage Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Crystal: definitions and types of Brillouin zones Consider a crystal with Ncells unit cells at positions R and Ns/c sites per unit cell at positions tA with respect to the center of the cell R, so that the total number of sites is Nsites = Ncells · Ns/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The Bloch states with a definite sublattice are, therefore: |φA k ⟩ = 1 √Ncells � R eik·(R+tA)|φA R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C9) The projection of the Wannier functions onto plane-waves reads: φA R(p) = ⟨p|φA R⟩ = 1 √Nsites e−ip·(R+tA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C10) 19 1 e i2 /3 ei2 /3 e i2 /3 ei2 /3 e i2 /3 ei2 /3 Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' BZ types for the honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Colors are different for each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Red corresponds to a = 0 mod 3, and therefore a phase e−iG·tAB = eia2π/3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Blue represents a = 1 mod 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', a phase ei2π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Finally, green refers to a = 2 mod 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', a phase e−i2π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, the overlap between the Bloch states and the plane waves is: ⟨k + G|φA k ⟩ = 1 √Ns/c e−iG·tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C11) However, the band eigenvectors are combinations of these Bloch states in different sublattices: |ψn k⟩ = � A cnA k |φA k ⟩, (C12) and, therefore, their overlap with the plane waves reads: ⟨k + G|ψn k⟩ = 1 √Ns/c � A cnA k e−iG·tA, (C13) Let us now show that observables projected to a plane wave with momentum p = k+G depend on the phase factors e−iG·tAB, where tAB = tA −tB are the relative positions of the different sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For concreteness, let us start considering the simplest observable, that will be a building block for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' the spillage: the projector onto band n at crystal momentum k, P n k = ��ψn k⟩⟨ψn k ��: ⟨k + G ��P n k ��k + G⟩ = ��⟨k + G ��ψn k⟩ ��2 = 1 Ns/c � A,B cnA k � cnB tk �∗ e−iG·tAB = 1 Ns/c � �1 + � A̸=B cnA k � cnB k �∗ e−iG·tAB � � , (C14) which is different from tr [P n k ] = 1 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' These phase factors, which depend on G, lead to at least some BZs being inequivalent even if the orbitals are still Dirac deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, the types of BZs in the multi-site crystal can be classified by the set of phase factors � e−iG·tAB� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In general, some BZs become inequivalent whenever there is structure inside the unit cell, irrespective of whether it comes from spatially-extended orbitals or from several sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As an example, consider the honeycomb lattice, where there are Ns/c = 2 sublattices A and B such that tAB = −a � 0, 1/ √ 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The reciprocal lattice basis vectors are G1 = 4π/ √ 3a �√ 3/2, 1/2 � , and G2 = 4π/ √ 3a [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A general reciprocal lattice vector G = n1G1 + n2G2, with n1, n2 ∈ Z, satisfies G · tAB = −4π/3(2n2 + n1) = 2π/3 · 2(2n2 + n1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, e−iG·tAB = eia2π/3, with a ∈ Z3, so there are NBZs = 3 different types of BZs depending on the value of this phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' If we consider all possible momenta, from zero to infinity, then the multiplicity in momentum space of each type of BZ is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the other hand, if we only consider momenta up to a cutoff pmax, then the multiplicity in momentum space of each type of BZ can be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S7 shows the type of the first BZ and the six nearest-neighbour second BZs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Note that the first BZ has G = 0, and therefore it is always characterized by a = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', by a phase e−iG·tAB = eia2π/3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Crystal: recovering the exact results using plane waves We now ask the question of how to recover the exact values of the observables in the crystalline tight binding, this time using the plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We also keep in mind that we want to later extend our definitions to the amorphous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' First, we have to choose a basis of plane waves for this crystalline multi-site case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The tight-binding Hilbert space has dimension Nsites = Ns/c · Ncells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, a possibility is to select Ncells plane waves in Ns/c inequivalent BZs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Decomposing 20 the plane-wave momenta as p = k + G, we find that plane waves with different k are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, in contrast to the single-site case, plane waves with the same k but differing in a reciprocal lattice vector G are generically neither orthogonal nor equivalent in the crystalline case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' It is only when the differing reciprocal lattice vector G verify � e−iG·tAB� = {1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', when the BZs are equivalent, that the projected plane waves are equivalent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For instance, in the honeycomb lattice, where Ns/c = 2, we can choose the basis in the first BZ (G0 = 0) and in the G1 = 4π/ √ 3a(0, 1) BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this example, the overlap between plane waves is |⟨k + G0|k + G1⟩| = |⟨k|k + G1⟩| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, we have to use the formalism of non-orthogonal bases (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' [73]) and properly modify the quasi-Bloch spillage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Within this formalism, the closure relation reads: 1 = � k � GG′ ��k + G⟩ � S−1� G,G′ ⟨k + G′��, (C15) where the overlap matrix is defined as SG,G′ = ⟨k + G ��k + G′⟩, which depends only on the difference G′ − G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Also, the sums over the reciprocal lattice vectors G run over the Ns/c BZs chosen in the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the previous example of the honeycomb lattice, they would run over G0 = 0 and G1 = 4π/ √ 3a(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Using this expression for the closure relation, we can derive the expressions for the observables in this non-orthogonal plane-wave basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For example, the trace of the projector onto band n at crystal momentum k, tr [P n k ], becomes tr [P n k ]non-orth = � GG′ ⟨k + G ��P n k ��k + G′⟩ � S−1� G′,G , (C16) Importantly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C16) recovers the expected crystalline value tr [P n k ] = 1, irrespective of the chosen plane-wave basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Furthermore, in this non-orthogonal basis, the quasi-Bloch spillage is given by the appropriate generalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2a): γnon-orth qB (k) = 1 2 � k′ � G1G2G3G4 � αβ � P αβ k+G1,k′+G2 � S−1� G2,G3 P βα k′+G3,k+G4 � S−1� G4,G1 − −P αβ k+G1,k′+G2 � S−1� G2,G3 ˜P βα k′+G3,k+G4 � S−1� G4,G1 � + � P ↔ ˜P � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C17) Crucially, when comparing two crystals, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C17) exactly recovers the Bloch spillage, regardless of the plane wave basis chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Comparing an amorphous system to a crystal using the structural spillage: no-scattering approximation Let us now try to compute the structural spillage between a crystalline and an amorphous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Aside from the issues already discussed for the single-site case, here is where comparing two tight bindings with sites at different positions becomes problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The reason is that overlap between the plane waves is different in the crystal and in the amorphous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the crystal, as discussed in section C 3 a, some plane waves ��p + G⟩ are different states from ��p⟩, yet their overlap is non-zero, ⟨p ��p + G⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the amorphous system, in the limit of infinite size, all plane waves are inequivalent (as in the single-site case), and more significantly, they are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the structural spillage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C17), the crystalline and the amorphous projector appear sandwiched between the overlap matrices, but this overlap depends on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, we cannot apply the previous non-orthogonal formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As explained in the main text, this issue can be avoided by neglecting the momentum scattering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', by setting k′ = k and G′ = G in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Such approximation has been used previously to determine the topology of an amorphous system using other methods such as the effective Hamiltonian approach [14, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' It is also inspired by the fact that continuous translational symmetry is recovered after averaging over different disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Let us now write the expressions for the projector and the spillage within this approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the one hand, the trace of the projector into band n at crystal momentum k simplifies to: tr [P n k ]no scatt = � G ⟨k + G ��P n k ��k + G⟩, (C18) where the sums over the reciprocal lattice vectors G again run over the Ns/c BZs chosen in the plane wave basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the other hand, the corresponding expression for the structural quasi-Bloch spillage without scattering, which is obtained by setting k′ = k and G′ = G in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2a), reads: γno scatt qB (k) = 1 2 � G tr �� Pk+G − ˜Pk+G �2� , (C19) 21 where the trace acts over the internal degrees of freedom α, and, as in the main text, P αβ p = ⟨p|P|p⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C19) is not yet the definite expression of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) for the structural spillage in the tight-binding approximation, since it still suffers from a problem that we detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Taking into account different types of Brillouin zones In contrast to the single-site case, the values of the observables computed within this no-scattering approximation depend on the BZs chosen in the basis even in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' The reason is the presence of different types of BZs (see Appendix C 3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this section, we will provide a method to circumvent this issue based on the condition that, when applied to crystals, it leads to values as close as possible to the exact crystalline values, where rigorous proofs exist [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In short, our solution consists of computing a observable without scattering, performing an average over the NBZs different types of BZs, and then multiplying by the number of sites per unit cell Ns/c in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' First, let us show that our proposal recovers the correct crystalline result for the observables that depend only on one projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, the BZ-averaged Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C18) representing the trace of the projector into the band n at crystal momentum k becomes: tr [P n k ]BZ av no scatt = Ns/c NBZs � a∈BZs ⟨k + Ga ��P n k ��k + Ga⟩ = 1 + � A̸=B cnA k � cnB k �∗ � 1 NBZs � a∈BZs e−iGa·tAB � = 1, (C20) where the sum over a runs over a representative BZ of each type, and we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C14) and the fact that the term inside the square brackets vanishes identically for A ̸= B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' If there is a finite number NBZs of BZ types, this term vanishes because the NBZs phases e−iGa·tAB are the 1/NBZs roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' If there are infinite BZ types, which might occur, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', if the sites are located at a generic nonsymmetric Wyckoff position incommensurate with the reciprocal lattice vectors, then this term vanishes due to the infinite sum of a continuum of phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the example of the honeycomb lattice, where NBZs = 3 and e−iGa·tAB = eia2π/3 with a ∈ Z3 if A ̸= B, and e−iGa·tAB = 1 if A = B, we obtain, as expected: 1 3 � a=0,1,2 e−iGa·tAB = δAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C21) We have also verified that the correct crystalline results are obtained numerically in our bismuthene and Bi bilayer tight-binding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S8 shows the number of occupied states per unit cell � n∈occ tr [P n k ]BZ av no scatt at k = 0 as a function of the onsite SOC for crystalline bismuthene and Bi bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In both models, this number of occupied states (or filling) is constant and equal to 4 and 6, as expected, since they correspond to half-filling in bismuthene and Bi bilayer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Note that the filling artificially deviates from these values close to the topological transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, this is an artifact stemming from the finite KPM resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, this artifact only appears close to the transition, which is where the bulk gap is smaller, and therefore is where the required precision to obtain the correct results is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' We have checked that the deviations from the exact filling shrink when increasing the KPM precision and the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In summary, we have shown that, by averaging over the BZ types and multiplying by Ns/c, we recover the correct values in the crystal for the quantities that involve the trace of one projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This exact result is recovered despite neglecting both the scattering by different reciprocal lattice vectors and the non-orthogonality of the plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This means that the scattering does not play a crucial role in the quantities that involve the trace of only one projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Structural spillage without scattering in the tight-binding approximation Now, let us consider quantities that involve the trace of two projectors, such as the spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Unlike in the quantities involving just one projector, here scattering plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, we will show that scattering should be included to obtain the exact result in the crystalline limit (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2b), where the sum over G′ represents the scattering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, as explained in Appendix C 3 c, the scattering has to be neglected in order to be able to use the structural spillage to compare amorphous and crystalline systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Nevertheless, we will also show that, even if the crystalline results are not exactly recovered, our method gives reasonably good results, which allows the structural spillage to work as a topological indicator also in the tight-binding approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Consider, the trace of (P n k )2, which should be equal to one if P n k is a projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' If we include scattering and average over 22 (a) (b) Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Sum over occupied bands of the trace of one and two projectors, � n∈occ tr [P n k ]BZ av no scatt and � n∈occ tr � (P n k )2�BZ av no scatt, as a function of onsite SOC, computed using the formalism of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C20) and (C23) at k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (a) Bismuthene crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (b) Bi bilayer crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the one hand, the filling � n∈occ tr [P n k ]BZ av no scatt recovers the exact crystalline result, except close to the transition due to finite precision effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the other hand, the trace of the projector square � n∈occ tr � (P n k )2�BZ av no scatt, which should be equal to the filling, is just slightly (∼ 8 − 25%) smaller due to neglecting the momentum scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Brillouin zones this exact condition is fulfilled for the crystal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' as can be checked explicitly: tr � (P n k )2�BZ av scatt = Ns/c NBZs � a∈BZs Ns/c NBZs � a′∈BZs � ⟨k + Ga ��P n k ��k + Ga + Ga′⟩⟨k + Ga + Ga′��P n k ��k + Ga⟩ � = = � A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='D cnA k � cnB k �∗ cnC k � cnD k �∗ 1 NBZs � a∈BZs e−iGa·(tAB+tCD) 1 NBZs � a′∈BZs e−iGa′·tCB = = � A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='D cnA k ��cnB k ��2 � cnD k �∗ 1 NBZs � a∈BZs e−iGa·tAD = � A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='B ��cnA k ��2��cnB k ��2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C22) However, including scattering is not possible in general, unlike BZ averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' As explained above, the scattering cannot be taken into account when the two projectors belong to systems with a different lattice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, when computing two-projector quantities we still perform the BZ average on the external sum over Ga, but are forced to neglect the scattering resummation over Ga′: tr � (P n k )2�BZ av no scatt = Ns/c NBZs � a∈BZs � ⟨k + Ga ��P n k ��k + Ga⟩⟨k + Ga ��P n k ��k + Ga⟩ � = = 1 Ns/c � A,B,C,D cnA k � cnB k �∗ cnC k � cnD k �∗ 1 NBZs � a∈BZs e−iGa·(tAB+tCD) = = 1 Ns/c � A,B,C,D cnA k � cnB k �∗ cnC k � cnD k �∗ δtAB+tCD,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C23) Although this equation does not exactly recover the crystalline value, we have numerically verified that the sum over occupied bands of this Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C23), � n∈occ tr[(P n k )2]BZ av no scatt, gives values just ∼ 8 − 25% smaller than � n∈occ tr[P n k ]BZ av no scatt in the crystal, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, we take this as a reasonable approximation, especially taking into account that this quantity can also be computed when one of the projectors corresponds to an amorphous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Applying this method to the structural quasi-Bloch spillage, we arrive at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In order to implement the tight-binding spillage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) we need to account for a final detail: the choice of a representative BZ of each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This is a requirement because we introduced the average over BZ types in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C20)-(C23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To perform this average, one has to select one representative for each type of BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To this end, let us consider the example of the honeycomb lattice relevant to our Bi models, which has NBZs = 3 types of BZ, as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Due to the argument which lead us to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C8) in Appendix C 2 c, the optimal criterium for choosing the BZ representatives is to consider the ones whose reciprocal lattice vector is smaller in modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For example, the first BZ will always be chosen as the representative of the BZs characterized by a phase eiG·tAB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' There can still be several options, such as the three possibilities for the BZs with phases eiG·tAB = e±i2π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this case, one can choose any of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' A better choice however is to perform an angular average over Zn tr[Pn] neoco 8 cell 6 : states 4 # 2 1 2 入/t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='neocc 12 cell 9 : states 6 # 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='0 入/t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='23 them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Indeed, while the crystal is anisotropic, the amorphous structure is effectively isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In particular, although the total traces in the crystal are exactly the same in all equivalent BZs, some orbital-resolved quantities might vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For instance, in the honeycomb lattice, if the occupied eigenstate at G = 4π/ √ 3(0, 1) is of py character, the eigenstate at the threefold rotated ˆC3G = 4π/ √ 3(− √ 3/2, −1/2) is of the threefold rotated −( √ 3/2)px − (1/2)py character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' On the other hand, for sufficiently large samples, amorphous structures are expected to be isotropic in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Therefore, one would ideally perform an angular average over the G corresponding to equivalent BZs with the same modulus, but pointing in a different direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In the honeycomb lattice, the quantity corresponding to the BZs with phase eiG·tAB = e+i2π/3 would be an average over the three BZs shown in blue in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Consequently, when the corresponding crystal displays a honeycomb lattice, the angle-averaged Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) for the structural quasi-Bloch spillage in the tight-binding approximation reads: γTB qB (k) = 2 3 � � � 1 2tr �� Pk+G0 − ˜Pk+G0 �2� + 1 3 � Gm 1 1 2tr �� Pk+Gm 1 − ˜Pk+Gm 1 �2� + 1 3 � Gm 2 1 2tr �� Pk+Gm 2 − ˜Pk+Gm 2 �2�� � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C24) where: G0 = 0 ⇒ e−iG0·tAB = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C25) � � � G0 1 = 4π/ √ 3(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 1) G1 1 = ˆC3G0 1 = 4π/ √ 3(− √ 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' −1/2) G2 1 = ( ˆC3)2G0 1 = 4π/ √ 3( √ 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' −1/2) � � � ⇒ e−iGm 1 ·tAB = ei2π/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C26) � � � G0 2 = 4π/ √ 3(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' −1) G1 2 = ˆC3G0 2 = 4π/ √ 3( √ 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 1/2) G2 2 = ( ˆC3)2G0 2 = 4π/ √ 3(− √ 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 1/2) � � � ⇒ e−iGm 2 ·tAB = e−i2π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C27) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (C24) is a specific instance of the general Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) that we used for computing the spillage in our bismuthene and Bi bilayer tight-binding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, we have also checked that in these models, for the system sizes considered, performing the angular average or not does not noticeably change the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In summary, our proposed method for computing two-projector quantities, such as the structural spillage, consists of neglecting the momentum scattering, performing an average over the different types of BZs, and multiplying by the number of sites per unit cell in the corresponding crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Applying this method to the structural quasi-Bloch spillage, we arrive at the final expression for the structural spillage in the tight-binding approximation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To conclude, we highlight that, in the specific case when the number of types of BZs is infinite or very large, (3) would involve reciprocal lattice vectors |G| ≫ 2π/a, with a the crystalline lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this case, as in the single-site case, we may introduce a momentum cutoff and consider only the reciprocal lattice vectors G smaller than this cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Phase transition criterion in the tight-binding approximation In this section we define our criterion to choose the topological transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' To this end it is important to note first that, as mentioned above, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) does not exactly recover the values of the Bloch spillage when applied to two crystals with and without SOC, because we neglected scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, we have numerically verified that it results in similar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In particular, the maximum spillage without scattering is max � γTB qB (k = 0) � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='5 in the two models, which is a factor of 4/3 smaller than the exact spillage max [γqB(k = 0)] = 2 that would be recovered after considering the scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This is related to the fact that � n∈occ tr � (P n k=0)2�BZ av no scatt is a factor of 4/3 smaller than � n∈occ tr [P n k=0]BZ av no scatt in the topological and trivial phases for the bismuthene and Bi bilayer tight-binding models, respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' There is no reason to believe that this factor is universal, and thus we consider it model dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' With this in mind, in order to identify the topological phases in a tight-binding phase diagram, we take the criterion that the topological transition occurs when the quasi-Bloch spillage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (3) equals to half the maximum value of the spillage between two topologically different crystals when scattering is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In both our models, this critical value equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, in general, this critical value of the tight-binding structural spillage will be model-dependent, and must be determined in a case-to-case basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' 24 Appendix D: Absence of a corresponding crystal: spin-orbit plane-wave spillage One of our assumptions for applying the structural quasi-Bloch spillage of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2)-(3) is that there exists a crystalline structure with similar local environments to the non-crystalline one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' While this is a quite generic feature [37], there are also some amorphous and quasicrystalline structures whose local environment is different to any crystalline phase of the same material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this case, while the structural quasi-Bloch spillage could still be calculated, it would probably not be very indicative of the topology, since many possibly trivial band inversions could occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' In this case, one could again resort to computing the spin-orbit Bloch spillage comparing an amorphous supercell with and without SOC, as proposed for crystals by Liu and Vanderbilt [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, as mentioned in the main text, this would always be a large quantity due to the big size of the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Liu and Vanderbilt proposed to fix this issue by analyzing valence- and conduction-band-resolved spillages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, these are not gauge-invariant, and a careful analysis is required to discern the topological character using this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' These solutions are not practical from the point of view of a performing high-throughput screening of amorphous materials, where it is desirable to define a quantity that is easily implemented and analyzed using ab-initio codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For such cases without a crystalline counterpart, we propose instead a plane-wave-resolved spin-orbit spillage comparing an amorphous system with and without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' This spin-orbit plane-wave spillage γpw(p) is defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (2a) but without the sum over crystalline reciprocal lattice vectors G: γpw(p) = 1 2 � p′ � αβ � P αβ p,p′P βα p′,p − P αβ p,p′ ˜P βα p′,p � + � P ↔ ˜P � (D1) where p and p′ are plane-wave momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' For a supercell Gamma calculation in DFT, p and p′ would be the supercell reciprocal lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' Now, since both systems that are being compared have the same structure, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' (D1) can also be applied within a tight-binding approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} +page_content=' However, for the latter approximation, one could first compute the much more efficient plane-wave spillage without scattering, which would read: γno scatt pw (p) = 1 2tr �� Pp − ˜Pp �2� , (D2) We however leave the benchmarking of the plane-wave spillage for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQf0gJJ/content/2301.02686v1.pdf'} diff --git a/PdE5T4oBgHgl3EQfYg_i/vector_store/index.faiss b/PdE5T4oBgHgl3EQfYg_i/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..75330bd91a022f71ce00210fb1122226d7ef4a4f --- /dev/null +++ b/PdE5T4oBgHgl3EQfYg_i/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6a0f136fc0c366badab9eca18c628bb5ef13dabd1c31ef5239fb39461f66baaf +size 7733293 diff --git a/QdE2T4oBgHgl3EQfVwfD/content/tmp_files/2301.03827v1.pdf.txt b/QdE2T4oBgHgl3EQfVwfD/content/tmp_files/2301.03827v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a932751011525e084b7d0a95a85694d83b505cd --- /dev/null +++ b/QdE2T4oBgHgl3EQfVwfD/content/tmp_files/2301.03827v1.pdf.txt @@ -0,0 +1,1607 @@ +arXiv:2301.03827v1 [q-bio.GN] 10 Jan 2023 +Longest common subsequence algorithms +and applications in determining +transposable genes +Yue Wang1,2,* +1Department of Computational Medicine, University of California, +Los Angeles, California, United States of America +2School of Mathematical Sciences, Peking University, Beijing, China +*E-mail address: yuew@g.ucla.edu. ORCID: 0000-0001-5918-7525 +Abstract +Given several number sequences, determining the longest common +subsequence is a classical problem in computer science. Transposons +are nucleotide sequences in DNA that can change their positions. Many +transposons are shorter than a general gene. When we restrict to nu- +cleotide sequences that form complete genes, we can still find genes +that change their relative locations in a genome. Thus for different +individuals of the same species, the orders of genes might be different. +A practical problem is to determine such transposable genes in given +gene sequences. Through an intuitive rule, we transform the biological +problem of determining transposable genes into a rigorous mathemati- +cal problem of determining the longest common subsequence. Depend- +ing on whether the gene sequence is linear (each sequence has a fixed +head and tail) or circular (we can choose any gene as the head, and +the previous one is the tail), and whether genes have multiple copies, +we classify the problem of determining transposable genes into four +scenarios: (1) linear sequences without duplicated genes; (2) circular +sequences without duplicated genes; (3) linear sequences with dupli- +cated genes; (4) circular sequences with duplicated genes. With the +help of graph theory, we design fast algorithms for different scenar- +ios. +Specifically, we study the situation where the longest common +subsequence is not unique. +KEY WORDS: transposon, gene sequence, algorithm, graph +1 + +1 +Introduction +The nucleotide sequence can be changed by various events, such as inver- +sion, insertion, deletion, and duplication [28]. Such rearrangement events +lead to the existence of transposons (also called transposable elements or +jumping genes), which are DNA sequences that can change their relative +positions within the genome. Transposons were first discovered in maize by +Barbara McClintock [41]. Transposons have various types: long terminal re- +peats (LTR) retrotransposons, Dictyostelium intermediate repeat sequence +(DIRS)-like elements, Penelope-like elements (PLE), long interspersed ele- +ments (LINE), short interspersed elements (SINE), terminal inverted repeats +(TIR), Helitrons, etc. [40]. +Transposons are common in various species. For the human genome, the +proportion of transposons is approximately 44%, although most of trans- +posons are inactive [42]. Transposons can participate in controlling gene +expression [80], and they are related to several diseases, such as cancer [13], +hemophilia [33], and porphyria [45]. Transposons can drive rapid pheno- +typic variations, which cause complicated cell behaviors [78, 48, 47, 11, 29]. +Transposons can be used to detect cancer drivers [49] and potential therapies +[2]. Transposons are also essential for the development of Oxytricha trifallax +[50], antibiotic resistance of bacteria [3], and the proliferation of various cells +[53, 76, 14]. With the presence of transposons, the regulation between genes +might be affected, which is a challenge for inferring the structures of gene +regulatory networks [72] and general transcriptome analysis [58, 79]. +When transposons have been determined, we can use them to compare +the genomes of different species, and such comparisons can be combined with +other measurements between species, such as metrics on developmental trees +[68]. Such comparisons can be also extended to different tissues to help with +the prediction of tissue transplantation experiments [73]. Besides, for some +species, cells at different positions have different gene expression patterns, +which might be related to transposons [70]. +Many transposons are as short as 102 − 103 base pairs, shorter than a +general gene [52]. To determine such short transposons, one needs to analyze +the original AGCT nucleotide sequences. There have been many algorithms +developed to determine short transposons from nucleotide sequences, such as +MELT (Mobile Element Locator Tool) [18], ERVcaller (Endogenous Retro- +Virus caller) [10], and TEMP2 (Transposable Elements Movements Present +2) [77]. +Different algorithms may only determine certain types of trans- +posons. For more details, readers may refer to other papers [51, 20]. They +use raw DNA sequencing data, which only contain imperfect information +2 + +about the true DNA sequence, and the data quality depends on some factors +that vary across different datasets [17]. Besides, they need a corresponding +genome or reference transposon libraries. +There are gross DNA changes that associate with many genes, also called +genomic rearrangements [21]. Such rearrangements include inversion, trans- +position, fusion, and fission [8]. To determine such gross genomic rearrange- +ments, one first needs to convert nucleotide sequences into gene sequences +by annotation. +For two different gene sequences, the general idea of de- +termining rearrangements is to calculate the minimal number of operations +required for transforming one sequence into the other [60]. +This defines +an editing distance between gene sequences, which can be used to compare +the evolution distance between species and construct the phylogenetic tree +[59]. +There have been many algorithms developed to determine genomic +rearrangements. +They consider different scenarios: whether the gene se- +quence is linear or circular, whether genes have unique labels, and what +operations can be taken. Kececioglu and Sankoff only consider inversion for +linear sequences with unique gene labels [34]; Blanchette et al. consider in- +version and transposition for circular sequences with unique gene labels [6]; +Tesler considers inversion, transposition, fusion, and fission for linear and +circular sequences with unique gene labels [60]; Terauds and Sumner study +circular sequences with representation theory tools [59]; Bohnenk¨amper et +al. consider linear and circular sequences with possibly duplicated labels +[8]. +There are also systematic pipelines for determining rearrangements +from whole-genome assemblies [19, 43]. Nevertheless, these methods con- +sider large-scale rearrangements, and minimize the number of operations to +transform one gene sequence into the other, not concrete genes that can +change their locations. Besides, these methods only compare two gene se- +quences, not more. Their results depend on the set of possible operations, +which is somewhat arbitrary. +In this paper, we consider a mesoscopic scenario between the genomic +rearrangement situation and the short transposon situation: Given accu- +rately annotated gene sequences (not nucleotide sequences) from different +individuals, determine individual genes (not short nucleotide segments or +long gene strands) that can change their locations (transposable). This pro- +vides a qualitative description for the stability of genes, which can guide gene +editing [65] and phylogenetics [32]. The proportion of fixed genes quanti- +fies the robustness of the genome. We aim at minimizing the number of +genes to move. When there are only two gene sequences, this is equivalent +to calculating genomic arrangements, where the only allowed operation is +single-gene transposition. +3 + +In the copy-paste (duplication) case and deletion case, we can compare +the numbers of copies of genes for different individuals to determine the +transposable genes that have changed their copy numbers. In the inver- +sion case, we can check the direction of genes to determine transposable +genes that have changed their orientations [38]. In the cut-paste (insertion) +case, the compositions of gene sequences are the same, but the orders of +genes differ. It is not straightforward to uniquely determine which genes +have changed their relative locations. Instead, we can consider the comple- +ment of transposable genes, which keep their relative locations and form a +common subsequence of gene sequences from different individuals. Notice +that genes in a subsequence does not need to be adjacent in the original +sequences, different from a substring. We aim at explaining the difference +among gene sequences with minimal transposable genes, meaning that we +want to maximize the length of the complement of transposable genes. Thus +we define the transposable genes to be the complement of the longest com- +mon subsequence. Given raw nucleotide sequences, we first transform them +into gene sequences. Then we apply our algorithms to find the longest com- +mon subsequence, and the complement is transposable genes. If the longest +common subsequence is not unique, we also need to determine which genes +are more conserved and appear in all longest common subsequences. +It is common to use the length of the longest common subsequence as a +quantitative score for comparing DNA sequences [12, 26, 81]. The longest +common subsequence has also been used to define ultraconserved elements +[54] or remove incongruent markers [16]. +Determining the longest common subsequence is a classical problem in +computer science. +Various scenarios for this problem have been studied. +Here we list Scenarios A-E, where the first two are more commonly stud- +ied. For more works in these scenarios, readers may refer to more thorough +reviews [5, 24, 74]. Scenario A considers two sequences with possibly re- +peated genes, and the sequence length is n. The goal is to find the longest +common subsequence, where the length is count by gene copies. This can +be solved by dynamic programming with O(n2) time complexity and O(n) +space complexity [23], but O(n2−ǫ) time complexity for any ǫ > 0 is im- +possible [4]. This also can be solved with o(n) space complexity and O(n3) +time complexity [35]. In Scenario B, there are m sequences with possibly +repeated genes, and the sequence length is n. The goal is to find the longest +common subsequence, where the length is count by gene copies. A standard +dynamic programming algorithm has O(nm) time complexity [7]. +There +have been other faster algorithms [64, 44, 27]. This scenario is equivalent to +the maximum clique problem in graph theory, which is NP-hard [39], but +4 + +has fast exact and heuristic algorithms [30, 37, 69]. Scenario C considers 2 +sequences with possibly repeated genes, and the sequence length is n. The +goal is to find the longest common subsequence, where each gene appears at +most once. This scenario is NP-hard [1]. Scenario D is similar to Scenario +B, but only consider common subsequences that contain or do not contain +certain strings [66, 46]. In Scenario E, the gene sequences are arc-annotated, +and the longest common subsequence should have the same arc annotation +in original sequences [31]. +In this paper, we consider four scenarios that are different from the +previously studied longest common subsequence problems. These four sce- +narios are determined by two factors: whether the considered species has +linear or circular gene sequences, and whether genes have multiple copies. +When genes have multiple copies, we only consider common subsequences +that consist of all or none of copies of the same gene. Scenario 1 has linear +sequences without duplicated genes; Scenario 2 has circular sequences with- +out duplicated genes; Scenario 3 has linear sequences with duplicated genes; +Scenario 4 has circular sequences with duplicated genes. +Most known methods only aim at finding one longest common subse- +quence. When the longest common subsequence is not unique, we also need +to classify whether a gene appears in all/some/none of the longest common +subsequences. Determining all longest common subsequences is too time- +consuming. To determine the relationship between genes and longest com- +mon subsequences, we develop corresponding algorithms with polynomial +time complexities for Scenarios 1,2 (Algorithms 2,4). +To our knowledge, +there are no other determinations of whether genes appear in all longest +common subsequences with polynomial complexities. +Scenarios 3,4 only +consider subsequences that consist of all or none copies of the same gene, +and calculate the length by genes. Therefore, they are different from the +classic Scenario B. We develop the equivalence of Scenario 3 with the maxi- +mum clique problems on graphs (Proposition 1). We prove that Scenario 4 is +between the maximum clique problems on graphs and the maximum clique +problems on 3-uniform hypergraphs (Propositions 2, 3). Although circular +sequences are commonly studied in the context of genomic rearrangements, +they are rare in the literature of longest common subsequence problems. +Therefore, our Algorithm 3 that finds a longest common subsequence for +Scenario 2 should also be novel. We test Algorithms 1,2,3,4 on the gene +sequences of different Escherichia coli individuals and find some possible +transposable genes. +If we only need to find one longest common subsequence, then Scenario +1 is a special case of Scenario B, and our method (Algorithm 1) is easily +5 + +derived from standard algorithms. Scenarios 3,4 are equivalent to maximum +clique problems in graphs and hypergraphs, which are NP-hard. These prop- +erties are also similar to Scenario B. Although there have been numerous +algorithms for the maximum clique problem [75], for the sake of complete- +ness, we design fast heuristic algorithms (Algorithms 5,6) and test them to +find that they only fail in rare cases. +We proposed the idea of using the longest common subsequence to find +transposable genes and Algorithm 1 in a previous paper [32], where Algo- +rithm 1 was applied to study the “core-gene-defined genome organizational +framework” (the complement of transposable genes) in various bacteria, and +found that for different species, the transposable gene distribution and de- +velopmental traits are correlated. +This paper considers other situations +(especially when the longest common subsequence is not unique), and can +be regarded as a theoretical sequel of that previous paper. Algorithm 1 is +contained in this paper for the sake of completeness. +In sum, our main contributions are Algorithms 2,3,4 in Scenarios 1,2 and +Propositions 1, 2, 3 in Scenarios 3,4. +We first describe the setup for the problem of determining transposable +genes and transform it into the problem of finding the longest common subse- +quence. In the following four sections, we transform them into corresponding +graph theory problems and design algorithms. We finish with some discus- +sions. All the algorithms in this paper have been implemented in Python. +For the code and data files, see https://github.com/YueWangMathbio/Transposon. +2 +Setup +Given raw DNA sequencing data, the first step is to transform them into +gene sequences. This can be done with various genome annotation tools +[57, 9]. For simplicity, we replace the gene names by numbers 1, . . . , n. +For some species, the DNA is a line [56]. We can represent this DNA as a +linear gene sequence of distinct numbers that represent genes: (1, 2, 3, 4). If +some genes change their transcriptional orientations, we can simply detect +them and handle the remaining genes. Now a linear DNA naturally has +a direction (from 5’ end to 3’ end), thus (1, 2, 3, 4) and (4, 3, 2, 1) are two +different gene sequences. +Consider two linear gene sequences from different individuals: (1, 2, 3, 4) +and (1, 4, 2, 3). We can intuitively detect that gene 4 changes its relative +position, and should be regarded as a transposable gene. However, changing +the positions of genes 2, 3 can also transform one sequence into the other. +6 + +The reason that we think gene 4 (not genes 2, 3) changes its relative position +is that the number of genes we need to move is smaller. However, the number +of genes that change their relative locations is difficult to determine. We can +consider the complement of transposable genes, i.e., genes that do not change +their relative positions. These fixed genes can be easily defined as the longest +common subsequence of given gene sequences. Here a common subsequence +consists of some genes (not necessarily adjacent, different from a substring) +that keep their relative orders in the original sequences. Thus transposable +genes are the complement of this longest common subsequence. Notice that +the longest common subsequence might not be unique. We classify genes by +their relations with the longest common subsequence(s). The motivation of +classifying transposable genes with respect to the intersection and union of +longest common subsequences is similar to defining essential variables with +Markov boundaries in causal inference [71]. +Definition 1. A gene is proper-transposable if it is not contained in any +longest common subsequence. A gene is non-transposable if it is contained +in every longest common subsequence. A gene is quasi-transposable if it +is contained in some but not all longest common subsequences. +In the example of (1, 2, 3, 4) and (1, 4, 2, 3), the unique longest com- +mon subsequence is (1, 2, 3). Thus 4 is proper-transposable, and 1, 2, 3 are +non-transposable. In the following, we consider other scenarios, where the +proper/quasi/non-transposable genes still follow Definition 1, but the defi- +nition of the longest common subsequence differs. +For some species, the DNA is a circle, not a line [63]. A circular DNA +also has a natural direction (from 5’ end to 3’ end), and we use the clock- +wise direction to represent this natural direction. In the circular sequence +scenario, a common subsequence is a circular sequence that can be obtained +from each circular gene sequence by deleting some genes. See Fig. 1 for two +circular gene sequences and their longest common subsequence. Notice that +we can rotate each circular sequence for a better match. +1 +2 +3 +3 +1 +2 +1 +2 +6 +5 +4 +5 +4 +6 +5 +4 +Figure 1: Two circular gene sequences without duplicated genes and their +longest common subsequence, corresponding to Scenario 2. +7 + +A gene might have multiple copies (duplicated) in a gene sequence [25]. +Notice that the definition of the transposable gene is a gene (specific DNA +sequence) that has the ability to change its position, not a certain copy of +a gene that changes its position. This means transposable genes should be +defined for genes, not gene copies. Thus we should only consider common +subsequences that consist of all or none copies of the same gene. +When +calculating the length of a common subsequence, we should count genes, +not gene copies. Consider two linear sequences (4, 1, 2, 1, 1, 3, 2, 4, 1, 1) and +(4, 1, 2, 3, 1, 1, 2, 1, 1, 4). If we consider any subsequences, the longest com- +mon subsequence is (4, 1, 2, 1, 1, 2, 1, 1); if we only consider subsequences that +contain all or none copies of the same gene, but count the length by copies, +the longest common subsequence is (1, 2, 1, 1, 2, 1, 1); if we only consider +subsequences that contain all or none copies of the same gene, and count +the length by genes, the unique longest common subsequence is (4, 2, 3, 2, 4), +and gene 1 is proper-transposable. +When we consider circular gene sequences with duplicated genes, we +should still only consider subsequences that consist of all or none copies +of the same gene, and calculate the length by genes. Notice that circular +sequences can be rotated. See Fig. 2 for two circular gene sequences with +duplicated genes and their longest common subsequence. +1 +2 +1 +3 +1 +3 +1 +2 +3 +2 +3 +2 +1 +2 +2 +1 +Figure 2: +Two circular gene sequences with duplicated genes and their +longest common subsequence, corresponding to Scenario 4. +We have turned the problem of determining transposable genes into find- +ing the longest common subsequence of several gene sequences. Depending +on whether the gene sequences are linear or circular, and whether genes have +multiple copies, the problem can be classified into four scenarios: +Scenario 1: Consider m linear sequences of genes 1, . . . , n, where each gene +has only one copy in each sequence. Determine the longest linear sequence +that is a common subsequence of these m sequences. +Scenario 2: Consider m circular sequences of genes 1, . . . , n, where each +gene has only one copy in each sequence. Determine the longest circular +sequence that is a common subsequence of these m sequences. Here circular +sequences can be rotated. +8 + +Scenario 3: Consider m linear sequences of genes 1, . . . , n, where each +gene can have multiple copies in each sequence. +Determine the longest +linear sequence that is a common subsequence of these m sequences. Only +consider subsequences that consist of all or none copies of the same gene, +and calculate the length by genes. +Scenario 4: Consider m circular sequences of genes 1, . . . , n, where each +gene can have multiple copies in each sequence. +Determine the longest +circular sequence that is a common subsequence of these m sequences. Only +consider subsequences that consist of all or none copies of the same gene, +and calculate the length by genes. Here circular sequences can be rotated. +These four scenarios correspond to different algorithms, and will be dis- +cussed separately. +3 +Linear sequences without duplicated genes +In Scenario 1, consider m linear gene sequences, where each sequence con- +tains n genes 1, . . . , n. Each gene has only one copy. For such permutations +of 1, . . . , n, we need to find the longest common subsequence. +3.1 +A graph representation of the problem +Brute-force searching that tests whether each subsequence appears in all +sequences is not applicable, since the time complexity is exponential in n. +To develop a polynomial algorithm, we first design an auxiliary directed +graph G. +Definition 2. For m linear sequences with n non-duplicated genes, the cor- +responding auxiliary graph G is a directed graph, where each vertex is a +gene gi, and there is a directed edge from gi to gj if and only if gi appears +before gj in all m sequences. +A directed path g1 → g2 → g3 → · · · → g4 → g5 in G corresponds to a +common subsequence (g1, g2, g3, . . . , g4, g5) of m sequences, and vice versa. +We add 0 to the head of each sequence and n + 1 to the tail. Then the +longest common subsequence must start at 0 and end at n+1. The problem +of finding the longest common subsequence becomes finding the longest path +from 0 to n + 1 in G. See Fig. 3 for an example of using the auxiliary graph +to determine transposable genes. This auxiliary graph G has no directed +loop (acyclic). If there exists a loop g1 → g2 → g3 → · · · → g4 → g1, then +g1 is prior to g4 and g4 is prior to g1 in all sequences, a contradiction. +9 + +0 +� ������� +������� +�❃❃❃❃❃❃❃❃ +�✏✏✏✏✏✏✏✏✏✏✏✏✏✏ +�✳✳✳✳✳✳✳✳✳✳✳✳✳✳ +� +1 +� +� +�◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +�✳✳✳✳✳✳✳✳✳✳✳✳✳✳ +2 +� ♣♣♣♣♣♣♣♣♣♣♣♣♣ +♣♣♣♣♣♣♣♣♣♣♣♣♣ +�✏✏✏✏✏✏✏✏✏✏✏✏✏✏ +3 +�❃❃ +❃❃ +❃❃ +❃ +❃❃ +❃❃ +❃❃ +❃ +4 +��������� +5 +Figure 3: The auxiliary graph G of two sequences ([0], 1, 2, 3, 4, [5]) and +([0], 1, 4, 2, 3, [5]). +The unique longest path (double arrows) from 0 to 5 +is 0 → 1 → 2 → 3 → 5, meaning that the unique longest common se- +quence is ([0], 1, 2, 3, [5]). Thus 1, 2, 3 are non-transposable, and 4 is proper- +transposable. +3.2 +Find the longest path +Determining the longest path between two vertices in a directed acyclic +graph can be solved by a standard dynamic programming algorithm. For a +vertex gi ∈ {0, 1, . . . , n}, consider the longest path from gi to n + 1. Since +there exists an edge gi → n + 1, and G is acyclic, this longest path exists. If +the longest path is not unique, assign one arbitrarily. +Definition 3. Define F+(gi) to be the length of the longest path from gi to +n + 1 in G, and H+(gi) to be the vertex next to gi in this path. +F+ and H+ can be calculated recursively: For one gene gi, consider all +genes gj with an edge gi → gj in G. The gene gj with the largest F+(gj) is +assigned to be H+(gi), and F+(gi) = F+(gj) + 1. If gl → n + 1 is the only +edge that starts from gene gl, then F+(gl) = 1, and H+(gl) = n+1. In other +words, +H+(gi) = +argmax +{gj with gi→gj} +F+(gj); +F+(gi) = 1 + F+[H+(gi)]. +Then 0 → H+(0) → H2 ++(0) → H3 ++(0) → · · · → Hf−1 ++ +(0) → Hf ++(0) = n + 1, +denoted by L0, is a longest path in G. Here f = F+(0), and Hi ++ is the ith +iteration of H+. +10 + +3.3 +Test the uniqueness of the longest path +To test whether quasi-transposable genes exist, we need to check the unique- +ness of this longest path. +Definition 4. For gi ∈ {1, . . . , n, n + 1}, define F−(gi) to be the length of +the longest path from 0 to gi in G, and H−(gi) to be the vertex prior to gi +in this path. +F− and H− can be calculated similar to F+ and H+. We can see that +F+(gi) + F−(gi) is the length of +0 = HF−(gi) +− +(gi) → HF−(gi)−1 +− +(gi) → · · · → H−(gi) → gi +→ H+(gi) → · · · → HF+(gi)−1 ++ +(gi) → HF+(gi) ++ +(gi) = n + 1, +a longest path from 0 through gi to n + 1. For gi /∈ L0, if F+(gi) + F−(gi) < +F+(0), then gi is proper-transposable; if F+(gi) + F−(gi) = F+(0), then +gi is quasi-transposable. If every gi /∈ L0 is proper-transposable, then the +longest common subsequence is unique, and all genes in L0 (excluding the +auxiliary 0 and n + 1) are non-transposable. The procedure of determining +transposable genes stops here. Otherwise, the longest common subsequence +is not unique, and we need to find quasi-transposable genes in L0. +3.4 +Find quasi-transposable genes +When determining all quasi-transposable genes g1, . . . , gk not in L0, as de- +scribed above, we construct corresponding longest paths L1, . . . , Lk from 0 +to n + 1, where each Li passes through gi. We claim that a gene gj ∈ L0 +is non-transposable if and only if gj is contained in all L1, . . . , Lk. To prove +this, we need the following lemma. +Lemma 1. In Scenario 1 of linear sequences without duplicated genes, each +quasi-transposable gene gi has a corresponding quasi-transposable gene gj, +so that no longest common subsequence can contain both gi and gj. +If a gene gj ∈ L0 is non-transposable, then it is contained in all L1, . . . , Lk. +If gj ∈ L0 is quasi-transposable, by Lemma 1, there is a quasi-transposable +gene gl /∈ L0 which is mutual-exclusive with gj, in the sense that gl and gj +cannot appear in the same longest common subsequence. The correspond- +ing longest path Ll contains gl, thus cannot contain gj. This proves our +approach to determine the quasi-transposable genes in L0. +11 + +Proof of Lemma 1. Fix a quasi-transposable gene gi. It is contained in a +longest path Li, which contains all non-transposable genes. Thus for each +non-transposable gene g∗, there is an edge between g∗ and gi in G. Assume +gi has no such mutual-exclusive quasi-transposable gene gj. Then there is +an edge (direction unknown) in G between gi and each quasi-transposable +gene gj. Choose a longest path L∗ in G that does not contain gi. Whether +gj ∈ L∗ is a non-transposable gene or a quasi-transposable gene, there is +an edge between gj and gi. Determine the first gene gk in L∗ that has an +edge gi → gk. Since there is an edge gi → n + 1, gk exists. Since there +is an edge 0 → gi, gk ̸= 0. Denote the previous gene of gk in L∗ by gl, +then gl exists, and there is an edge gl → gi. +Thus we construct a path +0 → · · · → gl → gi → gk → · · · → n + 1, which is longer than the longest +path, a contradiction. Thus gi has a mutual-exclusive quasi-transposable +gene gj. +3.5 +Algorithms and complexities +We summarize the above method as Algorithms 1,2. +If we have known +that the longest common subsequence is unique, then we just need to apply +Algorithm 1, so that genes in L0 are non-transposable, and genes not in +L0 are proper-transposable. We have reported Algorithm 1 previously [32, +67]. +Algorithm 1 is kept here to make the story complete. +Assume we +have m sequences with length n, and the length of the longest common +subsequence is n − k. The time complexities of Steps 2-5 in Algorithm 1 are +O(m), O(mn2), O(n), O(n). The time complexities of Step 2 and Step 3 in +Algorithm 2 are O(k) and O(kn). Since k ≤ n, the overall time complexity of +determining transposable genes in Scenario 1 by Algorithms 1,2 is O(mn2). +The space complexity is trivially O(mn + n2). +3.6 +Applications on experimental data +We test Algorithms 1,2 on Escherichia coli gene sequences. From NCBI +sequencing database, we obtain gene sequences of three individuals of E. +coli strain ST540 (GenBank CP007265.1, GenBank CP007390.1, GenBank +CP007391.1) and three individuals of E. coli strain ST2747 (GenBank CP007392.1, +GenBank CP007393.1, GenBank CP007394.1). +All three sequences of ST540 start with gene dnaA and end with gene +rpmH. We can regard them as linear gene sequences. We remove genes that +appear more than once in one sequence, and remove genes that do not ap- +pear in all three sequences. After applying Algorithms 1,2 on these three +12 + +1. Input +m linear sequences of genes 1, . . . , n. No duplicated genes. +2. Modify the sequences: +Add 0 to the head, and n + 1 to the tail of each sequence +3. Construct the auxiliary graph G: +Vertices of G are all the genes 1, . . . , n +For each pair of genes gi, gj +If gi is prior to gj in all m sequences +Add a directed edge gi → gj in G +End of if +End of for +4. Calculate F+(·) and H+(·) for each gene gi in 0, 1, . . . , n recursively; +calculate F−(·) and H−(·) for each gene gi in 1, . . . , n, n + 1 +recursively: +H+(gi) = +argmax +{gj with gi→gj} +F+(gj) +% If gj with gi → gj that maximizes F+(gj) is not unique, choose +one randomly +F+(gi) = 1 + F+[H+(gi)] +H−(gi) = +argmax +{gj with gj→gi} +F−(gj) +% If argmax is not unique, choose one randomly +F−(gi) = 1 + F−[H−(gi)] +5. Construct a longest path L0 from 0 to n + 1: +0 → H+(0) → H2 ++(0) → H3 ++(0) → · · · → Hf−1 ++ +(0) → Hf ++(0) = n + 1 +% Here f = F+(0), and Hi ++ is the ith iteration of H+ +6. Output F+(·), H+(·), F−(·), H−(·), L0 +Algorithm 1: Detailed workflow of determining proper-transposable +genes and quasi-transposable genes in Scenario 1, preparation stage. +13 + +1. Input +F+(·), H+(·), F−(·), H−(·), L0 calculated from Algorithm 1 +Denote all genes not in L0 by g1, . . . , gk +2. For each gene gi in g1, . . . , gk +If F+(gi) + F−(gi) < F+(0) +Output gi is a proper-transposable gene +Else +Output gi is a quasi-transposable gene +End of if +End of for +3. If all genes in g1, . . . , gk are proper-transposable +Output all genes in L0 are non-transposable +Else +For each gene gi in g1, . . . , gk +Use H+(·) and H−(·) to construct Li, a longest path from 0 to +n + 1 that passes gi. +End of for +For each gene gj in L0 (excluding auxiliary 0 and n + 1) +If gj is contained in all L1, . . . , Lk +Output gj is non-transposable +Else +Output gj is quasi-transposable +End of if +End of for +End of if +4. Output: whether each gene is proper/quasi/non-transposable +Algorithm 2: Detailed workflow of determining proper-transposable +genes and quasi-transposable genes in Scenario 1, output stage. +14 + +sequences, there are 301 non-transposable genes, 4 quasi-transposable genes +(hpaC, iraD, fbpC, psiB), and 263 proper-transposable genes. The reason for +the large amount of proper-transposable genes is that sequence CP007265.1 +is significantly different from the other two. +After removing it and ap- +plying Algorithms 1,2 to the remaining two sequences (CP007390.1 and +CP007391.1), there are 564 non-transposable genes and 4 quasi-transposable +genes (hpaC, iraD, fbpC, psiB). Therefore, some genes in hpaC, iraD, fbpC, +psiB are likely to translocate. +All three sequences of ST2747 start with gene glnG and end with gene +hemG. We can regard them as linear gene sequences. +We remove genes +that appear more than once in one sequence, and remove genes that do not +appear in all three sequences. After applying Algorithms 1,2 on these three +sequences, all 573 genes are non-transposable. +4 +Circular sequences without duplicated genes +In Scenario 2, consider m circular gene sequences, where each sequence +contains n genes 1, . . . , n. Each gene has only one copy in each sequence. For +such circular permutations of 1, . . . , n, we need to find the longest common +subsequence. Assume the length of the longest common subsequence is n−k. +4.1 +Find a longest common subsequence +We first randomly choose a gene gi. Cut all circular sequences at gi and +expand them to be linear sequences. For example, the circular sequences in +Fig. 1 cut at 1 are correspondingly (1, 2, 3, 4, 5, 6) and (1, 2, 6, 4, 5, 3). Using +Algorithm 1, we can find Li that begins with gi, which is a longest common +subsequence of all expanded linear sequences. In the above example, the +longest common linear subsequence starting from 1 is (1, 2, 4, 5). +If gi is +a non-transposable gene or a quasi-transposable gene, then Li (glued back +to a circle) is a longest common circular subsequence. If gi is a proper- +transposable gene, then Li is shorter than the longest common circular sub- +sequence. In Fig. 1, gene 1 is non-transposable, and (1, 2, 4, 5) (glued) is the +longest common circular subsequence. +We do not know if Li (glued) is a longest common subsequence (whether +containing gi or not) for all circular sequences. If there is a longer common +subsequence, it should contain genes that are not in Li. +Consider four +variables L, g, C, and S, whose initial values are Li, gi, the length of Li, and +the complement of Li. These variables contain information on the longest +common linear subsequence that we have found during this procedure. +15 + +Choose a gene gj in S, and cut all circular gene sequences at gj. Apply +Algorithm 1 to find Lj, which is the longest in common subsequences that +contain gj. If the length of Lj is larger than C, set L to be Lj, set g to +be gj, set C to be the length of Lj, and set S to be the complement of Lj. +Otherwise, keep L, g, C, and S still. +Choose another gene gl in S which has not been chosen before, and +repeat this procedure. This procedure terminates when all genes in S have +been chosen and cut. Denote the final values of L, g, C, and S by L0, g0, +C0, and S0. Here S0 is the complement of L0. +During this procedure, if the current g is a proper-transposable gene, +then S contains a non-transposable gene or a quasi-transposable gene, which +has not been chosen. +Thus L, g, C, S will be further updated. +If the +current g is a non-transposable gene or a quasi-transposable gene, then C +has reached its maximum, and L, g, C, S will not be further updated. This +means L0 is a longest common circular subsequence, and C0 is the length +of the longest common subsequence, n − k. Also, the total number of genes +being chosen and cut is k+1. All k genes in S0 and g0 are chosen and cut. A +gene gt in L0 (excluding g0) is non-transposable or quasi-transposable, and +cannot be chosen and cut. The reason is that it cannot be chosen before g0 +is chosen (only proper-transposable genes can be chosen before g0 is chosen), +and it cannot be chosen after g0 is chosen (gt /∈ S0). +4.2 +Determine quasi-transposable genes +For each gene gp ∈ S0, apply Algorithm 1 to calculate Cp, the length of the +longest common subsequence that contains gp. If Cp < C0, gp is a proper- +transposable gene. Otherwise, Cp = C0 means gp is a quasi-transposable +gene. We have found all proper-transposable genes. If all genes in S0 are +proper-transposable, then all genes in L0 are non-transposable, and the +procedure terminates. +If S0 contains quasi-transposable genes, then L0 also has quasi-transposable +genes. To determine quasi-transposable genes in L0, we need the following +lemma. +Lemma 2. In Scenario 2, choose a quasi-transposable gene gp and cut the +circular sequences at gp to obtain linear sequences. A proper-transposable +gene for the circular sequences is also a proper-transposable gene for the +linear sequences; a non-transposable gene for the circular sequences is also +a non-transposable gene for the linear sequences. +Proof. Consider a longest common subsequence Lp for linear sequences cut +16 + +at gp. Since gp is a quasi-transposable gene, the length of Lp is also n − k, +meaning that Lp is also a longest common subsequence for circular se- +quences. Now, this lemma is proved by the definition of proper/quasi/non- +transposable gene. +If a gene gr in L0 is non-transposable for the circular sequences, then gr is +a non-transposable gene for linear sequences cut at each quasi-transposable +gene gq ∈ S0. +If a gene gs in L0 is quasi-transposable for the circular +sequences, then there is a longest common circular subsequence Lt that +does not contain gs, meaning that Lt contains a quasi-transposable gene gt +not in L0. Then gs is a proper/quasi-transposable gene for linear sequences +cut at gt. +Therefore, we can use the following method to determine quasi-transposable +genes in L0. For each quasi-transposable gene gq ∈ S0, cut at gq and ap- +ply Algorithms 1,2 to determine if each gene in L0 is proper/quasi/non- +transposable for the linear gene sequences cut at gq. A gene gr ∈ L0 is non- +transposable for the circular sequences if and only if it is non-transposable +for linear sequences cut at any quasi-transposable gene gq ∈ S0. A gene +gs ∈ L0 is quasi-transposable for the circular sequences if and only if it is +proper/quasi-transposable for linear sequences cut at some quasi-transposable +gene gq ∈ S0. +When we have determined all quasi-transposable genes in S0, it might +be tempting to apply a simpler approach to determine quasi-transposable +genes in L0: For each quasi-transposable gene gq ∈ S0, cut at gq and apply +Algorithm 1 to find a longest common subsequence Lq. A gene in L0 is +non-transposable if and only if it appears in all such Lq. This approach is +valid only if the following conjecture holds, which is similar to Lemma 1: +Conjecture 1. In Scenario 2 of circular sequences without duplicated genes, +each quasi-transposable gene gi has a corresponding quasi-transposable gene +gj, so that no longest common subsequence can contain both gi and gj. +However, Conjecture 1 does not hold. See Fig. 4 for a counterexample. +All genes are quasi-transposable. +Any two quasi-transposable genes are +contained in a longest common subsequence (length 3). Thus the simplified +approach above does not work. +We summarize the above method as Algorithms 3,4. If we have known +that the longest common subsequence is unique, then we just need to apply +Algorithm 3, so that genes in S0 are proper-transposable, and genes not in S0 +are non-transposable. Assume we have m sequences with length n, and the +length of the longest common subsequence is n − k. The time complexities +17 + +1 +2 +3 +1 +2 +6 +1 +2 +7 +8 +4 +3 +5 +6 +8 +7 +6 +5 +4 +7 +8 +5 +4 +3 +Figure 4: A counterexample with three circular sequences that fails Conjec- +ture 1. +of Step 2 and Step 3 in Algorithm 3 are O(mn2) and O(kmn2). The time +complexities of Step 2 in Algorithm 4 is O(kmn2). The overall time com- +plexity of determining transposable genes in Scenario 2 by Algorithms 3,4 +is O(kmn2). The space complexity is trivially O(mn + n2). +4.3 +Applications on experimental data +Similar to Subsection 3.6, we test Algorithms 3,4 on Escherichia coli gene +sequences. From NCBI sequencing database, we obtain gene sequences of +three individuals of E. coli strain ST540 (GenBank CP007265.1, GenBank +CP007390.1, GenBank CP007391.1) and three individuals of E. coli strain +ST2747 (GenBank CP007392.1, GenBank CP007393.1, GenBank CP007394.1). +We regard all three sequences of ST540 as circular gene sequences. We +remove genes that appear more than once in one sequence, and remove genes +that do not appear in all three sequences. After applying Algorithms 3,4 +on these three sequences, there are 389 non-transposable genes, 50 quasi- +transposable genes, and 129 proper-transposable genes. The reason for the +large amount of proper-transposable genes is that sequence CP007265.1 is +significantly different from the other two. After removing it and applying Al- +gorithms 3,4 to the remaining two sequences (CP007390.1 and CP007391.1), +there are 564 non-transposable genes and 4 quasi-transposable genes (hpaC, +iraD, fbpC, psiB). Therefore, some genes in hpaC, iraD, fbpC, psiB are likely +to translocate. +We regard all three sequences of ST2747 as circular gene sequences. We +remove genes that appear more than once in one sequence, and remove genes +that do not appear in all three sequences. After applying Algorithms 3,4 on +these three sequences, all 573 genes are non-transposable genes. +18 + +1. Input +m circular sequences of genes 1, . . . , n, where each gene has only +one copy in each sequence +2. Choose a gene gi randomly +Cut all circular sequences at gi and expand them to be linear +sequences +Apply Algorithm 1 to find Li, a longest common subsequence in the +expanded linear sequences +Set C to be the length of Li, and set S to be the complement of Li +3. While S has a gene gj that has not been chosen and cut +Cut all circular sequences at gj and apply Algorithm 1 to find Lj +Denote the length of Lj by Cj +If Cj > C +Update C to be Cj, and update S to be the complement of Lj +End of if +End of while +Denote the final C by C0, and denote the final S by S0 +4. Output C0 and S0 +Algorithm 3: Detailed workflow of determining proper-transposable +genes and quasi-transposable genes in Scenario 2, preparation stage. +19 + +1. Input +m circular sequences of genes 1, . . . , n, where each gene has only +one copy in each sequence; C0 and S0 calculated from Algorithm 3 +2. For each gene gl ∈ S0 +Cut all circular sequences at gl and expand them to be linear +sequences +Apply Algorithm 1 to find Ll, a longest common subsequence in +the expanded linear sequences. +Denote the length of Ll by Cl +If Cl < C0 +Output gl is a proper-transposable gene +Else +Output gl is a quasi-transposable gene +Cut all circular sequences at gl and apply Algorithms 1,2 to +find all proper/quasi-transposable genes for linear gene sequences +starting at gl +Output genes not in S0 but being proper/quasi-transposable for +such linear sequences are quasi-transposable for circular sequences +End of if +End of for +Output other genes that have not been determined to be +proper/quasi-transposable are all non-transposable +3. Output: whether each gene is proper/quasi/non-transposable +Algorithm 4: Detailed workflow of determining proper-transposable +genes and quasi-transposable genes in Scenario 2, output stage. +20 + +5 +Linear sequences with duplicated genes +In Scenario 3, consider m linear gene sequences, where each sequence con- +tains different numbers of copies of n genes 1, . . . , n. We need to find the +longest common subsequence. Here we only consider common subsequences +that consist of all or none copies of the same gene, and the subsequence +length is calculated by genes, not gene copies. +5.1 +A graph representation of the problem +Similar to Scenario 1, we construct an auxiliary graph G, where each vertex +is a gene (not a copy of a gene). However, in this case, the auxiliary graph is +undirected: There is an undirected edge between gene gi and gene gj if and +only if all the copies of gi and gj keep their relative locations in all sequences. +For example, consider two sequences (1, 2, 3, 2, 3, 4, 5) and (2, 1, 3, 3, 2, 4, 5). +For gene pair 1, 3, the corresponding sequences are (1, 3, 3) and (1, 3, 3), +meaning that there is an edge between 1 and 3. +For gene pair 1, 2, the +corresponding sequences are (1, 2, 2) and (2, 1, 2), meaning that there is no +edge between 1 and 2. See Fig. 5 for the auxiliary graph in this case. +1 +❃❃❃❃❃❃❃❃ +✳✳✳✳✳✳✳✳✳✳✳✳✳✳ +✏✏✏✏✏✏✏✏✏✏✏✏✏✏ +2 +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +3 +♣♣♣♣♣♣♣♣♣♣♣♣♣♣ +4 +5 +Figure 5: +The auxiliary graph G of two sequences (1, 2, 3, 2, 3, 4, 5) and +(2, 1, 3, 3, 2, 4, 5). The unique largest complete subgraph is {1, 3, 4, 5}, mean- +ing that the unique longest common sequence is (1, 3, 3, 4, 5). Thus 1, 3, 4, 5 +are non-transposable genes, and 2 is a proper-transposable gene. +Definition 5. A subgraph of G consists of some genes g1, . . . , gl and the +edges between them. +In a subgraph, if there is an edge between any two +genes, this subgraph is called a complete subgraph (also called a clique). +Definition 6. In graph G, the degree of a gene g is the number of edges +linking g. In a complete graph of p genes, where any two genes have an edge +in between, each gene has degree p − 1. +21 + +Definition 7. If all copies of genes g1, . . . , gl keep their relative locations in +all linear sequences, we say that g1, . . . , gl form a common subsequence. +The following Lemma 3 shows that there is a bijection between common +subsequences and complete subgraphs in G. The problem of determining the +longest common subsequence now becomes determining the largest complete +subgraph of G. +Lemma 3. In Scenario 3, construct the auxiliary graph G from gene se- +quences. If g1, . . . , gk form a complete subgraph in G, then g1, . . . , gk form a +common subsequence, and vice versa. +Proof. If g1, . . . , gl form a common subsequence, then there is an edge in +G between any two genes in g1, . . . , gl, meaning that they form a complete +subgraph. +For the other direction, only consider copies of g1, . . . , gk in these se- +quences. +If g1, . . . , gk do not form a common subsequence, find the first +digit that such sequences differ. Assume gp and gq can both appear in this +digit. Then gp, gq cannot form a common subsequence, and there is no edge +between gp and gq. +We illustrate this proof with Fig. 5: For genes 2, 3, 4, the sequences are +(2, 3, 2, 3, 4) and (2, 3, 3, 2, 4). The third digit is different, where 2 and 3 can +both appear. Then the sequences for genes 2, 3, (2, 3, 2, 3) and (2, 3, 3, 2), +cannot match, and there is no edge between 2 and 3. +5.2 +A heuristic algorithm +The above discussion shows that given gene sequences, we can construct +an undirected graph G, so that there is a bijection between common subse- +quences and complete subgraphs. The inverse also holds: We can construct +corresponding gene sequences for a graph. +Lemma 4. Given an undirected graph G, we can construct two gene se- +quences, so that there is a bijection between common subsequences and com- +plete subgraphs. +Proof. Assume the graph has n genes. We start with two sequences (1, 2, . . . , n) +and(1, 2, . . . , n). For each pair of genes gi, gj, if there is no edge between +them in G, add gi, gj to the end of the first sequence, and gj, gi to the +end of the second sequence. Then gi, gj cannot both appear in a common +subsequence, and this operation does not affect other gene pairs. +22 + +For example, corresponding to Fig. 5, we start with (1, 2, 3, 4, 5) and +(1, 2, 3, 4, 5). +Since there is no edge between 1, 2, we add them to have +(1, 2, 3, 4, 5, 1, 2) and (1, 2, 3, 4, 5, 2, 1). Since there is no edge between 2, 3, +we add them to have (1, 2, 3, 4, 5, 1, 2, 2, 3) and (1, 2, 3, 4, 5, 2, 1, 3, 2). These +two sequences corresponds to Fig. 5. +Combining Lemma 3 and Lemma 4, we obtain the following result: +Proposition 1. Finding the longest common sequence in Scenario 3 is +equivalent to the maximum clique problem, which is NP-hard. +Proof. For an undirected graph, we can use Lemma 4 to construct corre- +sponding sequences. If we have the solution of finding the longest common +sequence in Scenario 3, then we can find the largest complete subgraph in +an extra polynomial time. +For gene sequences in Scenario 3, we can construct corresponding auxil- +iary graph. If we have the solution of finding the largest complete subgraph, +then we can use Lemma 3 to find the longest common sequence in Scenario +3 in an extra polynomial time. +Therefore, finding the longest common sequence in Scenario 3 and finding +the largest complete subgraph are equivalent. The problem of determining +the largest complete subgraph is just the maximum clique problem, which +is NP-hard [62]. Thus finding the longest common sequence in Scenario 3 +is also NP-hard. This means it is not likely to design an algorithm that +always correctly determines the longest common subsequence in polynomial +time. +We have transformed Scenario 3 into the maximum clique problem for +a graph G. There have been various algorithms for the maximum clique +problem [30, 37, 69], and readers may refer to a review for more details +[75]. For completeness, we propose a simple idea: In the auxiliary graph +G, repeatedly abandon the gene with the smallest degree (and also edges +linking this gene) until the remaining genes form a complete subgraph. See +Algorithm 5 for the details of this greedy heuristic method. This algorithm +is easy to understand, and can provide some intuition. We do not claim that +Algorithm 5 is comparable to other sophisticated algorithms. +We test Algorithm 5 on random graphs. Construct a random graph with +n genes, and any two genes have probability 0.5 to have an edge in between. +Use brute-force search to find the maximum clique, and compare its size with +the result of Algorithm 5. For each n ≤ 15, we repeat this for 10000 times, +and every time Algorithm 5 returns the correct result. Therefore, for small +23 + +1. Input +m linear sequences of genes 1, . . . , n, where each gene can have +multiple copies +2. Construct the auxiliary graph G: +Vertices of G are all the genes 1, . . . , n (not their copies) +For each pair of genes gi, gj +If all copies of gi and gj keep their relative locations in all m +sequences +Add an undirected edge between gi and gj in G +End of if +End of for +Calculate the degree for each gene in G +3. While true +Find a gene gi with the smallest degree di in G +% If the minimal gi is not unique, choose one randomly +If di + 1 is smaller than the number of genes in G +Delete gi and edges linking gi in G +Update the degrees of other genes +Else +% The remaining genes form a complete subgraph +Break the while loop +End of if +End of while +% The final G is a complete subgraph of the original G, and it is +likely to be the largest one +4. Output genes in the final G are not transposable, and genes not in +the final G are transposable +Algorithm 5: A heuristic method for detecting transposable genes in +Scenario 3. +24 + +random graphs, the 95% credible interval for the success rate of Algorithm 5 +is [0.9997, 1]. We can claim that Algorithm 5 is a good heuristic algorithm +that fails with a very small probability. Since finding the true maximum +clique requires exponentially slow brute-force search, we do not test on very +large graphs. +Nevertheless, Algorithm 5 does not always produce the correct result. +See Fig. 6 for a counterexample. Here genes 1, 2, 3, 4, 5, 6 have degree 4, while +genes 7, 8, 9, 10 have degree 3. When applying Algorithm 5, genes 7, 8, 9, 10 +are first abandoned, and the final result just has three genes, such as 1, 3, 5. +However, the largest complete graph is 7, 8, 9, 10. Besides, Algorithm 5 can +only determine one (possibly longest) common subsequence. Thus we cannot +determine the existence of quasi-transposable genes. +3 +❃❃❃❃❃❃❃❃ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +1 +�������� +❃❃❃❃❃❃❃❃ +4 +♣♣♣♣♣♣♣♣♣♣♣♣♣♣ +�������� +7 +❅ +❅ +❅ +❅ +❅ +❅ +❅ +❅ +8 +⑦⑦⑦⑦⑦⑦⑦ +5 +2 +6 +10 +9 +Figure +6: +The +auxiliary +graph +G +of +linear +sequences +(7, 8, 9, 10, 1, 1, 2, 3, 3, 4, 5, 5, 6) and (1, 2, 1, 3, 4, 3, 5, 6, 5, 7, 8, 9, 10). +This +counterexample fails Algorithm 5. +Assume we have m sequences with n genes. In general, the copy number +of a gene is small, and we can assume the length of each sequence is O(n). +The time complexities of Step 2 and Step 3 in Algorithm 5 are O(mn2) and +O(n2), and the overall time complexity is O(mn2). The space complexity is +trivially O(mn + n2). +6 +Circular sequences with duplicated genes +In Scenario 4, consider m circular gene sequences, where each sequence +contains different numbers of copies of n genes 1, . . . , n. We need to find the +longest common subsequence. Here we only consider common subsequences +that consist of all or none copies of the same gene, and the subsequence +length is calculated by genes, not gene copies. +We shall prove that finding the longest common subsequence in Scenario +4 is no easier than in Scenario 3. Thus Scenario 4 is also NP-hard. +Proposition 2. Finding the longest common subsequence in Scenario 4 is +NP-hard. +25 + +Proof. From Proposition 1, Scenario 3 is NP-hard, meaning that any NP +problem can be reduced to Scenario 3 in polynomial time. We just need to +prove that Scenario 3 can be reduced to Scenario 4 in polynomial time. +Given m linear sequences with n genes in Scenario 3, add genes n + +1, . . . , 2n + 1 to the end of each sequence, and glue each linear sequence +into a circular sequence. The longest common subsequence for these circular +sequence has the following properties: (1) it contains all genes n+1, . . . , 2n+ +1; (2) after cutting at n + 1 and removing genes n + 1, . . . , 2n + 1, the +remaining linear sequence is the longest common subsequence in Scenario 3. +(1) The longest common subsequence has at least n + 1 genes (n + +1, . . . , 2n + 1). Therefore, at least one gene in n + 1, . . . , 2n + 1 is included, +such as n+1. Since gene n+1 aligned in all sequences, n+2, . . . , 2n+1 are +also aligned, meaning that they are also in the longest common subsequence. +(2) After cutting and removing n + 1, . . . , 2n + 1, the remaining linear +sequence is a common subsequence in Scenario 3. If there is a longer common +subsequence, then that with n + 1, . . . , 2n + 1 should be a longer common +subsequence in Scenario 4, a contradiction. +Therefore, if we can find the longest common subsequence for these cir- +cular sequences, then we can find the longest common subsequence for linear +sequences in polynomial time. +Similar to Scenario 3, to find the longest common subsequence in Sce- +nario 4, we want to reduce it to a maximum clique problem. +However, +Lemma 3 does not hold in Scenario 4. For example, we can consider a cir- +cular sequence (1, 2, 3) and its mirror symmetry. These two sequences are +different, but any two genes form a common subsequence. However, inspired +by Lemma 3, we have the following conjecture, although we do not know if +it is correct or not. +Conjecture 2. In Scenario 4, if any three genes gi, gj, gl in g1, . . . , gk form +a common subsequence, then g1, . . . , gk form a common subsequence. +To solve Scenario 4,construct a 3-uniform hypergraph G as following +[15]: vertices are genes 1, . . . , n; there is a 3-hyperedge (undirected) that +links genes gi, gj, gk if and only if they form a common subsequence. +Proposition 3. If Conjecture 2 holds, then finding the longest common +sequence in Scenario 4 can be reduced to the maximum clique problem for +3-uniform hypergraphs. +Proof. If g1, . . . , gk form a common subsequence, then any three genes gi, gj, gl +has a 3-hyperedge, and g1, . . . , gk form a complete subgraph. If g1, . . . , gk +26 + +form a complete subgraph, then any three genes gi, gj, gl form a common +subsequence. By Conjecture 2 , this means g1, . . . , gk form a common sub- +sequence. Therefore, there is a bijection between common subsequence and +complete subgraph. +If we can find the maximum clique problem for 3- +uniform hypergraphs, then it corresponds to the longest common subse- +quence. +We have reduced Scenario 4 into the maximum clique problem for 3- +uniform hypergraphs, which is also NP-hard [75]. There have been some +algorithms for the maximum clique problem for 3-uniform hypergraphs [61, +55]. For completeness, we propose a simple idea: Repeatedly delete the gene +that has the smallest degree, until we have a complete subgraph that any +three genes have a 3-hyperedge that links them. We summarize this greedy +heuristic method as Algorithm 6. This algorithm is easy to understand, and +can provide some intuition. We do not claim that Algorithm 6 is comparable +to other sophisticated algorithms. +We test Algorithm 6 on random graphs. Construct a random graph with +n genes, and any two genes have probability 0.5 to have an edge in between. +Use brute-force search to find the maximum clique, and compare its size with +the result of Algorithm 6. For each n ≤ 15, we repeat this for 10000 times, +and every time Algorithm 6 returns the correct result. Therefore, for small +random graphs, the 95% credible interval for the success rate of Algorithm 6 +is [0.9997, 1]. We can claim that Algorithm 6 is a good heuristic algorithm +that fails with a very small probability. Since finding the true maximum +clique requires exponentially slow brute-force search, we do not test on very +large graphs. +Nevertheless, Algorithm 6 does not always produce the correct result. +See Fig. 7 for a counterexample. Here each gene in 1, 2, 3, 4, 5, 6 has degree +4, while each gene in 7, 8, 9, 10 has degree 3 .When applying Algorithm 6, +genes 7, 8, 9, 10 are first deleted, and the final result just has three genes, +such as (1, 3, 5). However, the longest common subsequence (7, 8, 9, 10) has +four genes. +Assume we have m sequences with n genes. In general, the copy number +of a gene is small, and we can assume the length of each sequence is O(n). +The time complexities of Step 2 and Step 3 in Algorithm 6 are O(mn3) and +O(n3), and the overall time complexity is O(mn3). The space complexity is +trivially O(mn + n3). +27 + +1. Input +m circular sequences of genes 1, . . . , n, where each gene can have +multiple copies +2. Construct the auxiliary graph G: +Vertices of G are all the genes 1, . . . , n (not their copies) +For each gene triple gi, gj, gk +If all copies of gi, gj, gk keep their relative locations in all m +sequences +Add a 3-hyperedge that links gi, gj, gk in G +End of if +End of for +3. While there exist three genes that do not share a 3-hyperedge +Calculate the degree for each gene in G +Delete the gene with the smallest degree and 3-hyperedges that +links this gene +% If there are multiple genes with the smallest degree, delete one +randomly +End of while +% After this while loop, any three genes form a common subsequence +% If Conjecture 2 holds, the remaining genes form a common +subsequence +4. Output remaining genes are not transposable, and other genes are +transposable +Algorithm 6: A heuristic method for detecting transposable genes in +Scenario 4. +28 + +1 +2 +7 +3 +4 +2 +1 +10 +4 +3 +10 +6 +5 +9 +8 +9 +5 +6 +8 +7 +1 +2 +9 +3 +4 +2 +1 +8 +4 +3 +8 +6 +5 +7 +10 +7 +5 +6 +10 +9 +Figure 7: Four circular sequences. +The longest common subsequence is +(7, 8, 9, 10). This counterexample fails Algorithm 6. +7 +Discussion +A gene gi might be missing in some sequences. Since gi is not in any longest +common subsequence, it should be a proper-transposable gene. This gene +can be directly removed before applying corresponding algorithms. +We can adopt a stricter definition of transposable genes to exclude a gene +which only changes its relative position in a few (no more than l, where l +is small enough) sequences. Then we should consider the longest sequence +which is a common subsequence of at least m − l sequences. We can run +the corresponding algorithm for every m − l sequences. Thus the total time +complexity will be multiplied by a factor of ml. +In Scenario 1 and Scenario 2 (linear/circular sequences without dupli- +cated genes), if each sequence has n genes, and the longest common sub- +sequence has length n − k, then there are at most k proper-transposable +genes. About quasi-transposable genes, inspired by Lemma 1, we have the +following guess. +Conjecture 3. Consider m linear/circular sequences with n genes without +multiple copies. Assume the length of the longest common subsequence is +n − k, and there are l proper-transposable genes. Then the number of quasi- +transposable genes is no larger than 2(k − l). +When l + 2(k − l) ≤ n, in both linear and circular scenarios, we can find +examples with 2(k − l) quasi-transposable genes. +29 + +8 +Conclusion +In this paper, we study the problem of determining transposable genes in +gene sequences, and design Algorithms 1–6 for different scenarios. To apply +those algorithms, one needs to apply genomic annotation tools to transform +raw DNA sequencing data into gene sequences, and replace gene names by +numbers. Those algorithms have at most O(mn3) time complexity, where +m is the number of sequences, and n is the number of genes. Thus they can +run in a reasonable time for most applications. We prove that the latter +two scenarios are NP-hard (Propositions 1,2), and propose two unresolved +problems (Conjectures 2,3) in discrete mathematics. +We start with gene sequences and determine translocated genes. There- +fore, short transposons (possibly shorter than a gene) cannot be determined. +Besides, we do not determine specific genomic rearrangement events. We +aim at determining which genes are able to translocate. +Specifically, we +study how many longest common subsequences contain a certain gene, as a +measure for its “stability”. This mesoscopic viewpoint can be intriguing for +understanding changes in genome. +The results in this paper are not limited to Scenarios 1–4. They can +be applied to other bioinformatics situations, or even other fields that need +discrete mathematics tools, such as text processing, compiler optimization, +data analysis, image analysis [22]. 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Genome Res. 27, 5 (2017), 787–792. +38 + diff --git a/QdE2T4oBgHgl3EQfVwfD/content/tmp_files/load_file.txt b/QdE2T4oBgHgl3EQfVwfD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..949203adbd1506b80903931df35a15584f58a855 --- /dev/null +++ b/QdE2T4oBgHgl3EQfVwfD/content/tmp_files/load_file.txt @@ -0,0 +1,1550 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf,len=1549 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='03827v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='GN] 10 Jan 2023 Longest common subsequence algorithms and applications in determining transposable genes Yue Wang1,2,* 1Department of Computational Medicine, University of California, Los Angeles, California, United States of America 2School of Mathematical Sciences, Peking University, Beijing, China E-mail address: yuew@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' ORCID: 0000-0001-5918-7525 Abstract Given several number sequences, determining the longest common subsequence is a classical problem in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Transposons are nucleotide sequences in DNA that can change their positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Many transposons are shorter than a general gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When we restrict to nu- cleotide sequences that form complete genes, we can still find genes that change their relative locations in a genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus for different individuals of the same species, the orders of genes might be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A practical problem is to determine such transposable genes in given gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Through an intuitive rule, we transform the biological problem of determining transposable genes into a rigorous mathemati- cal problem of determining the longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Depend- ing on whether the gene sequence is linear (each sequence has a fixed head and tail) or circular (we can choose any gene as the head, and the previous one is the tail), and whether genes have multiple copies, we classify the problem of determining transposable genes into four scenarios: (1) linear sequences without duplicated genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' (2) circular sequences without duplicated genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' (3) linear sequences with dupli- cated genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' (4) circular sequences with duplicated genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' With the help of graph theory, we design fast algorithms for different scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Specifically, we study the situation where the longest common subsequence is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' KEY WORDS: transposon, gene sequence, algorithm, graph 1 1 Introduction The nucleotide sequence can be changed by various events, such as inver- sion, insertion, deletion, and duplication [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Such rearrangement events lead to the existence of transposons (also called transposable elements or jumping genes), which are DNA sequences that can change their relative positions within the genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Transposons were first discovered in maize by Barbara McClintock [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Transposons have various types: long terminal re- peats (LTR) retrotransposons, Dictyostelium intermediate repeat sequence (DIRS)-like elements, Penelope-like elements (PLE), long interspersed ele- ments (LINE), short interspersed elements (SINE), terminal inverted repeats (TIR), Helitrons, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Transposons are common in various species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For the human genome, the proportion of transposons is approximately 44%, although most of trans- posons are inactive [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Transposons can participate in controlling gene expression [80], and they are related to several diseases, such as cancer [13], hemophilia [33], and porphyria [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Transposons can drive rapid pheno- typic variations, which cause complicated cell behaviors [78, 48, 47, 11, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Transposons can be used to detect cancer drivers [49] and potential therapies [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Transposons are also essential for the development of Oxytricha trifallax [50], antibiotic resistance of bacteria [3], and the proliferation of various cells [53, 76, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' With the presence of transposons, the regulation between genes might be affected, which is a challenge for inferring the structures of gene regulatory networks [72] and general transcriptome analysis [58, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When transposons have been determined, we can use them to compare the genomes of different species, and such comparisons can be combined with other measurements between species, such as metrics on developmental trees [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Such comparisons can be also extended to different tissues to help with the prediction of tissue transplantation experiments [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Besides, for some species, cells at different positions have different gene expression patterns, which might be related to transposons [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Many transposons are as short as 102 − 103 base pairs, shorter than a general gene [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' To determine such short transposons, one needs to analyze the original AGCT nucleotide sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' There have been many algorithms developed to determine short transposons from nucleotide sequences, such as MELT (Mobile Element Locator Tool) [18], ERVcaller (Endogenous Retro- Virus caller) [10], and TEMP2 (Transposable Elements Movements Present 2) [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Different algorithms may only determine certain types of trans- posons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For more details, readers may refer to other papers [51, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' They use raw DNA sequencing data, which only contain imperfect information 2 about the true DNA sequence, and the data quality depends on some factors that vary across different datasets [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Besides, they need a corresponding genome or reference transposon libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' There are gross DNA changes that associate with many genes, also called genomic rearrangements [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Such rearrangements include inversion, trans- position, fusion, and fission [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' To determine such gross genomic rearrange- ments, one first needs to convert nucleotide sequences into gene sequences by annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For two different gene sequences, the general idea of de- termining rearrangements is to calculate the minimal number of operations required for transforming one sequence into the other [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This defines an editing distance between gene sequences, which can be used to compare the evolution distance between species and construct the phylogenetic tree [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' There have been many algorithms developed to determine genomic rearrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' They consider different scenarios: whether the gene se- quence is linear or circular, whether genes have unique labels, and what operations can be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Kececioglu and Sankoff only consider inversion for linear sequences with unique gene labels [34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Blanchette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' consider in- version and transposition for circular sequences with unique gene labels [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Tesler considers inversion, transposition, fusion, and fission for linear and circular sequences with unique gene labels [60];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Terauds and Sumner study circular sequences with representation theory tools [59];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Bohnenk¨amper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' consider linear and circular sequences with possibly duplicated labels [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' There are also systematic pipelines for determining rearrangements from whole-genome assemblies [19, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Nevertheless, these methods con- sider large-scale rearrangements, and minimize the number of operations to transform one gene sequence into the other, not concrete genes that can change their locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Besides, these methods only compare two gene se- quences, not more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Their results depend on the set of possible operations, which is somewhat arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In this paper, we consider a mesoscopic scenario between the genomic rearrangement situation and the short transposon situation: Given accu- rately annotated gene sequences (not nucleotide sequences) from different individuals, determine individual genes (not short nucleotide segments or long gene strands) that can change their locations (transposable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This pro- vides a qualitative description for the stability of genes, which can guide gene editing [65] and phylogenetics [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The proportion of fixed genes quanti- fies the robustness of the genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We aim at minimizing the number of genes to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When there are only two gene sequences, this is equivalent to calculating genomic arrangements, where the only allowed operation is single-gene transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 3 In the copy-paste (duplication) case and deletion case, we can compare the numbers of copies of genes for different individuals to determine the transposable genes that have changed their copy numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In the inver- sion case, we can check the direction of genes to determine transposable genes that have changed their orientations [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In the cut-paste (insertion) case, the compositions of gene sequences are the same, but the orders of genes differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' It is not straightforward to uniquely determine which genes have changed their relative locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Instead, we can consider the comple- ment of transposable genes, which keep their relative locations and form a common subsequence of gene sequences from different individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Notice that genes in a subsequence does not need to be adjacent in the original sequences, different from a substring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We aim at explaining the difference among gene sequences with minimal transposable genes, meaning that we want to maximize the length of the complement of transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus we define the transposable genes to be the complement of the longest com- mon subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Given raw nucleotide sequences, we first transform them into gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then we apply our algorithms to find the longest com- mon subsequence, and the complement is transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If the longest common subsequence is not unique, we also need to determine which genes are more conserved and appear in all longest common subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' It is common to use the length of the longest common subsequence as a quantitative score for comparing DNA sequences [12, 26, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The longest common subsequence has also been used to define ultraconserved elements [54] or remove incongruent markers [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Determining the longest common subsequence is a classical problem in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Various scenarios for this problem have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here we list Scenarios A-E, where the first two are more commonly stud- ied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For more works in these scenarios, readers may refer to more thorough reviews [5, 24, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenario A considers two sequences with possibly re- peated genes, and the sequence length is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The goal is to find the longest common subsequence, where the length is count by gene copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This can be solved by dynamic programming with O(n2) time complexity and O(n) space complexity [23], but O(n2−ǫ) time complexity for any ǫ > 0 is im- possible [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This also can be solved with o(n) space complexity and O(n3) time complexity [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In Scenario B, there are m sequences with possibly repeated genes, and the sequence length is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The goal is to find the longest common subsequence, where the length is count by gene copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A standard dynamic programming algorithm has O(nm) time complexity [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' There have been other faster algorithms [64, 44, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This scenario is equivalent to the maximum clique problem in graph theory, which is NP-hard [39], but 4 has fast exact and heuristic algorithms [30, 37, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenario C considers 2 sequences with possibly repeated genes, and the sequence length is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The goal is to find the longest common subsequence, where each gene appears at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This scenario is NP-hard [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenario D is similar to Scenario B, but only consider common subsequences that contain or do not contain certain strings [66, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In Scenario E, the gene sequences are arc-annotated, and the longest common subsequence should have the same arc annotation in original sequences [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In this paper, we consider four scenarios that are different from the previously studied longest common subsequence problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' These four sce- narios are determined by two factors: whether the considered species has linear or circular gene sequences, and whether genes have multiple copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When genes have multiple copies, we only consider common subsequences that consist of all or none of copies of the same gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenario 1 has linear sequences without duplicated genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenario 2 has circular sequences with- out duplicated genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenario 3 has linear sequences with duplicated genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenario 4 has circular sequences with duplicated genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Most known methods only aim at finding one longest common subse- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When the longest common subsequence is not unique, we also need to classify whether a gene appears in all/some/none of the longest common subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Determining all longest common subsequences is too time- consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' To determine the relationship between genes and longest com- mon subsequences, we develop corresponding algorithms with polynomial time complexities for Scenarios 1,2 (Algorithms 2,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' To our knowledge, there are no other determinations of whether genes appear in all longest common subsequences with polynomial complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenarios 3,4 only consider subsequences that consist of all or none copies of the same gene, and calculate the length by genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, they are different from the classic Scenario B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We develop the equivalence of Scenario 3 with the maxi- mum clique problems on graphs (Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We prove that Scenario 4 is between the maximum clique problems on graphs and the maximum clique problems on 3-uniform hypergraphs (Propositions 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Although circular sequences are commonly studied in the context of genomic rearrangements, they are rare in the literature of longest common subsequence problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, our Algorithm 3 that finds a longest common subsequence for Scenario 2 should also be novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We test Algorithms 1,2,3,4 on the gene sequences of different Escherichia coli individuals and find some possible transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If we only need to find one longest common subsequence, then Scenario 1 is a special case of Scenario B, and our method (Algorithm 1) is easily 5 derived from standard algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenarios 3,4 are equivalent to maximum clique problems in graphs and hypergraphs, which are NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' These prop- erties are also similar to Scenario B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Although there have been numerous algorithms for the maximum clique problem [75], for the sake of complete- ness, we design fast heuristic algorithms (Algorithms 5,6) and test them to find that they only fail in rare cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We proposed the idea of using the longest common subsequence to find transposable genes and Algorithm 1 in a previous paper [32], where Algo- rithm 1 was applied to study the “core-gene-defined genome organizational framework” (the complement of transposable genes) in various bacteria, and found that for different species, the transposable gene distribution and de- velopmental traits are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This paper considers other situations (especially when the longest common subsequence is not unique), and can be regarded as a theoretical sequel of that previous paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Algorithm 1 is contained in this paper for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In sum, our main contributions are Algorithms 2,3,4 in Scenarios 1,2 and Propositions 1, 2, 3 in Scenarios 3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We first describe the setup for the problem of determining transposable genes and transform it into the problem of finding the longest common subse- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In the following four sections, we transform them into corresponding graph theory problems and design algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We finish with some discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' All the algorithms in this paper have been implemented in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For the code and data files, see https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='com/YueWangMathbio/Transposon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 2 Setup Given raw DNA sequencing data, the first step is to transform them into gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This can be done with various genome annotation tools [57, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For simplicity, we replace the gene names by numbers 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For some species, the DNA is a line [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can represent this DNA as a linear gene sequence of distinct numbers that represent genes: (1, 2, 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If some genes change their transcriptional orientations, we can simply detect them and handle the remaining genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Now a linear DNA naturally has a direction (from 5’ end to 3’ end), thus (1, 2, 3, 4) and (4, 3, 2, 1) are two different gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Consider two linear gene sequences from different individuals: (1, 2, 3, 4) and (1, 4, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can intuitively detect that gene 4 changes its relative position, and should be regarded as a transposable gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' However, changing the positions of genes 2, 3 can also transform one sequence into the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 6 The reason that we think gene 4 (not genes 2, 3) changes its relative position is that the number of genes we need to move is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' However, the number of genes that change their relative locations is difficult to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can consider the complement of transposable genes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', genes that do not change their relative positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' These fixed genes can be easily defined as the longest common subsequence of given gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here a common subsequence consists of some genes (not necessarily adjacent, different from a substring) that keep their relative orders in the original sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus transposable genes are the complement of this longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Notice that the longest common subsequence might not be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We classify genes by their relations with the longest common subsequence(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The motivation of classifying transposable genes with respect to the intersection and union of longest common subsequences is similar to defining essential variables with Markov boundaries in causal inference [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A gene is proper-transposable if it is not contained in any longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A gene is non-transposable if it is contained in every longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A gene is quasi-transposable if it is contained in some but not all longest common subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In the example of (1, 2, 3, 4) and (1, 4, 2, 3), the unique longest com- mon subsequence is (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus 4 is proper-transposable, and 1, 2, 3 are non-transposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In the following, we consider other scenarios, where the proper/quasi/non-transposable genes still follow Definition 1, but the defi- nition of the longest common subsequence differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For some species, the DNA is a circle, not a line [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A circular DNA also has a natural direction (from 5’ end to 3’ end), and we use the clock- wise direction to represent this natural direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In the circular sequence scenario, a common subsequence is a circular sequence that can be obtained from each circular gene sequence by deleting some genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 1 for two circular gene sequences and their longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Notice that we can rotate each circular sequence for a better match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 1 2 3 3 1 2 1 2 6 5 4 5 4 6 5 4 Figure 1: Two circular gene sequences without duplicated genes and their longest common subsequence, corresponding to Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 7 A gene might have multiple copies (duplicated) in a gene sequence [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Notice that the definition of the transposable gene is a gene (specific DNA sequence) that has the ability to change its position, not a certain copy of a gene that changes its position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This means transposable genes should be defined for genes, not gene copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus we should only consider common subsequences that consist of all or none copies of the same gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When calculating the length of a common subsequence, we should count genes, not gene copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Consider two linear sequences (4, 1, 2, 1, 1, 3, 2, 4, 1, 1) and (4, 1, 2, 3, 1, 1, 2, 1, 1, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If we consider any subsequences, the longest com- mon subsequence is (4, 1, 2, 1, 1, 2, 1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' if we only consider subsequences that contain all or none copies of the same gene, but count the length by copies, the longest common subsequence is (1, 2, 1, 1, 2, 1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' if we only consider subsequences that contain all or none copies of the same gene, and count the length by genes, the unique longest common subsequence is (4, 2, 3, 2, 4), and gene 1 is proper-transposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When we consider circular gene sequences with duplicated genes, we should still only consider subsequences that consist of all or none copies of the same gene, and calculate the length by genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Notice that circular sequences can be rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 2 for two circular gene sequences with duplicated genes and their longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 1 2 1 3 1 3 1 2 3 2 3 2 1 2 2 1 Figure 2: Two circular gene sequences with duplicated genes and their longest common subsequence, corresponding to Scenario 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We have turned the problem of determining transposable genes into find- ing the longest common subsequence of several gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Depending on whether the gene sequences are linear or circular, and whether genes have multiple copies, the problem can be classified into four scenarios: Scenario 1: Consider m linear sequences of genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, where each gene has only one copy in each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Determine the longest linear sequence that is a common subsequence of these m sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenario 2: Consider m circular sequences of genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, where each gene has only one copy in each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Determine the longest circular sequence that is a common subsequence of these m sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here circular sequences can be rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 8 Scenario 3: Consider m linear sequences of genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, where each gene can have multiple copies in each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Determine the longest linear sequence that is a common subsequence of these m sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Only consider subsequences that consist of all or none copies of the same gene, and calculate the length by genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Scenario 4: Consider m circular sequences of genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, where each gene can have multiple copies in each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Determine the longest circular sequence that is a common subsequence of these m sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Only consider subsequences that consist of all or none copies of the same gene, and calculate the length by genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here circular sequences can be rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' These four scenarios correspond to different algorithms, and will be dis- cussed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 3 Linear sequences without duplicated genes In Scenario 1, consider m linear gene sequences, where each sequence con- tains n genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Each gene has only one copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For such permutations of 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, we need to find the longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1 A graph representation of the problem Brute-force searching that tests whether each subsequence appears in all sequences is not applicable, since the time complexity is exponential in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' To develop a polynomial algorithm, we first design an auxiliary directed graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For m linear sequences with n non-duplicated genes, the cor- responding auxiliary graph G is a directed graph, where each vertex is a gene gi, and there is a directed edge from gi to gj if and only if gi appears before gj in all m sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A directed path g1 → g2 → g3 → · · · → g4 → g5 in G corresponds to a common subsequence (g1, g2, g3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , g4, g5) of m sequences, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We add 0 to the head of each sequence and n + 1 to the tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then the longest common subsequence must start at 0 and end at n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The problem of finding the longest common subsequence becomes finding the longest path from 0 to n + 1 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 3 for an example of using the auxiliary graph to determine transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This auxiliary graph G has no directed loop (acyclic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If there exists a loop g1 → g2 → g3 → · · · → g4 → g1, then g1 is prior to g4 and g4 is prior to g1 in all sequences, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 9 0 � ������� ������� �❃❃❃❃❃❃❃❃ �✏✏✏✏✏✏✏✏✏✏✏✏✏✏ �✳✳✳✳✳✳✳✳✳✳✳✳✳✳ � 1 � � �◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ �✳✳✳✳✳✳✳✳✳✳✳✳✳✳ 2 � ♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣ �✏✏✏✏✏✏✏✏✏✏✏✏✏✏ 3 �❃❃ ❃❃ ❃❃ ❃ ❃❃ ❃❃ ❃❃ ❃ 4 ��������� 5 Figure 3: The auxiliary graph G of two sequences ([0], 1, 2, 3, 4, [5]) and ([0], 1, 4, 2, 3, [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The unique longest path (double arrows) from 0 to 5 is 0 → 1 → 2 → 3 → 5, meaning that the unique longest common se- quence is ([0], 1, 2, 3, [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus 1, 2, 3 are non-transposable, and 4 is proper- transposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='2 Find the longest path Determining the longest path between two vertices in a directed acyclic graph can be solved by a standard dynamic programming algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For a vertex gi ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n}, consider the longest path from gi to n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since there exists an edge gi → n + 1, and G is acyclic, this longest path exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If the longest path is not unique, assign one arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Define F+(gi) to be the length of the longest path from gi to n + 1 in G, and H+(gi) to be the vertex next to gi in this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' F+ and H+ can be calculated recursively: For one gene gi, consider all genes gj with an edge gi → gj in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The gene gj with the largest F+(gj) is assigned to be H+(gi), and F+(gi) = F+(gj) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If gl → n + 1 is the only edge that starts from gene gl, then F+(gl) = 1, and H+(gl) = n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In other words, H+(gi) = argmax {gj with gi→gj} F+(gj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' F+(gi) = 1 + F+[H+(gi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then 0 → H+(0) → H2 +(0) → H3 +(0) → · · · → Hf−1 + (0) → Hf +(0) = n + 1, denoted by L0, is a longest path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here f = F+(0), and Hi + is the ith iteration of H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='3 Test the uniqueness of the longest path To test whether quasi-transposable genes exist, we need to check the unique- ness of this longest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For gi ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, n + 1}, define F−(gi) to be the length of the longest path from 0 to gi in G, and H−(gi) to be the vertex prior to gi in this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' F− and H− can be calculated similar to F+ and H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can see that F+(gi) + F−(gi) is the length of 0 = HF−(gi) − (gi) → HF−(gi)−1 − (gi) → · · · → H−(gi) → gi → H+(gi) → · · · → HF+(gi)−1 + (gi) → HF+(gi) + (gi) = n + 1, a longest path from 0 through gi to n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For gi /∈ L0, if F+(gi) + F−(gi) < F+(0), then gi is proper-transposable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' if F+(gi) + F−(gi) = F+(0), then gi is quasi-transposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If every gi /∈ L0 is proper-transposable, then the longest common subsequence is unique, and all genes in L0 (excluding the auxiliary 0 and n + 1) are non-transposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The procedure of determining transposable genes stops here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Otherwise, the longest common subsequence is not unique, and we need to find quasi-transposable genes in L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='4 Find quasi-transposable genes When determining all quasi-transposable genes g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk not in L0, as de- scribed above, we construct corresponding longest paths L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , Lk from 0 to n + 1, where each Li passes through gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We claim that a gene gj ∈ L0 is non-transposable if and only if gj is contained in all L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' To prove this, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In Scenario 1 of linear sequences without duplicated genes, each quasi-transposable gene gi has a corresponding quasi-transposable gene gj, so that no longest common subsequence can contain both gi and gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If a gene gj ∈ L0 is non-transposable, then it is contained in all L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If gj ∈ L0 is quasi-transposable, by Lemma 1, there is a quasi-transposable gene gl /∈ L0 which is mutual-exclusive with gj, in the sense that gl and gj cannot appear in the same longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The correspond- ing longest path Ll contains gl, thus cannot contain gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This proves our approach to determine the quasi-transposable genes in L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 11 Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Fix a quasi-transposable gene gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' It is contained in a longest path Li, which contains all non-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus for each non-transposable gene g∗, there is an edge between g∗ and gi in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Assume gi has no such mutual-exclusive quasi-transposable gene gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then there is an edge (direction unknown) in G between gi and each quasi-transposable gene gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Choose a longest path L∗ in G that does not contain gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Whether gj ∈ L∗ is a non-transposable gene or a quasi-transposable gene, there is an edge between gj and gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Determine the first gene gk in L∗ that has an edge gi → gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since there is an edge gi → n + 1, gk exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since there is an edge 0 → gi, gk ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Denote the previous gene of gk in L∗ by gl, then gl exists, and there is an edge gl → gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus we construct a path 0 → · · · → gl → gi → gk → · · · → n + 1, which is longer than the longest path, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus gi has a mutual-exclusive quasi-transposable gene gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='5 Algorithms and complexities We summarize the above method as Algorithms 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If we have known that the longest common subsequence is unique, then we just need to apply Algorithm 1, so that genes in L0 are non-transposable, and genes not in L0 are proper-transposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We have reported Algorithm 1 previously [32, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Algorithm 1 is kept here to make the story complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Assume we have m sequences with length n, and the length of the longest common subsequence is n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The time complexities of Steps 2-5 in Algorithm 1 are O(m), O(mn2), O(n), O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The time complexities of Step 2 and Step 3 in Algorithm 2 are O(k) and O(kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since k ≤ n, the overall time complexity of determining transposable genes in Scenario 1 by Algorithms 1,2 is O(mn2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The space complexity is trivially O(mn + n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='6 Applications on experimental data We test Algorithms 1,2 on Escherichia coli gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' From NCBI sequencing database, we obtain gene sequences of three individuals of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' coli strain ST540 (GenBank CP007265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1, GenBank CP007390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1, GenBank CP007391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1) and three individuals of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' coli strain ST2747 (GenBank CP007392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1, GenBank CP007393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1, GenBank CP007394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' All three sequences of ST540 start with gene dnaA and end with gene rpmH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can regard them as linear gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We remove genes that appear more than once in one sequence, and remove genes that do not ap- pear in all three sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' After applying Algorithms 1,2 on these three 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Input m linear sequences of genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' No duplicated genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Modify the sequences: Add 0 to the head, and n + 1 to the tail of each sequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Construct the auxiliary graph G: Vertices of G are all the genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n For each pair of genes gi, gj If gi is prior to gj in all m sequences Add a directed edge gi → gj in G End of if End of for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Calculate F+(·) and H+(·) for each gene gi in 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n recursively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' calculate F−(·) and H−(·) for each gene gi in 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, n + 1 recursively: H+(gi) = argmax {gj with gi→gj} F+(gj) % If gj with gi → gj that maximizes F+(gj) is not unique, choose one randomly F+(gi) = 1 + F+[H+(gi)] H−(gi) = argmax {gj with gj→gi} F−(gj) % If argmax is not unique, choose one randomly F−(gi) = 1 + F−[H−(gi)] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Construct a longest path L0 from 0 to n + 1: 0 → H+(0) → H2 +(0) → H3 +(0) → · · · → Hf−1 + (0) → Hf +(0) = n + 1 % Here f = F+(0), and Hi + is the ith iteration of H+ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Output F+(·), H+(·), F−(·), H−(·), L0 Algorithm 1: Detailed workflow of determining proper-transposable genes and quasi-transposable genes in Scenario 1, preparation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Input F+(·), H+(·), F−(·), H−(·), L0 calculated from Algorithm 1 Denote all genes not in L0 by g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For each gene gi in g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk If F+(gi) + F−(gi) < F+(0) Output gi is a proper-transposable gene Else Output gi is a quasi-transposable gene End of if End of for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If all genes in g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk are proper-transposable Output all genes in L0 are non-transposable Else For each gene gi in g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk Use H+(·) and H−(·) to construct Li, a longest path from 0 to n + 1 that passes gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' End of for For each gene gj in L0 (excluding auxiliary 0 and n + 1) If gj is contained in all L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , Lk Output gj is non-transposable Else Output gj is quasi-transposable End of if End of for End of if 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Output: whether each gene is proper/quasi/non-transposable Algorithm 2: Detailed workflow of determining proper-transposable genes and quasi-transposable genes in Scenario 1, output stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 14 sequences, there are 301 non-transposable genes, 4 quasi-transposable genes (hpaC, iraD, fbpC, psiB), and 263 proper-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The reason for the large amount of proper-transposable genes is that sequence CP007265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1 is significantly different from the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' After removing it and ap- plying Algorithms 1,2 to the remaining two sequences (CP007390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1 and CP007391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1), there are 564 non-transposable genes and 4 quasi-transposable genes (hpaC, iraD, fbpC, psiB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, some genes in hpaC, iraD, fbpC, psiB are likely to translocate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' All three sequences of ST2747 start with gene glnG and end with gene hemG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can regard them as linear gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We remove genes that appear more than once in one sequence, and remove genes that do not appear in all three sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' After applying Algorithms 1,2 on these three sequences, all 573 genes are non-transposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 4 Circular sequences without duplicated genes In Scenario 2, consider m circular gene sequences, where each sequence contains n genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Each gene has only one copy in each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For such circular permutations of 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, we need to find the longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Assume the length of the longest common subsequence is n−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1 Find a longest common subsequence We first randomly choose a gene gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Cut all circular sequences at gi and expand them to be linear sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For example, the circular sequences in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 1 cut at 1 are correspondingly (1, 2, 3, 4, 5, 6) and (1, 2, 6, 4, 5, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Using Algorithm 1, we can find Li that begins with gi, which is a longest common subsequence of all expanded linear sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In the above example, the longest common linear subsequence starting from 1 is (1, 2, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If gi is a non-transposable gene or a quasi-transposable gene, then Li (glued back to a circle) is a longest common circular subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If gi is a proper- transposable gene, then Li is shorter than the longest common circular sub- sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 1, gene 1 is non-transposable, and (1, 2, 4, 5) (glued) is the longest common circular subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We do not know if Li (glued) is a longest common subsequence (whether containing gi or not) for all circular sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If there is a longer common subsequence, it should contain genes that are not in Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Consider four variables L, g, C, and S, whose initial values are Li, gi, the length of Li, and the complement of Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' These variables contain information on the longest common linear subsequence that we have found during this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 15 Choose a gene gj in S, and cut all circular gene sequences at gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Apply Algorithm 1 to find Lj, which is the longest in common subsequences that contain gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If the length of Lj is larger than C, set L to be Lj, set g to be gj, set C to be the length of Lj, and set S to be the complement of Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Otherwise, keep L, g, C, and S still.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Choose another gene gl in S which has not been chosen before, and repeat this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This procedure terminates when all genes in S have been chosen and cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Denote the final values of L, g, C, and S by L0, g0, C0, and S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here S0 is the complement of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' During this procedure, if the current g is a proper-transposable gene, then S contains a non-transposable gene or a quasi-transposable gene, which has not been chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus L, g, C, S will be further updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If the current g is a non-transposable gene or a quasi-transposable gene, then C has reached its maximum, and L, g, C, S will not be further updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This means L0 is a longest common circular subsequence, and C0 is the length of the longest common subsequence, n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Also, the total number of genes being chosen and cut is k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' All k genes in S0 and g0 are chosen and cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A gene gt in L0 (excluding g0) is non-transposable or quasi-transposable, and cannot be chosen and cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The reason is that it cannot be chosen before g0 is chosen (only proper-transposable genes can be chosen before g0 is chosen), and it cannot be chosen after g0 is chosen (gt /∈ S0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='2 Determine quasi-transposable genes For each gene gp ∈ S0, apply Algorithm 1 to calculate Cp, the length of the longest common subsequence that contains gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If Cp < C0, gp is a proper- transposable gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Otherwise, Cp = C0 means gp is a quasi-transposable gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We have found all proper-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If all genes in S0 are proper-transposable, then all genes in L0 are non-transposable, and the procedure terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If S0 contains quasi-transposable genes, then L0 also has quasi-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' To determine quasi-transposable genes in L0, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In Scenario 2, choose a quasi-transposable gene gp and cut the circular sequences at gp to obtain linear sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A proper-transposable gene for the circular sequences is also a proper-transposable gene for the linear sequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' a non-transposable gene for the circular sequences is also a non-transposable gene for the linear sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Consider a longest common subsequence Lp for linear sequences cut 16 at gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since gp is a quasi-transposable gene, the length of Lp is also n − k, meaning that Lp is also a longest common subsequence for circular se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Now, this lemma is proved by the definition of proper/quasi/non- transposable gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If a gene gr in L0 is non-transposable for the circular sequences, then gr is a non-transposable gene for linear sequences cut at each quasi-transposable gene gq ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If a gene gs in L0 is quasi-transposable for the circular sequences, then there is a longest common circular subsequence Lt that does not contain gs, meaning that Lt contains a quasi-transposable gene gt not in L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then gs is a proper/quasi-transposable gene for linear sequences cut at gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, we can use the following method to determine quasi-transposable genes in L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For each quasi-transposable gene gq ∈ S0, cut at gq and ap- ply Algorithms 1,2 to determine if each gene in L0 is proper/quasi/non- transposable for the linear gene sequences cut at gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A gene gr ∈ L0 is non- transposable for the circular sequences if and only if it is non-transposable for linear sequences cut at any quasi-transposable gene gq ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A gene gs ∈ L0 is quasi-transposable for the circular sequences if and only if it is proper/quasi-transposable for linear sequences cut at some quasi-transposable gene gq ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When we have determined all quasi-transposable genes in S0, it might be tempting to apply a simpler approach to determine quasi-transposable genes in L0: For each quasi-transposable gene gq ∈ S0, cut at gq and apply Algorithm 1 to find a longest common subsequence Lq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A gene in L0 is non-transposable if and only if it appears in all such Lq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This approach is valid only if the following conjecture holds, which is similar to Lemma 1: Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In Scenario 2 of circular sequences without duplicated genes, each quasi-transposable gene gi has a corresponding quasi-transposable gene gj, so that no longest common subsequence can contain both gi and gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' However, Conjecture 1 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 4 for a counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' All genes are quasi-transposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Any two quasi-transposable genes are contained in a longest common subsequence (length 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus the simplified approach above does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We summarize the above method as Algorithms 3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If we have known that the longest common subsequence is unique, then we just need to apply Algorithm 3, so that genes in S0 are proper-transposable, and genes not in S0 are non-transposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Assume we have m sequences with length n, and the length of the longest common subsequence is n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The time complexities 17 1 2 3 1 2 6 1 2 7 8 4 3 5 6 8 7 6 5 4 7 8 5 4 3 Figure 4: A counterexample with three circular sequences that fails Conjec- ture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' of Step 2 and Step 3 in Algorithm 3 are O(mn2) and O(kmn2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The time complexities of Step 2 in Algorithm 4 is O(kmn2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The overall time com- plexity of determining transposable genes in Scenario 2 by Algorithms 3,4 is O(kmn2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The space complexity is trivially O(mn + n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='3 Applications on experimental data Similar to Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='6, we test Algorithms 3,4 on Escherichia coli gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' From NCBI sequencing database, we obtain gene sequences of three individuals of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' coli strain ST540 (GenBank CP007265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1, GenBank CP007390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1, GenBank CP007391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1) and three individuals of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' coli strain ST2747 (GenBank CP007392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1, GenBank CP007393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1, GenBank CP007394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We regard all three sequences of ST540 as circular gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We remove genes that appear more than once in one sequence, and remove genes that do not appear in all three sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' After applying Algorithms 3,4 on these three sequences, there are 389 non-transposable genes, 50 quasi- transposable genes, and 129 proper-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The reason for the large amount of proper-transposable genes is that sequence CP007265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1 is significantly different from the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' After removing it and applying Al- gorithms 3,4 to the remaining two sequences (CP007390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1 and CP007391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1), there are 564 non-transposable genes and 4 quasi-transposable genes (hpaC, iraD, fbpC, psiB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, some genes in hpaC, iraD, fbpC, psiB are likely to translocate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We regard all three sequences of ST2747 as circular gene sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We remove genes that appear more than once in one sequence, and remove genes that do not appear in all three sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' After applying Algorithms 3,4 on these three sequences, all 573 genes are non-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Input m circular sequences of genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, where each gene has only one copy in each sequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Choose a gene gi randomly Cut all circular sequences at gi and expand them to be linear sequences Apply Algorithm 1 to find Li, a longest common subsequence in the expanded linear sequences Set C to be the length of Li, and set S to be the complement of Li 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' While S has a gene gj that has not been chosen and cut Cut all circular sequences at gj and apply Algorithm 1 to find Lj Denote the length of Lj by Cj If Cj > C Update C to be Cj, and update S to be the complement of Lj End of if End of while Denote the final C by C0, and denote the final S by S0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Output C0 and S0 Algorithm 3: Detailed workflow of determining proper-transposable genes and quasi-transposable genes in Scenario 2, preparation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Input m circular sequences of genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, where each gene has only one copy in each sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' C0 and S0 calculated from Algorithm 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For each gene gl ∈ S0 Cut all circular sequences at gl and expand them to be linear sequences Apply Algorithm 1 to find Ll, a longest common subsequence in the expanded linear sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Denote the length of Ll by Cl If Cl < C0 Output gl is a proper-transposable gene Else Output gl is a quasi-transposable gene Cut all circular sequences at gl and apply Algorithms 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='2 to find all proper/quasi-transposable genes for linear gene sequences starting at gl Output genes not in S0 but being proper/quasi-transposable for such linear sequences are quasi-transposable for circular sequences End of if End of for Output other genes that have not been determined to be proper/quasi-transposable are all non-transposable 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Output: whether each gene is proper/quasi/non-transposable Algorithm 4: Detailed workflow of determining proper-transposable genes and quasi-transposable genes in Scenario 2, output stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 20 5 Linear sequences with duplicated genes In Scenario 3, consider m linear gene sequences, where each sequence con- tains different numbers of copies of n genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We need to find the longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here we only consider common subsequences that consist of all or none copies of the same gene, and the subsequence length is calculated by genes, not gene copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='1 A graph representation of the problem Similar to Scenario 1, we construct an auxiliary graph G, where each vertex is a gene (not a copy of a gene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' However, in this case, the auxiliary graph is undirected: There is an undirected edge between gene gi and gene gj if and only if all the copies of gi and gj keep their relative locations in all sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For example, consider two sequences (1, 2, 3, 2, 3, 4, 5) and (2, 1, 3, 3, 2, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For gene pair 1, 3, the corresponding sequences are (1, 3, 3) and (1, 3, 3), meaning that there is an edge between 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For gene pair 1, 2, the corresponding sequences are (1, 2, 2) and (2, 1, 2), meaning that there is no edge between 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 5 for the auxiliary graph in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 1 ❃❃❃❃❃❃❃❃ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ 2 ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 3 ♣♣♣♣♣♣♣♣♣♣♣♣♣♣ 4 5 Figure 5: The auxiliary graph G of two sequences (1, 2, 3, 2, 3, 4, 5) and (2, 1, 3, 3, 2, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The unique largest complete subgraph is {1, 3, 4, 5}, mean- ing that the unique longest common sequence is (1, 3, 3, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus 1, 3, 4, 5 are non-transposable genes, and 2 is a proper-transposable gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A subgraph of G consists of some genes g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gl and the edges between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In a subgraph, if there is an edge between any two genes, this subgraph is called a complete subgraph (also called a clique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In graph G, the degree of a gene g is the number of edges linking g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In a complete graph of p genes, where any two genes have an edge in between, each gene has degree p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 21 Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If all copies of genes g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gl keep their relative locations in all linear sequences, we say that g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gl form a common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The following Lemma 3 shows that there is a bijection between common subsequences and complete subgraphs in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The problem of determining the longest common subsequence now becomes determining the largest complete subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In Scenario 3, construct the auxiliary graph G from gene se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk form a complete subgraph in G, then g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk form a common subsequence, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gl form a common subsequence, then there is an edge in G between any two genes in g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gl, meaning that they form a complete subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For the other direction, only consider copies of g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk in these se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk do not form a common subsequence, find the first digit that such sequences differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Assume gp and gq can both appear in this digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then gp, gq cannot form a common subsequence, and there is no edge between gp and gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We illustrate this proof with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 5: For genes 2, 3, 4, the sequences are (2, 3, 2, 3, 4) and (2, 3, 3, 2, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The third digit is different, where 2 and 3 can both appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then the sequences for genes 2, 3, (2, 3, 2, 3) and (2, 3, 3, 2), cannot match, and there is no edge between 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='2 A heuristic algorithm The above discussion shows that given gene sequences, we can construct an undirected graph G, so that there is a bijection between common subse- quences and complete subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The inverse also holds: We can construct corresponding gene sequences for a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Given an undirected graph G, we can construct two gene se- quences, so that there is a bijection between common subsequences and com- plete subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Assume the graph has n genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We start with two sequences (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n) and(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For each pair of genes gi, gj, if there is no edge between them in G, add gi, gj to the end of the first sequence, and gj, gi to the end of the second sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then gi, gj cannot both appear in a common subsequence, and this operation does not affect other gene pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 22 For example, corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 5, we start with (1, 2, 3, 4, 5) and (1, 2, 3, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since there is no edge between 1, 2, we add them to have (1, 2, 3, 4, 5, 1, 2) and (1, 2, 3, 4, 5, 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since there is no edge between 2, 3, we add them to have (1, 2, 3, 4, 5, 1, 2, 2, 3) and (1, 2, 3, 4, 5, 2, 1, 3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' These two sequences corresponds to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Combining Lemma 3 and Lemma 4, we obtain the following result: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Finding the longest common sequence in Scenario 3 is equivalent to the maximum clique problem, which is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For an undirected graph, we can use Lemma 4 to construct corre- sponding sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If we have the solution of finding the longest common sequence in Scenario 3, then we can find the largest complete subgraph in an extra polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For gene sequences in Scenario 3, we can construct corresponding auxil- iary graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If we have the solution of finding the largest complete subgraph, then we can use Lemma 3 to find the longest common sequence in Scenario 3 in an extra polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, finding the longest common sequence in Scenario 3 and finding the largest complete subgraph are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The problem of determining the largest complete subgraph is just the maximum clique problem, which is NP-hard [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus finding the longest common sequence in Scenario 3 is also NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This means it is not likely to design an algorithm that always correctly determines the longest common subsequence in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We have transformed Scenario 3 into the maximum clique problem for a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' There have been various algorithms for the maximum clique problem [30, 37, 69], and readers may refer to a review for more details [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For completeness, we propose a simple idea: In the auxiliary graph G, repeatedly abandon the gene with the smallest degree (and also edges linking this gene) until the remaining genes form a complete subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' See Algorithm 5 for the details of this greedy heuristic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This algorithm is easy to understand, and can provide some intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We do not claim that Algorithm 5 is comparable to other sophisticated algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We test Algorithm 5 on random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Construct a random graph with n genes, and any two genes have probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='5 to have an edge in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Use brute-force search to find the maximum clique, and compare its size with the result of Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For each n ≤ 15, we repeat this for 10000 times, and every time Algorithm 5 returns the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, for small 23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Input m linear sequences of genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, where each gene can have multiple copies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Construct the auxiliary graph G: Vertices of G are all the genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n (not their copies) For each pair of genes gi, gj If all copies of gi and gj keep their relative locations in all m sequences Add an undirected edge between gi and gj in G End of if End of for Calculate the degree for each gene in G 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' While true Find a gene gi with the smallest degree di in G % If the minimal gi is not unique, choose one randomly If di + 1 is smaller than the number of genes in G Delete gi and edges linking gi in G Update the degrees of other genes Else % The remaining genes form a complete subgraph Break the while loop End of if End of while % The final G is a complete subgraph of the original G, and it is likely to be the largest one 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Output genes in the final G are not transposable, and genes not in the final G are transposable Algorithm 5: A heuristic method for detecting transposable genes in Scenario 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 24 random graphs, the 95% credible interval for the success rate of Algorithm 5 is [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='9997, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can claim that Algorithm 5 is a good heuristic algorithm that fails with a very small probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since finding the true maximum clique requires exponentially slow brute-force search, we do not test on very large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Nevertheless, Algorithm 5 does not always produce the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 6 for a counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here genes 1, 2, 3, 4, 5, 6 have degree 4, while genes 7, 8, 9, 10 have degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When applying Algorithm 5, genes 7, 8, 9, 10 are first abandoned, and the final result just has three genes, such as 1, 3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' However, the largest complete graph is 7, 8, 9, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Besides, Algorithm 5 can only determine one (possibly longest) common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus we cannot determine the existence of quasi-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 3 ❃❃❃❃❃❃❃❃ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 1 �������� ❃❃❃❃❃❃❃❃ 4 ♣♣♣♣♣♣♣♣♣♣♣♣♣♣ �������� 7 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ 8 ⑦⑦⑦⑦⑦⑦⑦ 5 2 6 10 9 Figure 6: The auxiliary graph G of linear sequences (7, 8, 9, 10, 1, 1, 2, 3, 3, 4, 5, 5, 6) and (1, 2, 1, 3, 4, 3, 5, 6, 5, 7, 8, 9, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This counterexample fails Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Assume we have m sequences with n genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In general, the copy number of a gene is small, and we can assume the length of each sequence is O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The time complexities of Step 2 and Step 3 in Algorithm 5 are O(mn2) and O(n2), and the overall time complexity is O(mn2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The space complexity is trivially O(mn + n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 6 Circular sequences with duplicated genes In Scenario 4, consider m circular gene sequences, where each sequence contains different numbers of copies of n genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We need to find the longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here we only consider common subsequences that consist of all or none copies of the same gene, and the subsequence length is calculated by genes, not gene copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We shall prove that finding the longest common subsequence in Scenario 4 is no easier than in Scenario 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus Scenario 4 is also NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Finding the longest common subsequence in Scenario 4 is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' From Proposition 1, Scenario 3 is NP-hard, meaning that any NP problem can be reduced to Scenario 3 in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We just need to prove that Scenario 3 can be reduced to Scenario 4 in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Given m linear sequences with n genes in Scenario 3, add genes n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , 2n + 1 to the end of each sequence, and glue each linear sequence into a circular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The longest common subsequence for these circular sequence has the following properties: (1) it contains all genes n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , 2n+ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' (2) after cutting at n + 1 and removing genes n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , 2n + 1, the remaining linear sequence is the longest common subsequence in Scenario 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' (1) The longest common subsequence has at least n + 1 genes (n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , 2n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, at least one gene in n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , 2n + 1 is included, such as n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since gene n+1 aligned in all sequences, n+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , 2n+1 are also aligned, meaning that they are also in the longest common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' (2) After cutting and removing n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , 2n + 1, the remaining linear sequence is a common subsequence in Scenario 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If there is a longer common subsequence, then that with n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , 2n + 1 should be a longer common subsequence in Scenario 4, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, if we can find the longest common subsequence for these cir- cular sequences, then we can find the longest common subsequence for linear sequences in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Similar to Scenario 3, to find the longest common subsequence in Sce- nario 4, we want to reduce it to a maximum clique problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' However, Lemma 3 does not hold in Scenario 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For example, we can consider a cir- cular sequence (1, 2, 3) and its mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' These two sequences are different, but any two genes form a common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' However, inspired by Lemma 3, we have the following conjecture, although we do not know if it is correct or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In Scenario 4, if any three genes gi, gj, gl in g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk form a common subsequence, then g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk form a common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' To solve Scenario 4,construct a 3-uniform hypergraph G as following [15]: vertices are genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' there is a 3-hyperedge (undirected) that links genes gi, gj, gk if and only if they form a common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If Conjecture 2 holds, then finding the longest common sequence in Scenario 4 can be reduced to the maximum clique problem for 3-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk form a common subsequence, then any three genes gi, gj, gl has a 3-hyperedge, and g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk form a complete subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk 26 form a complete subgraph, then any three genes gi, gj, gl form a common subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' By Conjecture 2 , this means g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , gk form a common sub- sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, there is a bijection between common subsequence and complete subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' If we can find the maximum clique problem for 3- uniform hypergraphs, then it corresponds to the longest common subse- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We have reduced Scenario 4 into the maximum clique problem for 3- uniform hypergraphs, which is also NP-hard [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' There have been some algorithms for the maximum clique problem for 3-uniform hypergraphs [61, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For completeness, we propose a simple idea: Repeatedly delete the gene that has the smallest degree, until we have a complete subgraph that any three genes have a 3-hyperedge that links them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We summarize this greedy heuristic method as Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This algorithm is easy to understand, and can provide some intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We do not claim that Algorithm 6 is comparable to other sophisticated algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We test Algorithm 6 on random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Construct a random graph with n genes, and any two genes have probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='5 to have an edge in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Use brute-force search to find the maximum clique, and compare its size with the result of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' For each n ≤ 15, we repeat this for 10000 times, and every time Algorithm 6 returns the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Therefore, for small random graphs, the 95% credible interval for the success rate of Algorithm 6 is [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='9997, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can claim that Algorithm 6 is a good heuristic algorithm that fails with a very small probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since finding the true maximum clique requires exponentially slow brute-force search, we do not test on very large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Nevertheless, Algorithm 6 does not always produce the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 7 for a counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Here each gene in 1, 2, 3, 4, 5, 6 has degree 4, while each gene in 7, 8, 9, 10 has degree 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='When applying Algorithm 6, genes 7, 8, 9, 10 are first deleted, and the final result just has three genes, such as (1, 3, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' However, the longest common subsequence (7, 8, 9, 10) has four genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Assume we have m sequences with n genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In general, the copy number of a gene is small, and we can assume the length of each sequence is O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The time complexities of Step 2 and Step 3 in Algorithm 6 are O(mn3) and O(n3), and the overall time complexity is O(mn3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The space complexity is trivially O(mn + n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Input m circular sequences of genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n, where each gene can have multiple copies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Construct the auxiliary graph G: Vertices of G are all the genes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' , n (not their copies) For each gene triple gi, gj, gk If all copies of gi, gj, gk keep their relative locations in all m sequences Add a 3-hyperedge that links gi, gj, gk in G End of if End of for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' While there exist three genes that do not share a 3-hyperedge Calculate the degree for each gene in G Delete the gene with the smallest degree and 3-hyperedges that links this gene % If there are multiple genes with the smallest degree, delete one randomly End of while % After this while loop, any three genes form a common subsequence % If Conjecture 2 holds, the remaining genes form a common subsequence 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Output remaining genes are not transposable, and other genes are transposable Algorithm 6: A heuristic method for detecting transposable genes in Scenario 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 28 1 2 7 3 4 2 1 10 4 3 10 6 5 9 8 9 5 6 8 7 1 2 9 3 4 2 1 8 4 3 8 6 5 7 10 7 5 6 10 9 Figure 7: Four circular sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The longest common subsequence is (7, 8, 9, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This counterexample fails Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 7 Discussion A gene gi might be missing in some sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Since gi is not in any longest common subsequence, it should be a proper-transposable gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This gene can be directly removed before applying corresponding algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can adopt a stricter definition of transposable genes to exclude a gene which only changes its relative position in a few (no more than l, where l is small enough) sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then we should consider the longest sequence which is a common subsequence of at least m − l sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We can run the corresponding algorithm for every m − l sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus the total time complexity will be multiplied by a factor of ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' In Scenario 1 and Scenario 2 (linear/circular sequences without dupli- cated genes), if each sequence has n genes, and the longest common sub- sequence has length n − k, then there are at most k proper-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' About quasi-transposable genes, inspired by Lemma 1, we have the following guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Consider m linear/circular sequences with n genes without multiple copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Assume the length of the longest common subsequence is n − k, and there are l proper-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Then the number of quasi- transposable genes is no larger than 2(k − l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' When l + 2(k − l) ≤ n, in both linear and circular scenarios, we can find examples with 2(k − l) quasi-transposable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 29 8 Conclusion In this paper, we study the problem of determining transposable genes in gene sequences, and design Algorithms 1–6 for different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' To apply those algorithms, one needs to apply genomic annotation tools to transform raw DNA sequencing data into gene sequences, and replace gene names by numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Those algorithms have at most O(mn3) time complexity, where m is the number of sequences, and n is the number of genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Thus they can run in a reasonable time for most applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We prove that the latter two scenarios are NP-hard (Propositions 1,2), and propose two unresolved problems (Conjectures 2,3) in discrete mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We start with gene sequences and determine translocated genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' There- fore, short transposons (possibly shorter than a gene) cannot be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Besides, we do not determine specific genomic rearrangement events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' We aim at determining which genes are able to translocate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Specifically, we study how many longest common subsequences contain a certain gene, as a measure for its “stability”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' This mesoscopic viewpoint can be intriguing for understanding changes in genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The results in this paper are not limited to Scenarios 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' They can be applied to other bioinformatics situations, or even other fields that need discrete mathematics tools, such as text processing, compiler optimization, data analysis, image analysis [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Besides, algorithms in this paper might be able to detect non-syntenic regions [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' There are some possible future directions: (1) prove Conjectures 2,3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' (2) extend Proposition 3 to find more efficient solutions to Scenario 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' (3) determine whether genes appear in all longest common subsequences in other similar scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Acknowledgments The author would like to thank Zhongkai Zhao for helping with designing Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' The author would like to thank Lucas B¨ottcher and anonymous reviewers for providing helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' References [1] Adi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', Braga, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', Fernandes, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', Puiu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', Luo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', Koren, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', Marc¸ais, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', Yorke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', Dvoˇr´ak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=', and Salzberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Hybrid assembly of the large and highly repetitive genome of aegilops tauschii, a progenitor of bread wheat, with the masurca mega-reads algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' Genome Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 27, 5 (2017), 787–792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} +page_content=' 38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE2T4oBgHgl3EQfVwfD/content/2301.03827v1.pdf'} diff --git a/S9E0T4oBgHgl3EQflAFl/content/tmp_files/2301.02480v1.pdf.txt b/S9E0T4oBgHgl3EQflAFl/content/tmp_files/2301.02480v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b46821f92ccf2ed8698b44bcbf50bb23c7fd078 --- /dev/null +++ b/S9E0T4oBgHgl3EQflAFl/content/tmp_files/2301.02480v1.pdf.txt @@ -0,0 +1,4276 @@ +Effective Medium Model for Graphene Superlattices with +Electrostatic and Magnetic Vector Potentials +David E. Fernandes +Instituto de Telecomunicações and Department of Electrical Engineering, University of Coimbra, 3030- +290 Coimbra, Portugal +E-mail: dfernandes@co.it.pt +Abstract +In this article we develop an effective medium model to characterize the electron wave +propagation in graphene based nanostructures with an electrostatic and magnetic vector +potentials imposed on their surface. We use a numerical algorithm to determine the +effective medium parameters of the heterostructure and calculate the electronic band +structure of the system. We apply our formalism to analyze superlattices with solely a +magnetic potential and reveal that the response of the structure remains reciprocal and is +characterized by a decrease in charge carrier’s velocity. We also study the response of +superlattices with both potentials superimposed on graphene and show that the response +of the system becomes nonreciprocal with a dispersion characterized by a tilted Dirac +cone. We demonstrate that it is possible to alternate between a type-I, type-II or type-III +Dirac cones by properly tuning the amplitude of the potentials. + + + + + + + +I. + Introduction +Graphene is a two-dimensional nanomaterial formed by carbon atoms that are +arranged in a honeycomb lattice [1-8]. Over the last decade this material has been on the +spotlight of condensed matter physics research due to its remarkable electronic +properties. By possessing a relativistic-spectrum, the low-energy electrons in graphene +have a linear dispersion and their propagation is determined by a massless Dirac +equation [3]. +It has been proposed that it is possible to achieve additional control over the +propagation characteristics of the electrons in graphene by modifying the original +material. These structures are known as graphene superlattices (GSLs) and may be +obtained by artificially introducing a new length scale into the system in the form of a +periodic potential, either by using an electrostatic potential [9-18] or magnetic vector +potential [19-28]. Superlattices characterized by electrostatic potentials may be realized +using different techniques, such as with periodically patterned gates, using a crystalline +substrate or with the deposition of adatoms on graphene’s surface [29-36]. On the other +hand, to obtain GSLs with a magnetic vector potential one can use nano-magnetic strips +[19-22] or strain-inducing modulations [37]. An electrostatic potential on the surface of +graphene can allow for an extreme anisotropic response which can lead to the super- +collimation of electron waves [17, 29-31]. Moreover, it can permit extreme wave +phenomena such as a perfect lens for matter waves [38, 39] or an electron wormhole +[40]. Conversely, a magnetic vector potential can also allow to tailor the electron wave +propagation by reducing the charge carriers velocity [22-28] or even providing a way to +tilt the energy dispersion of electrons in the medium [41], usually identified as a type-I +tilted Dirac cone [42]. Such type of response may be used for valley filtering in p-n +junctions [43] and to generate photocurrent [44]. + +The characterization of the propagation of electrons in superlattices with a magnetic +vector potential is typically done using a transfer matrix formalism [22-28] which can +limit the study to potentials characterized only by constant barriers. Interestingly, in the +works proposed in [17, 18, 38-40] the propagation of the electrons in the GSLs with an +electrostatic potential was studied under an effective medium formalism, so that +granular details of the potential are homogenized [38] and the structure is regarded as a +continuum characterized by some effective parameters. Such effective medium +techniques can vastly simply the analysis of the problem while simultaneously +providing invaluable insight into the physical phenomena taking place in the structure. +The main objective of this work is to develop an effective medium model for +superlattices characterized by both an electrostatic potential and a magnetic vector +potential. To determine the effective response of the superlattices we use a numerical +finite-difference time-domain (FDTD) algorithm based on the numerical tool proposed +in Ref. [17]. It is important to mention that FDTD numerical tools such as the ones +developed in [17, 18, 45, 46] have been widely used to determine the electron wave +propagation in graphene based nanomaterials. To begin with, we apply the numerical +algorithm to homogenize a GSL with a magnetic vector potential with a sinusoidal +spatial variation and show that, similar to what happens in GSLs with Krönig-Penney +type potentials [22-28], the response of the structure is isotropic, with the group velocity +of the charge carriers being smaller than in pristine graphene. We demonstrate that for +these superlattices the analysis of the problem can be vastly simplified by using an +effective Hamiltonian that discards the granular details of the potential and instead +considers an effective parameter that is independent of space. This effective parameter +may be regarded as an effective Fermi velocity whose value is only dependent on the +amplitude of the magnetic vector potential. We also determine the effective response of + +superlattices with both electrostatic and magnetic vector potentials with sinusoidal +spatial variations. Using our effective medium formalism, we demonstrate that the +interplay between the magnetic and electric potentials can give rise to an overall +nonreciprocal response whose energy dispersion is characterized by a Dirac cone tilted +along the direction perpendicular to the stratification of the potentials (type-I Dirac +cone). Moreover, we show that for propagation along such direction there is a wide +range of combinations of amplitude of the potentials for which the bulk eigenmodes can +flow along the same direction and, by properly tuning the amplitude of the potentials, it +is even possible to have eigenmodes with a null group velocity. Such dispersion +characteristic corresponds to a type-III Dirac cone [42, 47, 48], where one of the bands +is flat and the other has a linear dispersion. It has been proposed such dispersion can +enhance the superconducting gap in Weyl semimetals [49], and by using the flat band, +they can allow for a new platform to study the correlated phases in the structure [50]. +Importantly, a type-III Dirac cone marks the transition between type-I and type-II Dirac +cones. The type-II dispersion characteristic differs from the type-I from the fact that the +Fermi surface is no longer a point, but rather two-crossing lines [51-53]. Such +dispersion appears when one of the bands is tilted in such a way that the group velocity +of the associated energy eigenmode has the opposite sign that the corresponding value +in pristine graphene. +The article is organized as follows. In Sec. II we introduce the homogenization +formalism that will be used to characterize the effective medium response of the +graphene superlattices. In Sec. III we describe the numerical FDTD algorithm that is +used to determine the effective parameters of the superlattice. In Sec. IV the +homogenization formalism is applied to characterize the wave propagation in GSLs + +with solely a magnetic vector potential and in superlattices with both electrostatic and +magnetic potentials. Finally, the conclusions are drawn in Sec. V. +II. Effective Medium Model +In this work we study the electron wave propagation in graphene-based nanomaterials +characterized by an electrostatic potential and a magnetic vector potential. Near the K +point, the propagation of the charge carriers in graphene superlattices with electrostatic +and magnetic vector potentials may be described using the massless Dirac equation: +ˆ +i +H +t + += + ψ +ψ , + + + + +(1) +where +( ) +( +) +( ) +ˆ +F +H +v +i +q +V += + −  − ++ +σ +A r +r is the microscopic Hamiltonian operator, V is +the microscopic electric potential, q +e += − is the electron charge, A is the magnetic +vector potential, +6 +10 +/ +F +v +m s + + is the Fermi velocity in pristine graphene and +ˆ +ˆ +x +y += ++ +σ +σ x +σ y , with +, +x +y +σ σ the Pauli matrices. Moreover, + + +1 +2 +, +=   +ψ + is a two- +component pseudospinor, with each component of the pseudospinor associated with a +different trigonal sublattice of graphene. +Without any loss of generality, we consider that the microscopic potentials have a one- +dimensional (1D) spatial variation. Particularly we suppose that +( ) +( ) +V +V x += +r + and +( ) ˆ +y +A +x += +A +y . The proposed effective model could be readily extended to other (more +complex) types of spatial variations. +For the considered spatial variations of the electric and magnetic potentials Eq. (1) may +be re-written as: +( ) +( +) +( ) +ˆ +F +y +i +v +i +eA +x +V x +t + += + −  + + ++ + ψ +σ +y +ψ +ψ . + +(2) + +In Ref. [38] it was shown that provided the initial state of the system +( +) +0 +t = +ψ + is +macroscopic, so that +( +) +( +) +0 +0 +t +t += += += +ψ +ψ +, with + the spatial averaging operator +defined as +1 +i +F +Fe +dS +A +− + +=  +k r + (where we assume a spatial evolution of the type +ie + +k r ), +then the envelope of the pseudospinor can be accurately calculated from an effective +Hamiltonian of the system ˆ +ef +H , defined as ˆ +ˆ +ef +H +H += +ψ +ψ , which may be written as: +( +) +( +) +ˆ +, +, +ef +F +F +y +ef +ef +H +v +ev +A +V + + += + ++ + ++ +σ k +σ +k +k . + + +(3) +Here we used i = k , with k the pseudo-momentum, and introduced the effective +magnetic and electric potentials, +ef +A and +ef +V , which can generally be matrices. +Applying the spatial averaging operator to Eq. (2), with +i +t + + = − + + for a time evolution +of the type +i t +e + +− +, we obtain +ˆ +F +F +y +y +E +H +v +ev +A +V + += += + ++ + ++ +ψ +ψ +σ k ψ +ψ +ψ , + + +(4) +where +n +ik +n + = + +, with +, +n +x y += + and E +i +t + += + . +To determine the effective Hamiltonian for a fixed energy and pseudo-momentum +( +) +0 +0 +0 +, +x +y +k +k += +k + we can calculate the time evolution in the graphene nanomaterial of two +linear independent initial electronic states of the form: +( ) ( +) +0 +1 +1 +, +0 +0 +i +t +e +   += += +  +  +k +r +ψ +r +, + + + + +(5a) +( ) ( +) +0 +2 +0 +, +0 +1 +i +t +e +   += += +  +  +k +r +ψ +r +, + + + + +(5b) + +and use the Fourier transform and the spatial averaging operator to calculate +( ) +n +ψ +, +( ) +n +y +A ψ + and +( ) +n +Vψ + in the frequency and spatial domains for each initial state +( ) +n +ψ +, +with +1,2 +n = +. The effective Hamiltonian +( +) +ˆ +, +ef +H + k is then determined by calculating: +( +) +( ) +( ) +( ) +( ) +1 +1 +2 +1 +2 +ˆ +ˆ +ˆ +, +; +; +ef +H +H +H + +− + +  + += + + +  + +k +ψ +ψ +ψ +ψ + +(6) +Similarly, the effective electric and magnetic potentials can also be calculated using: +( ) +( ) +( ) +( ) +( ) +1 +1 +2 +1 +2 +; +; +ef +V +V +V + +− + +  + += + + +  + +ψ +ψ +ψ +ψ + + +(7) +( ) +( ) +( ) +( ) +( ) +1 +1 +2 +1 +2 +; +; +ef +A +A +A + +− + +  + += + + +  + +ψ +ψ +ψ +ψ + + +(8) +It is important to mention that the initial states are chosen in such a way that they are +not more localized than the characteristic period of the lattices, such that the pseudo- +momentum k0 associated with these states is within the first Brillouin minizone of the +superlattice. +III. Numerical Algorithm +The calculation of the effective Hamiltonian in the frequency domain requires the +calculation of the time evolution of the electronic states in Eq. 5a-b and its Fourier +transform. To determine the time-evolution of the initial electronic states we use a +FDTD (Finite Differences in the Time Domain) numerical algorithm. The algorithm is +based on the numerical tool developed in [17] which was successfully used to study the +transport properties of graphene superlattices characterized solely by an electrostatic +potential. We begin by separating the Dirac equation (2) for each component of the +pseudospinor: +1 +2 +1 +y +F +eA +V +v +i +t +x +y +i + + + + + += − +− ++ + + + + + + + + + + +, + + +(9a) + +2 +1 +2 +y +F +eA +V +v +i +t +x +y +i + + + + + += − ++ +− + + + + + + + + + + +. + + +(9b) +To obtain the time update equations in an explicit form we discretize the spatial domain +into a rectangular grid, such that the consecutive nodes along the x- and y-directions are +separated by a distance +x + and +y + , as depicted in Fig. 1a. Furthermore, each +component of the pseudospinor is sampled at instants of time separated by the time step +t + . +This +allows +us +to +write +the +pseudospinor +components +as +( +) +( +) +( +) +, , +, +, +, , +x +y +t +x y t +p +q +n +p q n + +=  + + + +  + and permits using a finite differences +method to calculate the partial derivatives in Eq(9) such that: +( ) +1 +1 +2 +2 +l +l +i +i +i + + + + + ++ +−  +− + + + + + + + + +  += + +, + + + + +(10) +with +, , +l +x y t += +. + +Fig. 1 (color online) a) The graphene superlattice spatial domain is discretized into a rectangular grid with +a finite number of nodes spaced by +x + along the x-direction and +y + along the y-direction. The +pseudospinor components +1 + and +2 + are defined on staggered subgrids so that +1 + is defined over the +nodes ( +) +,p q and +2 + is defined at the nodes ( +) +1/ 2, +1/ 2 +p +q ++ ++ +, shifted by a half-grid period. b) The +time domain is sampled at time intervals separated by a time step +t + . Similarly to the spatial domain + +a) +b) +A +n+1 +Un +I +3 +n+ +n+ +Ay +2 +2 +2 +2 +Ax +i+1discretization scheme, the pseudospinor components +1 + and +2 + are defined on staggered subgrids +shifted by +2 +t + +. + +We also consider that the two components of the pseudospinor are defined on staggered +grids in space and time so that +( +) +1 +, , +p q n + + and +2 +1 +1 +1 +, +, +2 +2 +2 +p +q +n + + + ++ ++ ++ + + + + +, as shown in +Fig. 1a-b. Applying these principles to Eqs. 9a-b leads to the following update +equations: +, +1 +1 +, +, +1 +2 +2 +1, , +1, , +2, , +1 +1 +2, +, +2 +2 +1 +2 +1 +1 +2, +, +2 +2 +1 +1 +1 +1 +2 +2 +2 +2 +1 +1 +1 +1 +2 +2 +2 +2 +p q +n +n +y +p q +p q +n +n +p q +t +p q +t +F +t +p q +p +q +x +y +n +p +q +x +y +x +y +eA +V +V +v +i +i +i +i +i ++ ++ ++ ++ ++ ++ +− ++ + + + + + + + + +− + +=  ++ + +− + ++ + ++ +− + + + + + + + + + + + + + + + + + + + + + + + + + − ++ + ++ ++ + + + + + + + + + + + + + + +1 +1 +2 +2 +1 +1 +1 +1 +2, +, +2, +, +2 +2 +2 +2 +1 +1 +2 +2 +n +n +p +q +p +q +x +y +i ++ ++ ++ +− +− +− + + + + +− +− + + + + + + + + + + + + + + + +(11a) +, +1 +1 +1 +1 +1 +, +, +1/2 +2 +2 +2 +2 +2 +1 +1 +1 +1 +1 +1 +1, +1, +1 +2, +, +2, +, +1, +, +2 +2 +2 +2 +2 +2 +1 +1 +1 +1 + +2 +2 +2 +2 +1 +1 +2 +2 +p q +p +q +p +q +n +y +n +n +n +t +t +F +t +p +q +p +q +p +q +p +q +x +y +x +y +V +V +eA +v +i +i +i +i ++ ++ ++ ++ ++ +− ++ ++ ++ ++ ++ ++ ++ ++ + + + + + + + + + + + + +− + +=  ++ + +− + +− + ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + − +− + + +1, , +1 +1, +1, +1, , +1 +1 +1 +1 + +2 +2 +2 +2 +n +n +n +p q +p +q +p q +x +y +x +y +i +i ++ ++ + + + + + + + + + +− + +− ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + (11b) +With +( +) +, +, +p q +x +y +V +V p +q += + + + and +( +) +, +, +y p q +y +x +y +A +A +p +q += + + +. Importantly, this discretization +scheme of the update equations requires the value of the pseudospinor components in +subgrid points where they are not defined. In particular, the value value +( +) +2 +,p q + + is +necessary in Eq. (11a), while the update equation 11b requires the value of +1 +1 +1 +, +2 +2 +p +q + + + ++ ++ + + + + +. To obtain such values we assume that the wavefunction varies +slowly in space so that the pseudospinor component values in points of space that lie +outside the grid nodes can be determined by the average of its neighboring nodes. In +that case one can use: + +( +) +2 +2 +2 +2 +2 +1 +1 +1 +1 +1 +1 +1 +1 +1 +, +, +, +, +, +4 +2 +2 +2 +2 +2 +2 +2 +2 +p q +p +q +p +q +p +q +p +q + + + + + + + + + + + + + ++ +− ++  +− ++ ++  ++ ++ ++  +− +− + + + + + + + + + + + + + + + + + + + + +, (12a) +and +( +) +( +) +( +) +( +) +( +) +1 +1 +1 +1 +1 +1 +1 +1 +, +1, +, +1 +1, +1 +, +2 +2 +4 +p +q +p +q +p q +p +q +p q + + + ++ ++ + + ++ ++  ++ ++  ++ ++ ++  + + + + +. (11b) +To determine the effective Hamiltonian it necessary to calculate the Fourier transform +of +( )t +ψ +, +( ) +y +A +t +ψ + and +( ) +V +t +ψ +. As shown in other works dealing with time-domain +homogenization techniques of artificial structured media [54], to ensure the +convergence of the Fourier transform for a given frequency  , at each time iteration +0,..., +n +N += + the quantities +( +) +n t + +ψ +, +( +) +y +A +n t + +ψ + and +( +) +V +n t + +ψ + must be multiplied +by a time decaying exponential of the form +n t +e  , with +i + + + + + += ++ +, so that the term + represents some small losses in the system. Importantly, the total number of +iterations N must be sufficiently high so that +0 +N t +e +  +. Typically, the total number of +iterations is on the order of +2 +t +N +  + + [54]. +At this point it is important to mention that in all calculations performed in this work we +consider that both potentials have the same spatial period a and that the distance +between adjacent nodes is the same, i.e. +x +y + =  =  . Moreover, we determine the time +evolution of the initial states in a region of space containing one period of the potentials +and apply Bloch boundary condition at the edge of the computational domain. In +Appendix A we analyze the stability conditions of the proposed FDTD algorithm and +show that the maximum value of the time-step depends both on the distance between +adjacent spatial nodes and the maximum amplitude of the magnetic potential. + +IV. Numerical results +In what follows, we use our homogenization algorithm to determine the band diagram +and effective medium parameters of graphene superlattices with electrostatic and +magnetic vector potentials. +It was shown in [38] that provided the microscopic Hamiltonian of the graphene +nanomaterial does not vary in time, the electronic band structure of the material close to +the K point can be computed directly from the effective Hamiltonian. This property is a +consequence of the energy eigenstates of the system calculated using ˆ +ef +H being equal to +the eigenstates of the microscopic Hamiltonian [38]. +In this work we consider an electrostatic potential with a sinusoidal-type spatial +variation of the form +( ) +2 +sin +osc +x +V x +V +a + + + += + + + + + and a periodic magnetic induction field +given by +( ) +1 +1 +2 +2 +ˆ +ˆ +cos +z +osc +x +B +x +A +a +a + + + + += += + + + + +B +z +z , so that it varies along the x-direction but is +oriented along the z-direction (perpendicular to the propagation plane). The magnetic +induction field it is on average null, i.e. +( ) +1 +1 0 +1 +0 +a +z +B +x dx +a += + +. Since the magnetic field is +related to the magnetic vector potential as +=  +B +A , it follows that +( ) ˆ +y +A +x += +A +y , with +( ) +2 +sin +y +osc +x +A +x +A +a + + + += + + + + +, which is also periodic with zero spatial average. +Similar to [38], to calculate the band diagram of the structures we expand the +effective potentials +ef +A , +ef +V as a Taylor series of the first order, such that: +( +) +( +) +( +) +( +) +,0 +,0 +, +,0 +ef +ef +ef +ef +x +y +x +y +A +A +A +A +k +k +k +k + + + + + + += ++ ++ + + +k + and + +(13) +( +) +( +) +( +) +( +) +,0 +,0 +, +,0 +ef +ef +ef +ef +x +y +x +y +V +V +V +V +k +k +k +k + + + + + + += ++ ++ + + +k +, + +(14) + +and use our numerical algorithm to calculate the effective Hamiltonian (given by Eq. 4). +Note that in general +( +) +, +ef +A + k and +( +) +, +ef +V + k are not scalars. +From hereon we consider that the spatial grid is discretized using a node spacing +50 +a + = + and the time step is +0.3 +t +F +v + = + +. Since the effective response of graphene +superlattices characterized solely by electrostatic potentials was already thoroughly +discussed in [17, 18, 38], we restrict our attention to superlattices with only magnetic +vector potential and structures with both magnetic and electrostatic potentials. +A) Superlattices with Magnetic Potential +We begin by calculating the response of superlattices characterized solely by a magnetic +vector potential. Particularly, we determine the effective potential +( +) +, +ef +A + k of the +nanomaterial for some amplitudes of the magnetic potential. For this GSL it is +immediate that the effective electric potential is null, i.e. +( +) +, +0 +ef +V + += +k +. +Our numerical results showed that to an excellent approximation +( +) +,0 +ef +y +A + + + +σ , with + a real value depicted in Fig. 2a. Interestingly, it is seen that for low-energy +excitations this value varies linearly with the energy on the carriers. Moreover, we also +verified that +( +) +,0 +ef +z +x +A +i +k + + + + + +σ , with +z +σ the Pauli matrix, and that +( +) +,0 +ef +y +A +k + + + + + +1. +Here  + + + are real constants almost independent of the carriers energy for low-energy +excitations, as shown in Fig. 2b-c, respectively, for some representative amplitudes of +the potential +osc +A +. + + +Fig. 2 Normalized effective parameters of the graphene superlattice as a function of the normalized +energy: a) ea +, b) e + and c) e +, for a magnetic vector potential with amplitude +1.0 +osc +eA a += + +(black line), +3.0 +osc +eA a += + (dark green line), +4.0 +osc +eA a += + (brown line) and +6.0 +osc +eA a += + (blue line). + +Hence, for low energy excitations, the effective magnetic potential may be +approximated by: +( +) +( ) +, +ef +y +x +z +y +A +i k +k + +  + + + ++ ++ +k +σ +σ +1 + + +(15) +Inserting this expression in Eq. 3, allows us to write the effective Hamiltonian as: +0 +ˆ +ef +F +H +E +v + + += ++ + +1 +σ k , + + + +(16) +with +0 +F +E +ev + + += + and +1 +1 +e +e + + + += − + − +. The energy dispersion of the superlattice +may then be calculated from the eigenvalue problem: +ˆ +ef +E +H + = + , + + + +(17) +and it is given by +2 +2 +0 +F +x +y +E +E +v +k +k + + +− + = ++ +. Since +0 + and  are weakly dependent on +the energy, the previous expression can be further simplified into +, +F eff +E +v + = +k , + + + +(18) +by defining an effective Fermi velocity +, +0 +1 +F eff +F +v +v + + + = +− +. Considering that +0 + is a +negative value (proportional to the slope of the curves in Fig. 2a) and that  is smaller +than unity because +, +0 +   +, it is expected that +, +F eff +v +can be significantly smaller than +the Fermi velocity. To verify the accuracy of our effective medium model we + +a) +b) +c) +1.0 +0.6 +0.6 +0.5 +eaa +eβ +ex +0.4 +0.4 +h +0.0 +h +h +0.2 +0.2 +0.5 +0.0 +0.0 +1.0 +-1 +0 +1 +N +-2 +-1 +0 +2 +-2 +-1 +0 +2 +Ea/hvF +Ea/hvF +Ea/hvroverlapped in Fig. 3a-b the “exact” band diagram for propagation along the x-direction +( +0 +yk = +) and the y-direction ( +0 +xk = +), with the corresponding results calculated using +our simplified effective formalism. The band diagram was calculated for a graphene +superlattice characterized by a magnetic potential with amplitude +4.0 +osc +eA a += +. As +seen, for low energy excitations both results have nearly exact agreement. + +Fig. 3 a) and b) Dispersion of the energy eigenstates of a graphene superlattice with a magnetic vector +potential +4.0 +osc +eA a += + for propagation along the x and y directions, respectively. The blue dashed +curves represent the “exact” energy dispersion of the GSL, the blue solid curves correspond to the +dispersion of the GSL calculated using the effective parameter +, +F eff +v + and the black solid curves show the +dispersion of pristine graphene. c) Normalized effective Fermi velocity +, +F eff +F +v +v as a function of the +normalized magnetic vector potential amplitude +osc +eA a +. +The results shown in Fig. 3a-b also show that even in the presence of a magnetic +potential, which the breaks time-reversal symmetry of the structure, the response of +these graphene superlattices remains isotropic and reciprocal for low energy excitations, +and without a bandgap. Moreover, we see that the group velocity of the carriers in the +superlattice is exactly equal to the effective Fermi velocity +, +, +, +1 +g x +g y +F eff +dE +v +v +v +dk += += += +. +Indeed, by comparing these results with the band diagram of pristine graphene, depicted +as black curves in Fig. 3a-b, it is confirmed that the effective Fermi velocity is smaller +than the Fermi velocity in pristine graphene. In in Fig. 3c we show the effect of the +amplitude of the magnetic potential on the effective Fermi velocity. The results reveal +that the carrier’s velocity can be severely reduced from the Fermi velocity as the + +a) +b) +( +k, =0 +k,=0 +1.0 +2 +0.8 +Ea +1 +Ea +0 +0 +hVF +hVF +VF +0.4 +1 +0.2 +.3 +0.0 +-2 +0 +1 +2 +3 +2 +-1 +0 +2 +0 +5 +10 +15 +20 +25 +k.a +k,a +eAsca/hamplitude of the magnetic vector potential increases. Hence, by precisely tailoring the +magnetic field distribution we can control the charge velocity in the medium. These +results go in line with previous studies [22-28] that demonstrated the effect of the +magnetic potential in the carrier velocity properties. + +B) Superlattices with Magnetic and Electric Potentials +The analysis we did in the previous section revealed that imposing a 1D magnetic +potential with zero-spatial average on the surface of graphene leads to a reciprocal and +isotropic response, wherein the charge carriers group velocity is decreased as the +amplitude of the magnetic potential increases. On the other hand, it was demonstrated in +[17, 38] that in graphene superlattices with 1D electrostatic potential with zero spatial +average, the transport properties of the electrons can also be modified so that they have +a preferred direction of propagation, i.e. the effective medium behaves as an anisotropic +medium. In what follows, we use our homogenization model to study the response of +superlattices characterized by both a 1D magnetic and a 1D electric potential with zero +spatial average. +As in the previous section, we start by calculating the effective potentials of the GSL +given by Eqs. (13)-(14). As a leading example, we consider a superlattice characterized +by a magnetic vector potential with amplitude +4 +osc +eA a += + and an electrostatic potential +with amplitude +5 +osc +F +V a +v = +. Our numerical results show to an excellent +approximation that the Taylor expansion of the effective potentials may be written as: +( +) +( +) +( +) +0 +0 +1 +1 +2 +2 +, +ef +y +x +y +x +y +y +A +k +k + + + + + + + += ++ ++ + ++ ++ ++ +k +1 +σ +σ +1 +σ +1 +σ + + +(19) +( +) +( +) +( +) +0 +0 +1 +1 +2 +2 +, +ef +y +x +y +x +y +y +V +k +k + + + + + + + += ++ ++ + ++ ++ ++ +k +1 +σ +σ +1 +σ +1 +σ +, + +(20) + +With +, +, +, +i +i +i +i +    , with +0,1,2 +i = +, real-valued scalars whose energy dependence is +shown in Fig. 4. + +Fig. 4 a) Normalized effective magnetic potential parameters +1 +0 +ea +− , +1 +0 +ea +− , +1 +1 +e +− , +1 +1 +e +− , +1 +2 +e +− , +1 +2 +e +− as a function of the normalized energy for a graphene superlattice for a characterized by a +magnetic vector potential with amplitude +4 +osc +eA a += + and an electrostatic potential with amplitude +5 +osc +F +V +a +v = + . b) Similar to a) but for the normalized effective electrostatic potential parameters +1 +0 +ea +− , +1 +0 +ea +− , +1 +1 +e +− , +1 +1 +e +− , +1 +2 +e +− , +1 +2 +e +− . +The results in Fig. 4 show that the zeroth order coefficients of the Taylor expansion of +the potentials +0 +0 +0 +0 +, +, +, + + +  vary linearly with the energy of the electrons. On the other +hand, the first order terms +, +, +, +i +i +i +i +    , with +1,2 +i = +, are almost independent of the +energy. Interestingly, our simulation results also show that the effective potentials are +linked to each other through the following relations: +0 +0 +1 +2 +2 +1 +0 +F +F +F +ev +e +e +c +E +E +v +v + + + + + + + + − + + + − += +, + +(21a) +0 +1 +2 +1 +F +ev +e +e +c +E + + + + − + += +, + + + + (21b) +0 +1 +2 +2 +F +F +c +E +v +v + + + + − + += +. + + + +(21c) +Surprisingly, these relations show that it is possible to describe the spatially dispersive +response of the potentials at the expense of the non-spatially dispersive terms. For this + +a) +1.0 +b) 1.0 +axo/hvr +eaα.h-1 +X: /hv, +0.5 +0.5 +eα,h-1 +S /hvr +eβh-1 +0.0 +0.0 +eα,h-l +eaβ,h-l +X / hvp +ad. /hv: +-0.5 +eβ,h-1 +-0.5 +0. / hvp +-1.0 +-1.0 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +Ea/hv +Ea/hvrexample, it is found that +0 +0.277 +c  − +, +1 +0.215 +c  − + and +2 +0.269 +c  − +. Using the effective +potentials (19)-(20) together with Eqs. 21a-c, we can write the effective Hamiltonian +(Eq. 3) as: +( +) +( +) +( +) +( +) +0 +1 +2 +1 +2 +1 +2 +0 +ˆ +2 +2 +ef +F +y +F +x +x +y +F +y +H +v +c E +c +c +E +c +c +v k +k +v +c +c +c += + ++ ++ ++ ++ +− ++ ++ ++ +σ k +σ +1 +σ +σ +1 .(22) +The energy dispersion of the modes supported in the superlattice is obtained from the +eigenvalue problem in Eq. (17) using the effective Hamiltonian given by Eq. (22). The +corresponding band diagram is shown in Fig. 5a and reveals that the response of the +superlattice is no longer reciprocal, with the band diagram being tilted along the +yk . +To have a better understanding of the structure’s nonreciprocal response effect in the +propagation of the electrons we focus our attention on propagation along the x- and y- +directions. Let us begin by studying the propagation along x ( +0, +0 +y +x +k +k += + + +). In that +case, the energy dispersion of the modes is simply given by: +( +) +( +) +1 +2 +2 +2 +0 +1 +2 +1 +4 +1 +F +x +v +c +c +k +E +c +c +c ++ +− + = +− +− +− + + + + +(23) +The corresponding band diagram is show in Fig. 5b, where we also depict the “exact” +band diagram. + +Fig. 5 a) Exact energy dispersion of the considered graphene superlattice b) and c) Dispersion of the +energy eigenstates for propagation along the x and y directions, respectively. The blue dashed curves +represent the “exact” energy dispersion of the GSL, the blue solid curves correspond to the dispersion of +the GSL calculated using the simplified effective medium formalism and the green solid curves show the +dispersion of pristine graphene. In all panels the GSL is characterized by a magnetic vector potential with +amplitude +4.0 +osc +eA a += + and an electrostatic potential with amplitude +5.0 +osc +F +V a +v = +. + + +(a) +k,a +-2 +(b) +(c) +2 +0 +k,=0 +2E +k.=0 +4 +2 +2 +1 +Ea +Ea +Ea +0 +hy. +hVe +hVr +2 +-2 +-2 +0 +2 +3 +-3 +-2 +0 +2 +k.a 0 +2 +k,a +k,aOur effective medium model results have a very good agreement with the exact band +diagram showing that for propagation along the x-direction the results are comparable to +the band diagram obtained for superlattices with solely a magnetic potential (see Fig. +3a). Indeed, for propagation along the x-direction, the response of the structure is +reciprocal and characterized by a group velocity smaller to that of pristine graphene +(whose +response +is +depicted +as +green +curves +in +Fig. +5a), +so +that +1 +, +0.77 +g x +x +F +v +E +k +v +− += + + + +. +To determine the band diagram for propagation perpendicular to the direction of the +stratification of the potentials, i.e. for propagation along the y-direction (with +0 +xk = +, +0 +yk  +), we follow a similar procedure and calculate the eigenvalue problem in Eq. (17) +considering the effective Hamiltonian given by Eq. (22) with +0 +xk = +. The problem +yields two solutions (eigenmodes), whose energy dispersion is given by: +( )1 +0 +1 +2 +0 +1 +2 +2 +1 +2 +1 +F +y +c +c +c +E +v k +c +c +c +− +− +− + = +− +− ++ + + +(24a) +( ) +2 +0 +1 +2 +0 +1 +2 +2 +1 +2 +1 +F +y +c +c +c +E +v k +c +c +c ++ ++ ++ + = +− +− +− ++ + + +(24b) +Clearly, for a fixed pseudo-momentum +yk both solutions are not symmetric. In Fig. 5c +we represent the band diagram calculated using the effective medium formalism and +overlap the results with both the exact band diagram of the superlattice and the band +diagram of pristine graphene. As seen, both results predict that the superlattice response +is vastly different from that of pristine graphene. The response of the structure is +nonreciprocal as for a fixed energy both bulk modes are characterized by wave vectors +yk that have the same sign. Additionally, the band diagram reveals that it is possible to +have unidirectional bulk modes as both bands have negative (but distinct) group +velocity, i.e. +( ) +( ) +( ) +( ) +1 +1 +2 +2 +1 +1 +, +, +g y +y +g y +y +v +E +k +v +E +k +− +− += + + + += + + +, consistent with a type-II Dirac + +cone dispersion characteristic. While one of the bands ( +( ) ( +) +1 +y +E +k +) follows closely the +original band of the pristine graphene, the other one ( +( ) ( +) +2 +y +E +k +) is tilted towards the +origin +0 +E = +. Indeed, our results suggest that it may be possible obtain an eigenmode +characterized by a flat band so that the group velocity is precisely equal to zero +, +0 +g y +v += +, +corresponding to a static wave. Importantly, this type of dispersion characteristic is +usually identified as a type-III Dirac cone [42] where the energy dispersion consists of +one flat band while the other has a liner dispersion [42, 47, 48]. Clearly, this is an effect +of the interplay between the electrostatic and magnetic vector potentials in the dynamics +of the charge carriers in the superlattice. In Fig. 6a we show the group velocities +( ) +( ) +1 +2 +, +, +, +, +, +g y +g y +g x +v +v +v + + + as a function of the amplitude of the magnetic potential for the fixed +amplitude of the electric potential +5 +osc +F +V a +v = +. It is seen that increasing the amplitude +of the magnetic potential decreases the group velocity along the x-direction, similar to +the previous studied superlattice with +0 +osc +V += +. On the other hand, the effects of +changing +osc +A + in the group velocities +( ) +( ) +1 +2 +, +, +, +g y +g y +v +v + are far more pronounced. To begin with, +we note that when +0 +osc +A += + the response is that of a superlattice with electric potential, +which is characterized by an anisotropic response [17, 38], so that +, +g y +v + can significantly +smaller than the Fermi velocity. Moreover, we see the group velocity +( )1 +, +g y +v + decreases as +the amplitude of the magnetic potential increases, reaching a minimum for +4.71 +osc +eA a + +, at which point it is equal to the Fermi velocity, and then it starts +increasing. In contrast, for the other eigenmode, its group velocity +( ) +2 +, +g y +v + decreases as +osc +A + increases, reaching a null value for a magnetic potential with amplitude +3.36 +osc +eA a + +. Such combination of amplitudes leads to dispersion characterized by a + +type-III Dirac cone, which crucially, marks the transition point where the dispersion +changes from a type-I Dirac cone (for +3.36 +osc +eA a + +) into a type-II (when +3.36 +osc +eA a + +), where both eigenmodes flow along the same direction. +To have a complete understanding of the interplay between both potentials in the +carriers velocity in the superlattice, we numerically calculated the group velocities +( ) +( ) +1 +2 +, +, +, +, +, +g x +g y +g y +v +v +v + + + while simultaneously varying +osc +A + and +osc +V +. The results are shown in Fig. +6b-d, respectively. + +Fig. 6 a) Normalized group velocities along the x-direction +, +g x +F +v +v and along the y-direction +( )1 +, +g y +F +v +v +and +( ) +2 +, +g y +F +v +v of the bulk eigenmodes of a GSL is characterized by an electrostatic potential with + +(a) +(b) +V +g.x +1.0 +V +6 +0.5 +5 +VF0.0 +4 +hVF +0.5 +0 +3 +-1.0 +0 +2 +4 +6 +8 +2 +eA +.a/h +1 +0 +0 +2 +4 +6 +8 +eA..a/h +(c) +(d) +,(1) +VF +V +g.y/ +g. +6 +6 +5 +5 +V +4 +hVF +hVF +0 +3 +3 +2 +2 +1I +1F +OE +0 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +eA +.a/n +eA. +.a/namplitude +5.0 +osc +F +V a +v = + as a function of the normalized amplitude of the magnetic vector potential +osc +eA a +. b) Normalized group velocities along the x-direction +, +g x +F +v +v of the bulk eigenmodes as a +function of the normalized amplitudes of the magnetic vector potential +osc +eA a + and electrostatic +potential +osc +F +V +a +v . c ) and d) Similar to b) but for the normalized group velocities of the bulk +eigenmodes that propagate along the y-direction +( )1 +, +g y +F +v +v and +( ) +2 +, +g y +F +v +v , respectively. +The results depicted in Fig. 6b reveal that +, +g x +v + is unaffected by a variation in +osc +V + when +0 +osc +A += + due to the Klein tunneling effect, and that when +0 +osc +A + + the carriers’ velocity +is decreased. Interestingly, our results suggest that for a fixed magnetic potential +, +g x +v + +tends to increase as the amplitude of the electric potential increases. +For propagation along the y-direction, the response of the structure to variations in both +potentials is more complex. In Fig. 6c it is seen that the group velocity +( )1 +, +g y +v + follows +closely the response of pristine graphene when both potentials are similarly valued +(using normalized quantities) but decreases significantly when one of the potentials is +significantly stronger than the other. On the other hand, the results associated with the +propagation properties of the other eigenmode, shown in Fig. 6d, show that +( ) +2 +, +g y +v + is +progressively reduced when both +osc +A + and +osc +V + increase. Hence, there is a vast +combination of potentials that can result in static waves ( +( ) +2 +, +0 +g y +v += +), i.e. a dispersion +characteristic consistent with a type-III Dirac cone, and also bulk unidirectional +eigenmodes ( +( ) +( ) +1 +2 +, +, +, +0 +g y +g y +v +v + +) (type-II Dirac cone), so that by properly tuning the potentials +we are able to precisely control the direction of propagation of the carriers in the +superlattice. + +V. + Conclusions +We developed a homogenization model to determine the effective response of +superlattices with magnetic vector potential and electrostatic potential. We used a +numerical FDTD algorithm to apply this formalism and study the propagation properties +of the charge carriers in graphene superlattices characterized by 1D magnetic and +electric potentials with a sinusoidal-type spatial variation. We demonstrated that when +the GSL has only a magnetic vector potential, the effective Hamiltonian of the structure +can be drastically simplified by neglecting the granular details of the potential and +considering solely an effective Fermi velocity that is smaller to that of pristine +graphene. We also demonstrated that when both potentials are present in the superlattice +the response of the structure becomes nonreciprocal and is characterized by a dispersion +characteristic consisting of a tilted Dirac cone. Particularly, we showed that for +propagation perpendicular to the stratification of the potentials the GSL supports two +eigenmodes whose energy dispersion, for a fixed pseudo-momentum, is not linked by +an odd symmetry. We showed that in such materials we can obtain extreme wave +phenomena such as energy flat bands and regimes where both eigenmodes flow along +the same direction. We envision that by properly tuning the potentials the proposed +GSL structure can be operated in regimes characterized by type-I, type-II or type-III +Dirac cones. +Appendix A: Stability of the FDTD Algorithm +In a FDTD numerical algorithm the calculations remain stable provided the time step is +small enough, below a give threshold [55]. In what follows we address the stability of +the proposed numerical FDTD algorithm to determine the time evolution of the waves +in the graphene superlattices. For simplicity we assume that the medium is spatially + +homogeneous (V and +y +A are independent of the spatial coordinates) in the update +equations 11a-b. Our aim is to characterize the stationary states of the system. Thus, we +look for plane-wave type solutions of Eq. (11) with: +1, +1, +1, , +1/2 +1/2 +2, +1/2, +1/2 +2, +1/2, +1/2 +n +n +p +q +p q +p +n +n +p +q +p +q + ++ ++ ++ ++ ++ +− ++ + + + + + + += + + + + + + + + + + + + + + +, + + +(A1a) +1, , +1 +1, , +1/2 +1/2 +2, +1/2, +1/2 +2, +1/2, +1/2 +n +n +p q +p q +q +n +n +p +q +p +q + ++ ++ ++ ++ ++ ++ +− + + + + + + += + + + + + + + + + + + + + + +, + + + +(A1b) +where +p +i +p +e + + = + and +q +i +q +e + + = + are the spatial phase-shifts between consecutive nodes. +Furthermore, we consider a time variation of the type +1 +1, , +1, , +n +n +p q +p q + ++ + +=  + and +1/2 +1/2 +2, , +2, , +n +n +p q +p q + ++ +− + +=  + where  is a function of the spatial phase-shifts ( +) +, +p +q + + +. Hence, the +proposed FDTD algorithm is stable as long as +1 +  for arbitrary values of +p + and +q + +with +1 +p +q + + += += . Inserting Eq. A1a-b into Eq. 11a-b and using simple mathematical +manipulations we obtain the following system written in a matrix form: +( +) +( +) +1, , +1 +2 +1 +1 +2, +, +2 +2 +1 +1 +1 +2 +0 +1 +1 +2 +n +p q +t +F +t +n +p +q +F +t +t +V +v +D +i +V +v +D +i + + + + + +− +− ++ ++ ++ + + + + + + + +− − + ++ + + + + + + + + + + += + + + +  + +  + + +− − + ++ + + + +  + + + + + +, with +(A2) +( +) +1 +1 +1 +1 +1 +1 +1 +1 +1 +4 +1 +1 +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +2 +2 +y +q +p +p +q +p +q +p +q +x +y +x +y +x +y +x +y +eA +D +i +i +i +i + + +  + + +  +− +− +− +− +− +− +− +− +− += ++ ++ ++ ++ + + + + + + + + + +− +− ++ ++ ++ +− +− + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +, +(A3) +( +) +1 +4 +1 +1 +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +2 +2 +y +q +p +p +q +p +q +q +p +x +y +x +y +x +y +x +y +eA +D +i +i +i +i + + +  +  + + ++ = − ++ ++ ++ ++ + + + + + + + + + ++ +− +− ++ +− +− ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +. (A4) + +To verify the stability of the algorithm we calculate the characteristic equation of the +problem, obtained from the kernel of Eq. (A2), which is given by +( +) +( +) +2 +2 +1 +1 +1 +0 +2 +t +F +t +V +v +D D +i + + + +− ++ + + +− − + ++ +− + += + + + + +, + + +(A5) +Using +p +i +p +e + + = + and +q +i +q +e + + = + it can be shown that: +2 +2 +2 +2 +2 +2 +2 +2 +1 +4 +cos +sin +cos +cos +2sin +2 +2 +2 +2 +2 +q +p +p +q +q +y +x +y +x +y +eA +D D + + + + + +− ++ + + + + += +−  +−  + ++ + + + + +   + + + + + + +. +(A6) +The nontrivial solutions  of characteristic equation are then: +( +) +2 +2 +2 +2 +2 +1 +1 +1 +2 +2 +B +B +C +B +C +C +i + + + + + + + += +− − ++ + +− +− + + + + + + +− + + +, + + + (A7) +with +2 +t +V +C = + and +( +) +2 +2 +0 +F +t +B +v +D D +− ++ += − + + + real-valued parameters. From this result, it +is simple to check that if +2 +2 +1 +0 +2 +B +C + + − +−  + + + + + then +( +) +1/2 +2 +2 +2 +2 +2 +2 +2 +1 +1 +1 +1 +2 +4 +1 +B +B +C +B +C +C + + + + + + + += ++ +− ++ ++ +− += + + + + + + ++ + + + + + + + + +. + + (A8) +Thus, the algorithm is stable when +2 +2 +1 +0 +2 +B +C + + − +−  + + + + +. This condition is equivalent to +( +) +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +cos +sin +cos +cos +sin +1 +2 +2 +2 +2 +2 +2 +2 +q +p +p +q +q +x +y +y +x +y +t +F +t +eA +V +v + + + + + + + +  + + + + + ++  + ++ + ++ + + + + + + + + + + + + + + + + + +(25) +The above inequality should be satisfied for all +p + and +q + . In particular, it is enough to +ensure that: +( +) +2 +2 +2 +2 +2 +2 +1 +1 +1 +2 +y +x +y +F +t +x +y +eA +v + + + + + +  + + + + + + + + + + +  + + + + + + + +(26) + +If we consider equally spaced nodes, i.e. +x +y + =  =  and time steps given by +t +F +v  + + = +, where  is a real valued positive constant, this condition is equivalent to: +2 +1 +1 +1 +2 +eA +  + + ++ + + + + + + + + + + +(27) +Hence, for these superlattices the numerical algorithm stability will depend on the +amplitude of the magnetic vector potential. 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Antennas Propagat. 14, 302 (1966). + + + diff --git a/S9E0T4oBgHgl3EQflAFl/content/tmp_files/load_file.txt b/S9E0T4oBgHgl3EQflAFl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2a13550a4c5969e9125479698fbfd5ca792e223d --- /dev/null +++ b/S9E0T4oBgHgl3EQflAFl/content/tmp_files/load_file.txt @@ -0,0 +1,939 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf,len=938 +page_content='Effective Medium Model for Graphene Superlattices with Electrostatic and Magnetic Vector Potentials David E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Fernandes Instituto de Telecomunicações and Department of Electrical Engineering, University of Coimbra, 3030- 290 Coimbra, Portugal E-mail: dfernandes@co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='pt Abstract In this article we develop an effective medium model to characterize the electron wave propagation in graphene based nanostructures with an electrostatic and magnetic vector potentials imposed on their surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We use a numerical algorithm to determine the effective medium parameters of the heterostructure and calculate the electronic band structure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We apply our formalism to analyze superlattices with solely a magnetic potential and reveal that the response of the structure remains reciprocal and is characterized by a decrease in charge carrier’s velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We also study the response of superlattices with both potentials superimposed on graphene and show that the response of the system becomes nonreciprocal with a dispersion characterized by a tilted Dirac cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We demonstrate that it is possible to alternate between a type-I, type-II or type-III Dirac cones by properly tuning the amplitude of the potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Introduction Graphene is a two-dimensional nanomaterial formed by carbon atoms that are arranged in a honeycomb lattice [1-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Over the last decade this material has been on the spotlight of condensed matter physics research due to its remarkable electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' By possessing a relativistic-spectrum, the low-energy electrons in graphene have a linear dispersion and their propagation is determined by a massless Dirac equation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' It has been proposed that it is possible to achieve additional control over the propagation characteristics of the electrons in graphene by modifying the original material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' These structures are known as graphene superlattices (GSLs) and may be obtained by artificially introducing a new length scale into the system in the form of a periodic potential, either by using an electrostatic potential [9-18] or magnetic vector potential [19-28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Superlattices characterized by electrostatic potentials may be realized using different techniques, such as with periodically patterned gates, using a crystalline substrate or with the deposition of adatoms on graphene’s surface [29-36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' On the other hand, to obtain GSLs with a magnetic vector potential one can use nano-magnetic strips [19-22] or strain-inducing modulations [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' An electrostatic potential on the surface of graphene can allow for an extreme anisotropic response which can lead to the super- collimation of electron waves [17, 29-31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Moreover, it can permit extreme wave phenomena such as a perfect lens for matter waves [38, 39] or an electron wormhole [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Conversely, a magnetic vector potential can also allow to tailor the electron wave propagation by reducing the charge carriers velocity [22-28] or even providing a way to tilt the energy dispersion of electrons in the medium [41], usually identified as a type-I tilted Dirac cone [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Such type of response may be used for valley filtering in p-n junctions [43] and to generate photocurrent [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The characterization of the propagation of electrons in superlattices with a magnetic vector potential is typically done using a transfer matrix formalism [22-28] which can limit the study to potentials characterized only by constant barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Interestingly, in the works proposed in [17, 18, 38-40] the propagation of the electrons in the GSLs with an electrostatic potential was studied under an effective medium formalism, so that granular details of the potential are homogenized [38] and the structure is regarded as a continuum characterized by some effective parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Such effective medium techniques can vastly simply the analysis of the problem while simultaneously providing invaluable insight into the physical phenomena taking place in the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The main objective of this work is to develop an effective medium model for superlattices characterized by both an electrostatic potential and a magnetic vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To determine the effective response of the superlattices we use a numerical finite-difference time-domain (FDTD) algorithm based on the numerical tool proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' It is important to mention that FDTD numerical tools such as the ones developed in [17, 18, 45, 46] have been widely used to determine the electron wave propagation in graphene based nanomaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To begin with, we apply the numerical algorithm to homogenize a GSL with a magnetic vector potential with a sinusoidal spatial variation and show that, similar to what happens in GSLs with Krönig-Penney type potentials [22-28], the response of the structure is isotropic, with the group velocity of the charge carriers being smaller than in pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We demonstrate that for these superlattices the analysis of the problem can be vastly simplified by using an effective Hamiltonian that discards the granular details of the potential and instead considers an effective parameter that is independent of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' This effective parameter may be regarded as an effective Fermi velocity whose value is only dependent on the amplitude of the magnetic vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We also determine the effective response of superlattices with both electrostatic and magnetic vector potentials with sinusoidal spatial variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Using our effective medium formalism, we demonstrate that the interplay between the magnetic and electric potentials can give rise to an overall nonreciprocal response whose energy dispersion is characterized by a Dirac cone tilted along the direction perpendicular to the stratification of the potentials (type-I Dirac cone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Moreover, we show that for propagation along such direction there is a wide range of combinations of amplitude of the potentials for which the bulk eigenmodes can flow along the same direction and, by properly tuning the amplitude of the potentials, it is even possible to have eigenmodes with a null group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Such dispersion characteristic corresponds to a type-III Dirac cone [42, 47, 48], where one of the bands is flat and the other has a linear dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' It has been proposed such dispersion can enhance the superconducting gap in Weyl semimetals [49], and by using the flat band, they can allow for a new platform to study the correlated phases in the structure [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Importantly, a type-III Dirac cone marks the transition between type-I and type-II Dirac cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The type-II dispersion characteristic differs from the type-I from the fact that the Fermi surface is no longer a point, but rather two-crossing lines [51-53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Such dispersion appears when one of the bands is tilted in such a way that the group velocity of the associated energy eigenmode has the opposite sign that the corresponding value in pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' II we introduce the homogenization formalism that will be used to characterize the effective medium response of the graphene superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' III we describe the numerical FDTD algorithm that is used to determine the effective parameters of the superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' IV the homogenization formalism is applied to characterize the wave propagation in GSLs with solely a magnetic vector potential and in superlattices with both electrostatic and magnetic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Finally, the conclusions are drawn in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Effective Medium Model In this work we study the electron wave propagation in graphene-based nanomaterials characterized by an electrostatic potential and a magnetic vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Near the K point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' the propagation of the charge carriers in graphene superlattices with electrostatic and magnetic vector potentials may be described using the massless Dirac equation: ˆ i H t \uf0b6 = \uf0b6 ψ ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (1) where ( ) ( ) ( ) ˆ F H v i q V = \uf0d7 − \uf0d1 − + σ A r r is the microscopic Hamiltonian operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' V is the microscopic electric potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' q e = − is the electron charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' A is the magnetic vector potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 6 10 / F v m s \uf0bb is the Fermi velocity in pristine graphene and ˆ ˆ x y = + σ σ x σ y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' with ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' x y σ σ the Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Moreover, \uf07b \uf07d 1 2 , = \uf059 \uf059 ψ is a two- component pseudospinor, with each component of the pseudospinor associated with a different trigonal sublattice of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Without any loss of generality, we consider that the microscopic potentials have a one- dimensional (1D) spatial variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Particularly we suppose that ( ) ( ) V V x = r and ( ) ˆ y A x = A y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The proposed effective model could be readily extended to other (more complex) types of spatial variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' For the considered spatial variations of the electric and magnetic potentials Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (1) may be re-written as: ( ) ( ) ( ) ˆ F y i v i eA x V x t \uf0b6 = \uf0d7 − \uf0d1 + \uf0d7 + \uf0b6 ψ σ y ψ ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (2) In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' [38] it was shown that provided the initial state of the system ( ) 0 t = ψ is macroscopic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' so that ( ) ( ) 0 0 t t = = = ψ ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' with the spatial averaging operator defined as 1 i F Fe dS A − \uf0d7 = \uf0f2 k r (where we assume a spatial evolution of the type ie \uf0d7 k r ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' then the envelope of the pseudospinor can be accurately calculated from an effective Hamiltonian of the system ˆ ef H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' defined as ˆ ˆ ef H H = ψ ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' which may be written as: ( ) ( ) ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ef F F y ef ef H v ev A V \uf077 \uf077 = \uf0d7 + \uf0d7 + σ k σ k k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (3) Here we used i\uf0d1 = k , with k the pseudo-momentum, and introduced the effective magnetic and electric potentials, ef A and ef V , which can generally be matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Applying the spatial averaging operator to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (2), with i t \uf077 \uf0b6 = − \uf0b6 for a time evolution of the type i t e \uf077 − , we obtain ˆ F F y y E H v ev A V \uf073 = = \uf0d7 + \uf0d7 + ψ ψ σ k ψ ψ ψ , (4) where n ik n \uf0b6 = \uf0b6 , with , n x y = and E i t \uf0b6 = \uf0b6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To determine the effective Hamiltonian for a fixed energy and pseudo-momentum ( ) 0 0 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' x y k k = k we can calculate the time evolution in the graphene nanomaterial of two linear independent initial electronic states of the form: ( ) ( ) 0 1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 0 0 i t e \uf0d7 \uf0e6 \uf0f6 = = \uf0e7 \uf0f7 \uf0e8 \uf0f8 k r ψ r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (5a) ( ) ( ) 0 2 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 0 1 i t e \uf0d7 \uf0e6 \uf0f6 = = \uf0e7 \uf0f7 \uf0e8 \uf0f8 k r ψ r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (5b) and use the Fourier transform and the spatial averaging operator to calculate ( ) n ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ( ) n y A ψ and ( ) n Vψ in the frequency and spatial domains for each initial state ( ) n ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' with 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='2 n = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The effective Hamiltonian ( ) ˆ , ef H \uf077 k is then determined by calculating: ( ) ( ) ( ) ( ) ( ) 1 1 2 1 2 ˆ ˆ ˆ , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ef H H H \uf077 − \uf0e9 \uf0f9 \uf0e9 \uf0f9 = \uf0d7 \uf0eb \uf0fb \uf0eb \uf0fb k ψ ψ ψ ψ (6) Similarly, the effective electric and magnetic potentials can also be calculated using: ( ) ( ) ( ) ( ) ( ) 1 1 2 1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ef V V V \uf077 − \uf0e9 \uf0f9 \uf0e9 \uf0f9 = \uf0d7 \uf0eb \uf0fb \uf0eb \uf0fb ψ ψ ψ ψ (7) ( ) ( ) ( ) ( ) ( ) 1 1 2 1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ef A A A \uf077 − \uf0e9 \uf0f9 \uf0e9 \uf0f9 = \uf0d7 \uf0eb \uf0fb \uf0eb \uf0fb ψ ψ ψ ψ (8) It is important to mention that the initial states are chosen in such a way that they are not more localized than the characteristic period of the lattices, such that the pseudo- momentum k0 associated with these states is within the first Brillouin minizone of the superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Numerical Algorithm The calculation of the effective Hamiltonian in the frequency domain requires the calculation of the time evolution of the electronic states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 5a-b and its Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To determine the time-evolution of the initial electronic states we use a FDTD (Finite Differences in the Time Domain) numerical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The algorithm is based on the numerical tool developed in [17] which was successfully used to study the transport properties of graphene superlattices characterized solely by an electrostatic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We begin by separating the Dirac equation (2) for each component of the pseudospinor: 1 2 1 y F eA V v i t x y i \uf0e6 \uf0f6 \uf0b6\uf059 \uf0b6 \uf0b6 = − − + \uf059 + \uf059 \uf0e7 \uf0f7 \uf0b6 \uf0b6 \uf0b6 \uf0e8 \uf0f8 , (9a) 2 1 2 y F eA V v i t x y i \uf0e6 \uf0f6 \uf0b6\uf059 \uf0b6 \uf0b6 = − + − \uf059 + \uf059 \uf0e7 \uf0f7 \uf0b6 \uf0b6 \uf0b6 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (9b) To obtain the time update equations in an explicit form we discretize the spatial domain into a rectangular grid, such that the consecutive nodes along the x- and y-directions are separated by a distance x \uf044 and y \uf044 , as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Furthermore, each component of the pseudospinor is sampled at instants of time separated by the time step t \uf044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' This allows us to write the pseudospinor components as ( ) ( ) ( ) , , , , , , x y t x y t p q n p q n \uf059 = \uf059 \uf044 \uf044 \uf044 \uf0ba \uf059 and permits using a finite differences method to calculate the partial derivatives in Eq(9) such that: ( ) 1 1 2 2 l l i i i \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf059 + − \uf059 − \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0b6 \uf059 = \uf044 , (10) with , , l x y t = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1 (color online) a) The graphene superlattice spatial domain is discretized into a rectangular grid with a finite number of nodes spaced by x \uf044 along the x-direction and y \uf044 along the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The pseudospinor components 1 \uf059 and 2 \uf059 are defined on staggered subgrids so that 1 \uf059 is defined over the nodes ( ) ,p q and 2 \uf059 is defined at the nodes ( ) 1/ 2, 1/ 2 p q + + , shifted by a half-grid period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' b) The time domain is sampled at time intervals separated by a time step t \uf044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Similarly to the spatial domain a) b) A n+1 Un I 3 n+ n+ Ay 2 2 2 2 Ax i+1discretization scheme, the pseudospinor components 1 \uf059 and 2 \uf059 are defined on staggered subgrids shifted by 2 t \uf044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We also consider that the two components of the pseudospinor are defined on staggered grids in space and time so that ( ) 1 , , p q n \uf059 and 2 1 1 1 , , 2 2 2 p q n \uf0e6 \uf0f6 \uf059 + + + \uf0e7 \uf0f7 \uf0e8 \uf0f8 , as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1a-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Applying these principles to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 9a-b leads to the following update equations: ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1 2 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2 2 1 2 1 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 p q n n y p q p q n n p q t p q t F t p q p q x y n p q x y x y eA V V v i i i i i + + + + + + − + \uf0e6 \uf0f6 \uf0e9 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf059 − \uf044 = \uf059 + \uf044 − \uf044 + \uf059 + − \uf059 \uf0e7 \uf0f7 \uf0ea \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf044 \uf044 \uf0ea \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0eb \uf0e8 \uf0f8 \uf0e6 \uf0f6 \uf0e6 \uf0f6 − + \uf059 + + \uf0e7 \uf0f7 \uf0e7 \uf0e7 \uf0f7 \uf0e7 \uf044 \uf044 \uf044 \uf044 \uf0e8 \uf0f8 \uf0e8 \uf0f8 1 1 2 2 1 1 1 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2 2 2 2 1 1 2 2 n n p q p q x y i + + + − − − \uf0f9 \uf0e6 \uf0f6 \uf059 − − \uf059 \uf0fa \uf0f7 \uf0e7 \uf0f7 \uf0f7 \uf0e7 \uf0f7 \uf044 \uf044 \uf0fa \uf0e8 \uf0f8 \uf0fb (11a) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1 1 1 1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1/2 2 2 2 2 2 1 1 1 1 1 1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 2 p q p q p q n y n n n t t F t p q p q p q p q x y x y V V eA v i i i i + + + + + − + + + + + + + + \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e9 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf059 − \uf044 = \uf059 + \uf044 − \uf044 − \uf059 + + \uf059 \uf0e7 \uf0f7 \uf0ea \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf044 \uf044 \uf0ea \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0eb \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e6 − − \uf044 \uf044 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1 1 1 1 2 2 2 2 n n n p q p q p q x y x y i i + + \uf0f9 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf059 + − \uf059 − + \uf059 \uf0fa \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf044 \uf044 \uf044 \uf044 \uf0fa \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0fb (11b) With ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' p q x y V V p q = \uf044 \uf044 and ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' y p q y x y A A p q = \uf044 \uf044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Importantly, this discretization scheme of the update equations requires the value of the pseudospinor components in subgrid points where they are not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In particular, the value value ( ) 2 ,p q \uf059 is necessary in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (11a), while the update equation 11b requires the value of 1 1 1 , 2 2 p q \uf0e6 \uf0f6 \uf059 + + \uf0e7 \uf0f7 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To obtain such values we assume that the wavefunction varies slowly in space so that the pseudospinor component values in points of space that lie outside the grid nodes can be determined by the average of its neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In that case one can use: ( ) 2 2 2 2 2 1 1 1 1 1 1 1 1 1 , , , , , 4 2 2 2 2 2 2 2 2 p q p q p q p q p q \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf059 \uf0bb \uf059 + − + \uf059 − + + \uf059 + + + \uf059 − − \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 , (12a) and ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 , 1, , 1 1, 1 , 2 2 4 p q p q p q p q p q \uf0e6 \uf0f6 \uf059 + + \uf0bb \uf059 + + \uf059 + + \uf059 + + + \uf059 \uf0e7 \uf0f7 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (11b) To determine the effective Hamiltonian it necessary to calculate the Fourier transform of ( )t ψ , ( ) y A t ψ and ( ) V t ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' As shown in other works dealing with time-domain homogenization techniques of artificial structured media [54], to ensure the convergence of the Fourier transform for a given frequency \uf077 , at each time iteration 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=', n N = the quantities ( ) n t \uf044 ψ , ( ) y A n t \uf044 ψ and ( ) V n t \uf044 ψ must be multiplied by a time decaying exponential of the form n t e\uf077\uf0a2\uf0a2 \uf044 , with i \uf077 \uf077 \uf077 \uf0a2 \uf0a2\uf0a2 = + , so that the term \uf077\uf0a2\uf0a2 represents some small losses in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Importantly, the total number of iterations N must be sufficiently high so that 0 N t e\uf077\uf0a2\uf0a2 \uf044 \uf0bb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Typically, the total number of iterations is on the order of 2 t N \uf070 \uf077\uf0a2\uf0a2 \uf0bb \uf044 [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' At this point it is important to mention that in all calculations performed in this work we consider that both potentials have the same spatial period a and that the distance between adjacent nodes is the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' x y \uf044 = \uf044 = \uf044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Moreover, we determine the time evolution of the initial states in a region of space containing one period of the potentials and apply Bloch boundary condition at the edge of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In Appendix A we analyze the stability conditions of the proposed FDTD algorithm and show that the maximum value of the time-step depends both on the distance between adjacent spatial nodes and the maximum amplitude of the magnetic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Numerical results In what follows, we use our homogenization algorithm to determine the band diagram and effective medium parameters of graphene superlattices with electrostatic and magnetic vector potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' It was shown in [38] that provided the microscopic Hamiltonian of the graphene nanomaterial does not vary in time, the electronic band structure of the material close to the K point can be computed directly from the effective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' This property is a consequence of the energy eigenstates of the system calculated using ˆ ef H being equal to the eigenstates of the microscopic Hamiltonian [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In this work we consider an electrostatic potential with a sinusoidal-type spatial variation of the form ( ) 2 sin osc x V x V a \uf070 \uf0e6 \uf0f6 = \uf0e7 \uf0f7 \uf0e8 \uf0f8 and a periodic magnetic induction field given by ( ) 1 1 2 2 ˆ ˆ cos z osc x B x A a a \uf070 \uf070 \uf0e6 \uf0f6 = = \uf0e7 \uf0f7 \uf0e8 \uf0f8 B z z , so that it varies along the x-direction but is oriented along the z-direction (perpendicular to the propagation plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The magnetic induction field it is on average null, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ( ) 1 1 0 1 0 a z B x dx a = \uf0f2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Since the magnetic field is related to the magnetic vector potential as = \uf0d1\uf0b4 B A , it follows that ( ) ˆ y A x = A y , with ( ) 2 sin y osc x A x A a \uf070 \uf0e6 \uf0f6 = \uf0e7 \uf0f7 \uf0e8 \uf0f8 , which is also periodic with zero spatial average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Similar to [38],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' to calculate the band diagram of the structures we expand the effective potentials ef A ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ef V as a Taylor series of the first order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' such that: ( ) ( ) ( ) ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 ef ef ef ef x y x y A A A A k k k k \uf077 \uf077 \uf077 \uf077 \uf0b6 \uf0b6 = + + \uf0b6 \uf0b6 k and (13) ( ) ( ) ( ) ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 ef ef ef ef x y x y V V V V k k k k \uf077 \uf077 \uf077 \uf077 \uf0b6 \uf0b6 = + + \uf0b6 \uf0b6 k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (14) and use our numerical algorithm to calculate the effective Hamiltonian (given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Note that in general ( ) , ef A \uf077 k and ( ) , ef V \uf077 k are not scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' From hereon we consider that the spatial grid is discretized using a node spacing 50 a \uf044 = and the time step is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='3 t F v \uf044 = \uf044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Since the effective response of graphene superlattices characterized solely by electrostatic potentials was already thoroughly discussed in [17, 18, 38], we restrict our attention to superlattices with only magnetic vector potential and structures with both magnetic and electrostatic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' A) Superlattices with Magnetic Potential We begin by calculating the response of superlattices characterized solely by a magnetic vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Particularly, we determine the effective potential ( ) , ef A \uf077 k of the nanomaterial for some amplitudes of the magnetic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' For this GSL it is immediate that the effective electric potential is null, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ( ) , 0 ef V \uf077 = k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Our numerical results showed that to an excellent approximation ( ) ,0 ef y A \uf077 \uf061 \uf0bb σ , with \uf061 a real value depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Interestingly, it is seen that for low-energy excitations this value varies linearly with the energy on the carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Moreover, we also verified that ( ) ,0 ef z x A i k \uf077 \uf062 \uf0b6 \uf0bb \uf0b6 σ , with z σ the Pauli matrix, and that ( ) ,0 ef y A k \uf077 \uf063 \uf0b6 \uf0bb \uf0b6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Here \uf062 \uf063 \uf0bb are real constants almost independent of the carriers energy for low-energy excitations, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2b-c, respectively, for some representative amplitudes of the potential osc A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2 Normalized effective parameters of the graphene superlattice as a function of the normalized energy: a) ea\uf061 , b) e\uf062 and c) e\uf063 , for a magnetic vector potential with amplitude 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 osc eA a = (black line), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 osc eA a = (dark green line), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 osc eA a = (brown line) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 osc eA a = (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Hence, for low energy excitations, the effective magnetic potential may be approximated by: ( ) ( ) , ef y x z y A i k k \uf077 \uf061 \uf077 \uf062 \uf063 \uf0bb + + k σ σ 1 (15) Inserting this expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 3, allows us to write the effective Hamiltonian as: 0 ˆ ef F H E v \uf075 \uf075 = + \uf0d7 1 σ k , (16) with 0 F E ev \uf075 \uf061 = and 1 1 e e \uf075 \uf062 \uf063 = − \uf0bb − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The energy dispersion of the superlattice may then be calculated from the eigenvalue problem: ˆ ef E H \uf059 = \uf0d7\uf059 , (17) and it is given by 2 2 0 F x y E E v k k \uf075 \uf075 − = + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Since 0 \uf075 and \uf075 are weakly dependent on the energy, the previous expression can be further simplified into , F eff E v = k , (18) by defining an effective Fermi velocity , 0 1 F eff F v v \uf075 \uf075 = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Considering that 0 \uf075 is a negative value (proportional to the slope of the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2a) and that \uf075 is smaller than unity because , 0 \uf062 \uf063 \uf03e , it is expected that , F eff v can be significantly smaller than the Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To verify the accuracy of our effective medium model we a) b) c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='5 eaa eβ ex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='4 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 h h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 1 0 1 N 2 1 0 2 2 1 0 2 Ea/hvF Ea/hvF Ea/hvroverlapped in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 3a-b the “exact” band diagram for propagation along the x-direction ( 0 yk = ) and the y-direction ( 0 xk = ), with the corresponding results calculated using our simplified effective formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The band diagram was calculated for a graphene superlattice characterized by a magnetic potential with amplitude 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 osc eA a = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' As seen, for low energy excitations both results have nearly exact agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 3 a) and b) Dispersion of the energy eigenstates of a graphene superlattice with a magnetic vector potential 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 osc eA a = for propagation along the x and y directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The blue dashed curves represent the “exact” energy dispersion of the GSL, the blue solid curves correspond to the dispersion of the GSL calculated using the effective parameter , F eff v and the black solid curves show the dispersion of pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' c) Normalized effective Fermi velocity , F eff F v v as a function of the normalized magnetic vector potential amplitude osc eA a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 3a-b also show that even in the presence of a magnetic potential, which the breaks time-reversal symmetry of the structure, the response of these graphene superlattices remains isotropic and reciprocal for low energy excitations, and without a bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Moreover, we see that the group velocity of the carriers in the superlattice is exactly equal to the effective Fermi velocity , , , 1 g x g y F eff dE v v v dk = = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Indeed, by comparing these results with the band diagram of pristine graphene, depicted as black curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 3a-b, it is confirmed that the effective Fermi velocity is smaller than the Fermi velocity in pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 3c we show the effect of the amplitude of the magnetic potential on the effective Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The results reveal that the carrier’s velocity can be severely reduced from the Fermi velocity as the a) b) ( k, =0 k,=0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='8 Ea 1 Ea 0 0 hVF hVF VF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 2 0 1 2 3 2 1 0 2 0 5 10 15 20 25 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='a k,a eAsca/hamplitude of the magnetic vector potential increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Hence, by precisely tailoring the magnetic field distribution we can control the charge velocity in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' These results go in line with previous studies [22-28] that demonstrated the effect of the magnetic potential in the carrier velocity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' B) Superlattices with Magnetic and Electric Potentials The analysis we did in the previous section revealed that imposing a 1D magnetic potential with zero-spatial average on the surface of graphene leads to a reciprocal and isotropic response, wherein the charge carriers group velocity is decreased as the amplitude of the magnetic potential increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' On the other hand, it was demonstrated in [17, 38] that in graphene superlattices with 1D electrostatic potential with zero spatial average, the transport properties of the electrons can also be modified so that they have a preferred direction of propagation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' the effective medium behaves as an anisotropic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In what follows, we use our homogenization model to study the response of superlattices characterized by both a 1D magnetic and a 1D electric potential with zero spatial average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' As in the previous section, we start by calculating the effective potentials of the GSL given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (13)-(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' As a leading example, we consider a superlattice characterized by a magnetic vector potential with amplitude 4 osc eA a = and an electrostatic potential with amplitude 5 osc F V a v = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Our numerical results show to an excellent approximation that the Taylor expansion of the effective potentials may be written as: ( ) ( ) ( ) 0 0 1 1 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ef y x y x y y A k k \uf077 \uf061 \uf062 \uf061 \uf062 \uf061 \uf062 = + + \uf0d7 + + + k 1 σ σ 1 σ 1 σ (19) ( ) ( ) ( ) 0 0 1 1 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ef y x y x y y V k k \uf077 \uf063 \uf064 \uf063 \uf064 \uf063 \uf064 = + + \uf0d7 + + + k 1 σ σ 1 σ 1 σ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (20) With ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' i i i i \uf061 \uf062 \uf063 \uf064 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' with 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='2 i = ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' real-valued scalars whose energy dependence is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 4 a) Normalized effective magnetic potential parameters 1 0 ea\uf061 − , 1 0 ea\uf062 − , 1 1 e\uf061 − , 1 1 e\uf062 − , 1 2 e\uf061 − , 1 2 e\uf062 − as a function of the normalized energy for a graphene superlattice for a characterized by a magnetic vector potential with amplitude 4 osc eA a = and an electrostatic potential with amplitude 5 osc F V a v = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' b) Similar to a) but for the normalized effective electrostatic potential parameters 1 0 ea\uf063 − , 1 0 ea\uf064 − , 1 1 e\uf063 − , 1 1 e\uf064 − , 1 2 e\uf063 − , 1 2 e\uf064 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 4 show that the zeroth order coefficients of the Taylor expansion of the potentials 0 0 0 0 , , , \uf061 \uf062 \uf063 \uf064 vary linearly with the energy of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' On the other hand, the first order terms , , , i i i i \uf061 \uf062 \uf063 \uf064 , with 1,2 i = , are almost independent of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Interestingly, our simulation results also show that the effective potentials are linked to each other through the following relations: 0 0 1 2 2 1 0 F F F ev e e c E E v v \uf061 \uf064 \uf061 \uf062 \uf063 \uf064 \uf0bb \uf0bb − \uf0bb \uf0bb \uf0bb − = , (21a) 0 1 2 1 F ev e e c E \uf062 \uf062 \uf061 \uf0bb − \uf0bb = , (21b) 0 1 2 2 F F c E v v \uf063 \uf063 \uf064 \uf0bb − \uf0bb = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (21c) Surprisingly, these relations show that it is possible to describe the spatially dispersive response of the potentials at the expense of the non-spatially dispersive terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' For this a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 axo/hvr eaα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='h-1 X: /hv, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='5 eα,h-1 S /hvr eβh-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 eα,h-l eaβ,h-l X / hvp ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' /hv: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='5 eβ,h-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' / hvp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 1 0 1 2 2 1 0 1 2 Ea/hv Ea/hvrexample, it is found that 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='277 c \uf0bb − , 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='215 c \uf0bb − and 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='269 c \uf0bb − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Using the effective potentials (19)-(20) together with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 21a-c, we can write the effective Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 3) as: ( ) ( ) ( ) ( ) 0 1 2 1 2 1 2 0 ˆ 2 2 ef F y F x x y F y H v c E c c E c c v k k v c c c = \uf0d7 + + + + − + + + σ k σ 1 σ σ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (22) The energy dispersion of the modes supported in the superlattice is obtained from the eigenvalue problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (17) using the effective Hamiltonian given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The corresponding band diagram is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 5a and reveals that the response of the superlattice is no longer reciprocal, with the band diagram being tilted along the yk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To have a better understanding of the structure’s nonreciprocal response effect in the propagation of the electrons we focus our attention on propagation along the x- and y- directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Let us begin by studying the propagation along x ( 0, 0 y x k k = \uf0b9 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In that case, the energy dispersion of the modes is simply given by: ( ) ( ) 1 2 2 2 0 1 2 1 4 1 F x v c c k E c c c + − = − − − (23) The corresponding band diagram is show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 5b, where we also depict the “exact” band diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 5 a) Exact energy dispersion of the considered graphene superlattice b) and c) Dispersion of the energy eigenstates for propagation along the x and y directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The blue dashed curves represent the “exact” energy dispersion of the GSL, the blue solid curves correspond to the dispersion of the GSL calculated using the simplified effective medium formalism and the green solid curves show the dispersion of pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In all panels the GSL is characterized by a magnetic vector potential with amplitude 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 osc eA a = and an electrostatic potential with amplitude 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 osc F V a v = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (a) k,a 2 (b) (c) 2 0 k,=0 2E k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='=0 4 2 2 1 Ea Ea Ea 0 hy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' hVe hVr 2 2 2 0 2 3 3 2 0 2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='a 0 2 k,a k,aOur effective medium model results have a very good agreement with the exact band diagram showing that for propagation along the x-direction the results are comparable to the band diagram obtained for superlattices with solely a magnetic potential (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Indeed, for propagation along the x-direction, the response of the structure is reciprocal and characterized by a group velocity smaller to that of pristine graphene (whose response is depicted as green curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 5a), so that 1 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='77 g x x F v E k v − = \uf0b6 \uf0b6 \uf0bb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To determine the band diagram for propagation perpendicular to the direction of the stratification of the potentials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' for propagation along the y-direction (with 0 xk = , 0 yk \uf0b9 ), we follow a similar procedure and calculate the eigenvalue problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (17) considering the effective Hamiltonian given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (22) with 0 xk = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The problem yields two solutions (eigenmodes), whose energy dispersion is given by: ( )1 0 1 2 0 1 2 2 1 2 1 F y c c c E v k c c c − − − = − − + (24a) ( ) 2 0 1 2 0 1 2 2 1 2 1 F y c c c E v k c c c + + + = − − − + (24b) Clearly, for a fixed pseudo-momentum yk both solutions are not symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 5c we represent the band diagram calculated using the effective medium formalism and overlap the results with both the exact band diagram of the superlattice and the band diagram of pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' As seen, both results predict that the superlattice response is vastly different from that of pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The response of the structure is nonreciprocal as for a fixed energy both bulk modes are characterized by wave vectors yk that have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Additionally, the band diagram reveals that it is possible to have unidirectional bulk modes as both bands have negative (but distinct) group velocity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ( ) ( ) ( ) ( ) 1 1 2 2 1 1 , , g y y g y y v E k v E k − − = \uf0b6 \uf0b6 \uf0b9 = \uf0b6 \uf0b6 , consistent with a type-II Dirac cone dispersion characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' While one of the bands ( ( ) ( ) 1 y E k ) follows closely the original band of the pristine graphene, the other one ( ( ) ( ) 2 y E k ) is tilted towards the origin 0 E = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Indeed, our results suggest that it may be possible obtain an eigenmode characterized by a flat band so that the group velocity is precisely equal to zero , 0 g y v = , corresponding to a static wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Importantly, this type of dispersion characteristic is usually identified as a type-III Dirac cone [42] where the energy dispersion consists of one flat band while the other has a liner dispersion [42, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Clearly, this is an effect of the interplay between the electrostatic and magnetic vector potentials in the dynamics of the charge carriers in the superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 6a we show the group velocities ( ) ( ) 1 2 , , , , , g y g y g x v v v as a function of the amplitude of the magnetic potential for the fixed amplitude of the electric potential 5 osc F V a v = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' It is seen that increasing the amplitude of the magnetic potential decreases the group velocity along the x-direction, similar to the previous studied superlattice with 0 osc V = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' On the other hand, the effects of changing osc A in the group velocities ( ) ( ) 1 2 , , , g y g y v v are far more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To begin with, we note that when 0 osc A = the response is that of a superlattice with electric potential, which is characterized by an anisotropic response [17, 38], so that , g y v can significantly smaller than the Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Moreover, we see the group velocity ( )1 , g y v decreases as the amplitude of the magnetic potential increases, reaching a minimum for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='71 osc eA a \uf0bb , at which point it is equal to the Fermi velocity, and then it starts increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In contrast, for the other eigenmode, its group velocity ( ) 2 , g y v decreases as osc A increases, reaching a null value for a magnetic potential with amplitude 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='36 osc eA a \uf0bb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Such combination of amplitudes leads to dispersion characterized by a type-III Dirac cone, which crucially, marks the transition point where the dispersion changes from a type-I Dirac cone (for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='36 osc eA a \uf03c ) into a type-II (when 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='36 osc eA a \uf03e ), where both eigenmodes flow along the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' To have a complete understanding of the interplay between both potentials in the carriers velocity in the superlattice, we numerically calculated the group velocities ( ) ( ) 1 2 , , , , , g x g y g y v v v while simultaneously varying osc A and osc V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 6b-d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 6 a) Normalized group velocities along the x-direction , g x F v v and along the y-direction ( )1 , g y F v v and ( ) 2 , g y F v v of the bulk eigenmodes of a GSL is characterized by an electrostatic potential with (a) (b) V g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 V 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='5 5 VF0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 4 hVF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='5 0 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 0 2 4 6 8 2 eA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='a/h 1 0 0 2 4 6 8 eA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='.a/h (c) (d) ,(1) VF V g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='y/ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 6 6 5 5 V 4 hVF hVF 0 3 3 2 2 1I 1F OE 0 0 2 4 6 8 0 2 4 6 8 eA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='a/n eA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='a/namplitude 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='0 osc F V a v = as a function of the normalized amplitude of the magnetic vector potential osc eA a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' b) Normalized group velocities along the x-direction , g x F v v of the bulk eigenmodes as a function of the normalized amplitudes of the magnetic vector potential osc eA a and electrostatic potential osc F V a v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' c ) and d) Similar to b) but for the normalized group velocities of the bulk eigenmodes that propagate along the y-direction ( )1 , g y F v v and ( ) 2 , g y F v v , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' The results depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 6b reveal that , g x v is unaffected by a variation in osc V when 0 osc A = due to the Klein tunneling effect, and that when 0 osc A \uf0b9 the carriers’ velocity is decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Interestingly, our results suggest that for a fixed magnetic potential , g x v tends to increase as the amplitude of the electric potential increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' For propagation along the y-direction, the response of the structure to variations in both potentials is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 6c it is seen that the group velocity ( )1 , g y v follows closely the response of pristine graphene when both potentials are similarly valued (using normalized quantities) but decreases significantly when one of the potentials is significantly stronger than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' On the other hand, the results associated with the propagation properties of the other eigenmode, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 6d, show that ( ) 2 , g y v is progressively reduced when both osc A and osc V increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Hence, there is a vast combination of potentials that can result in static waves ( ( ) 2 , 0 g y v = ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' a dispersion characteristic consistent with a type-III Dirac cone, and also bulk unidirectional eigenmodes ( ( ) ( ) 1 2 , , , 0 g y g y v v \uf03c ) (type-II Dirac cone), so that by properly tuning the potentials we are able to precisely control the direction of propagation of the carriers in the superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Conclusions We developed a homogenization model to determine the effective response of superlattices with magnetic vector potential and electrostatic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We used a numerical FDTD algorithm to apply this formalism and study the propagation properties of the charge carriers in graphene superlattices characterized by 1D magnetic and electric potentials with a sinusoidal-type spatial variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We demonstrated that when the GSL has only a magnetic vector potential, the effective Hamiltonian of the structure can be drastically simplified by neglecting the granular details of the potential and considering solely an effective Fermi velocity that is smaller to that of pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We also demonstrated that when both potentials are present in the superlattice the response of the structure becomes nonreciprocal and is characterized by a dispersion characteristic consisting of a tilted Dirac cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Particularly, we showed that for propagation perpendicular to the stratification of the potentials the GSL supports two eigenmodes whose energy dispersion, for a fixed pseudo-momentum, is not linked by an odd symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We showed that in such materials we can obtain extreme wave phenomena such as energy flat bands and regimes where both eigenmodes flow along the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' We envision that by properly tuning the potentials the proposed GSL structure can be operated in regimes characterized by type-I, type-II or type-III Dirac cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Appendix A: Stability of the FDTD Algorithm In a FDTD numerical algorithm the calculations remain stable provided the time step is small enough, below a give threshold [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In what follows we address the stability of the proposed numerical FDTD algorithm to determine the time evolution of the waves in the graphene superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' For simplicity we assume that the medium is spatially homogeneous (V and y A are independent of the spatial coordinates) in the update equations 11a-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Our aim is to characterize the stationary states of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Thus, we look for plane-wave type solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (11) with: 1, 1, 1, , 1/2 1/2 2, 1/2, 1/2 2, 1/2, 1/2 n n p q p q p n n p q p q \uf078 + + + + + − + \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf059 \uf059 = \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf059 \uf059 \uf0e8 \uf0f8 \uf0e8 \uf0f8 , (A1a) 1, , 1 1, , 1/2 1/2 2, 1/2, 1/2 2, 1/2, 1/2 n n p q p q q n n p q p q \uf078 + + + + + + − \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf059 \uf059 = \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf059 \uf059 \uf0e8 \uf0f8 \uf0e8 \uf0f8 , (A1b) where p i p e \uf071 \uf078 = and q i q e \uf071 \uf078 = are the spatial phase-shifts between consecutive nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Furthermore, we consider a time variation of the type 1 1, , 1, , n n p q p q \uf06c + \uf059 = \uf059 and 1/2 1/2 2, , 2, , n n p q p q \uf06c + − \uf059 = \uf059 where \uf06c is a function of the spatial phase-shifts ( ) , p q \uf078 \uf078 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Hence, the proposed FDTD algorithm is stable as long as 1 \uf06c \uf0a3 for arbitrary values of p \uf078 and q \uf078 with 1 p q \uf078 \uf078 = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' A1a-b into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 11a-b and using simple mathematical manipulations we obtain the following system written in a matrix form: ( ) ( ) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 1 2 1 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' 2 2 1 1 1 2 0 1 1 2 n p q t F t n p q F t t V v D i V v D i \uf06c \uf06c \uf06c \uf06c \uf06c − − + + + \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf059 − − \uf044 + \uf044 \uf0e7 \uf0f7 \uf0e7 \uf0f7\uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e7 \uf0f7 = \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf059 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf044 − − \uf044 + \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' with (A2) ( ) 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 y q p p q p q p q x y x y x y x y eA D i i i i \uf078 \uf078 \uf078 \uf078 \uf078 \uf078 \uf078 \uf078 − − − − − − − − − = + + + + \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 − − + + + − − \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf044 \uf044 \uf044 \uf044 \uf044 \uf044 \uf044 \uf044 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (A3) ( ) 1 4 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 y q p p q p q q p x y x y x y x y eA D i i i i \uf078 \uf078 \uf078 \uf078 \uf078 \uf078 \uf078 \uf078 + = − + + + + \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 + − − + − − + \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf044 \uf044 \uf044 \uf044 \uf044 \uf044 \uf044 \uf044 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (A4) To verify the stability of the algorithm we calculate the characteristic equation of the problem, obtained from the kernel of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (A2), which is given by ( ) ( ) 2 2 1 1 1 0 2 t F t V v D D i \uf06c \uf06c \uf06c − + \uf0e6 \uf0f6 − − \uf044 + − \uf044 = \uf0e7 \uf0f7 \uf0e8 \uf0f8 , (A5) Using p i p e \uf071 \uf078 = and q i q e \uf071 \uf078 = it can be shown that: 2 2 2 2 2 2 2 2 1 4 cos sin cos cos 2sin 2 2 2 2 2 q p p q q y x y x y eA D D \uf071 \uf071 \uf071 \uf071 \uf071 − + \uf0e9 \uf0f9 \uf0e6 \uf0f6 = − \uf044 − \uf044 \uf044 + \uf0ea \uf0fa \uf0e7 \uf0f7 \uf044 \uf044 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (A6) The nontrivial solutions \uf06c of characteristic equation are then: ( ) 2 2 2 2 2 1 1 1 2 2 B B C B C C i \uf06c \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0ea \uf0fa = − − + \uf0b1 − − \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 − \uf0eb \uf0fb , (A7) with 2 t V C = \uf044 and ( ) 2 2 0 F t B v D D − + = − \uf044 \uf03e real-valued parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' From this result, it is simple to check that if 2 2 1 0 2 B C \uf0e6 \uf0f6 − − \uf03c \uf0e7 \uf0f7 \uf0e8 \uf0f8 then ( ) 1/2 2 2 2 2 2 2 2 1 1 1 1 2 4 1 B B C B C C \uf06c \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0e6 \uf0f6 = + − + + − = \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0e7 \uf0f7 + \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' (A8) Thus, the algorithm is stable when 2 2 1 0 2 B C \uf0e6 \uf0f6 − − \uf03c \uf0e7 \uf0f7 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' This condition is equivalent to ( ) 2 2 2 2 2 2 2 2 2 2 cos sin cos cos sin 1 2 2 2 2 2 2 2 q p p q q x y y x y t F t eA V v \uf071 \uf071 \uf071 \uf071 \uf071 \uf0e6 \uf0f6 \uf044 \uf044 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf044 + \uf044 \uf044 + \uf03c + \uf044 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf044 \uf0e8 \uf0f8 \uf0e8 \uf0f8 (25) The above inequality should be satisfied for all p \uf071 and q \uf071 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In particular, it is enough to ensure that: ( ) 2 2 2 2 2 2 1 1 1 2 y x y F t x y eA v \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf044 + \uf044 \uf044 + \uf044 \uf03c \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf044 \uf044 \uf0e8 \uf0f8 \uf0e8 \uf0f8 (26) If we consider equally spaced nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' x y \uf044 = \uf044 = \uf044 and time steps given by t F v \uf061 \uf044 \uf044 = , where \uf061 is a real valued positive constant, this condition is equivalent to: 2 1 1 1 2 eA \uf061 \uf03c \uf0e6 \uf0f6 + \uf044 + \uf0e7 \uf0f7 \uf0e8 \uf0f8 (27) Hence, for these superlattices the numerical algorithm stability will depend on the amplitude of the magnetic vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' In case there is no magnetic vector potential ( 0 A = ) we regain the usual formula for the stability of the FDTD algorithm 2 t F v \uf044 \uf044 \uf03c [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' Acknowledgements This work was partially funded by the Institution of Engineering and Technology (IET) under the A F Harvey Research Prize 2018, by the Simons Foundation, by Instituto de Telecomunicações under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' UID/EEA/50008/2020 and SymBreak - UIDB/50008/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQflAFl/content/2301.02480v1.pdf'} +page_content=' E.' metadata={'source': 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0000000000000000000000000000000000000000..59aaab06fd9d9342191e8f54a351fd4f6f73acb8 --- /dev/null +++ b/S9E3T4oBgHgl3EQfDwmh/content/tmp_files/2301.04290v1.pdf.txt @@ -0,0 +1,1903 @@ +Modelling shear thinning polymer flooding using a dynamic +viscosity model +Prabir Daripaa∗ and Rohit Mishrab† +aDepartment of Mathematics +bMechanical Engineering +Texas A&M University +College Station, TX 77843-3368 +January 12, 2023 +Abstract +Two distinct effects that polymers exhibit are shear thinning and viscoelasticity. The shear thinning +effect is important as the polymers used in chemical enhanced oil recovery usually have this property. +We propose a novel approach to incorporate this shear thinning effect through an effective dynamic +viscosity of the shear thinning polysolution. The procedure of viscosity calculation of the polysolution, +although based on a very basic power law model, is based on empiric coefficients which depends on a +spatio-temporally evolving variable namely concentration of polymer. Since viscosity calculation is done +pointwise, the accuracy of the model is higher than what exists in literature. This method has been +integrated with an existing method for a Newtonian physics based model of porous media flows. The +solver uses a hybrid numerical method developed by Daripa & Dutta [1, 2, 3]. The above method solves a +system of coupled elliptic and transport equations modelling Darcy’s law based polymer flooding process +using a discontinuous finite element method and a modified method of characteristics. Simulations show +(i) competing effects of shear thinning and mobility ratio; (ii) injection conditions such as injection rate +and injected polymer concentration influence the choice of polymers to optimise cumulative oil recovery; +(iii) permeability affects the choice of polymer; (iii) dynamically evolving travelling viscosity waves; and +(v) shallow mixing regions of small scale viscous fingers in homogeneous porous media. This work shows +an effective yet easy approach to make design choices of polymers in any given flooding condition. +Article Highlights +• A dynamic viscosity framework for modeling enhanced oil recovery by shear thinning polymer flooding +is proposed. +• This framework is applied to an in-house surfactant-polymer flooding code. +• A new software that solves shear-thinning polymer flooding problem with or without surfactant has +been developed. +Keywords Shear thinning polymer flooding, Enhanced oil recovery, Power law fluids, Data driven modelling, +Immiscible two-phase flow +∗Author for correspondence (email:daripa@math.tamu.edu) +†email:rmishra@tamu.edu +1 +arXiv:2301.04290v1 [physics.flu-dyn] 11 Jan 2023 + +1 +Introduction +Multiphase multicomponent porous media flows are ubiquitous. They occur in environment, climate, sub- +surface, biomedical sciences, etc. just to name a few. They also occur in chemical enhanced oil recovery +(CEOR) by polymer and/or polymer-surfactant flooding, a topic related to the subject of this paper. In these +chemical floods, a polymer-thickened aqueous phase which we will henceforth call polysolution is injected to +displace oil. Assuming polysolution as a Newtonian fluid and neglecting capillary pressure, these flooding +processes are modeled by a nonlinear system of elliptic equation for pressure and hyperbolic equations for +pressure and saturation (see Daripa et al. [4]). An increase in viscosity, due to polymer, of the displacing fluid +inhibits fingering instability and increases the saturation behind the front sweeping the oil. In that study, +it has been established through nonlinear wave analysis and numerical simulation using a front tracking +method [5] that both of these effects of polymer improve oil recovery. This method is not applicable as it is +when capillary pressure is taken into account (in particular when surfactant is also used). +In Daripa & Dutta [2], authors extended the polymer model of Daripa et. al [4] to include the effects +of capillary pressure and also of another component, namely surfactant. This new model which we call +SP-model (‘S’ stands for surfactant and ‘P’ stands for polymer), is a coupled nonlinear system of elliptic +and transport equations. This SP model and the numerical method are very briefly reviewed in §2.1 and §3 +which is adequate for the development of the remaining part of this paper. Convergence of this numerical +method has been proven in Daripa & Dutta [3]. A Matlab code [1] available on Github has been developed +implementing this method. Using this code, numerical simulations have been performed to validate this +method qualitatively and quantitatively (see Daripa & Dutta [3]), and to evaluate the relative performance +of several floods in homogeneous and heterogeneous reservoirs. As we will see below, this model and the +numerical method are the building blocks for the data driven model and the method that incorporates +the shear thinning effect of polymer. In this paper, we are interested in including non-Newtonian effects +of polymer in SP-flooding model, develop a numerical method to numerically solve this new model, and +then perform numerical simulations to validate this method. In particular, we perform simulations with +polymer-flooding exclusively by setting the concentration of surfactant to zero in the SP model. +This +provides an understanding of data driven non-Newtonian effect of polymers on viscous fingering and oil +recovery properties of polymer floods. In turn, as we will see, this is helpful in the selection of a polymer for +maximum oil recovery at any given flooding condition. +Polymers that are commonly used in chemical enhanced oil recovery are usually stiff polymers of shear +thinning type. Thus, stability of such a displacement process where the displacing fluid is shear thinning plays +an important role. It is known from theoretical studies [6, 7] that shear thinning behavior of non-Newtonian +displacing fluids suppresses fingering instability. In these and many other studies, shear thinning fluids are +generally modeled as power-law fluids which depend on two indices, shear rate and density of polysolution. +These are taken as constants in linear stability studies. However, in practice such stability results are of +little use in CEOR since power law indices are functions of the type of polymer and its concentration which +change in space and time in the flow domain (see §2 and §3 below). Stability of such a displacement process +with time and space dependent parameters in the power law has not been studied to-date. Even if it were, +such stability results are likely to be of little value for practical applications to CEOR since values of these +parameters evolve in space-time and are not known a priori. For this and many applications related reasons, +there is a need to better understand and model the displacement processes with non-Newtonian fluids in +general. It appears that simulation is the only viable alternative for accurate prediction of flow features, +flow instabilities and other oil recovery performance measures. +Towards this end, it is worth mentioning some other works from a huge body of literature on CEOR. +CEOR methods increase oil recovery by altering fluid-fluid and/or fluid-rock interaction in the reservoir by +reducing interfacial tension (IFT) and/or altering the viscosity of the injected fluid for mobility control. +Another way the injected chemicals increase recovery is by altering the wettability of the rock to increase oil +permeability [8][9]. Pope et al. [10] reported that polymer flooding can increase oil recovery up to 12-30 % +original oil initially in-place (OOIP). The idea behind polymer flooding is to introduce an intermediate layer +of higher viscosity fluid which increases the cumulative oil recovered by reducing mobility ratio and thereby +2 + +reducing the finger formation of less viscous fluid (water) displacing the high viscous fluid (oil). Although this +method has been used widely in industry, it still lacks a fundamental understanding of the role of polymers +in CEOR. Sheng et al. [11] in their review paper on the status of polymer flooding listed different challenges +associated with polymer flooding. The major challenge is associated with the basic underlying physics of +polymer and its behavior at variable rheological conditions such as temperature, salinity, permeability among +others. Due to this highly variable nature of polymers it becomes a challenge in predicting oil recovery by +simulation or by experiments. +A polymer working in a specific set of conditions might not be viable in a different oil field completely. +The complexity around usage of polymer is so profound that there is no general consensus on a very basic +injection parameter, namely, the amount of polymer injected. Between 1970s to 1980s the amount of polymer +injected is about 100-200 ppm PV (pore volume) in Chinese projects which changed to 500 to 600 ppm PV +in early 1990s. This changed again in 2000s to 400-500ppm PV [12]. Niu et al. [13] reports that when the +amount of polymer injected is larger than 400ppm PV, incremental oil recovery becomes less sensitive to +the amount of polymer injected. Levitt et al. [14] finds a similar trend, namely increasing polymer (HPAM) +viscosity from 3 cP to 60 cP does not significantly change recovery from two pore volumes (PV) of tertiary +polymer injection. Apart from the above mentioned uncertainty, another reason polymer flooding may not be +the ideal flooding technique is the higher injection rate required to inject polymers. This is overcome by yet +another advanced CEOR method termed as the Surfactant-Polymer (SP) flooding. SP flooding is a chemical +oil recovery method that introduces a layer of surfactant before injecting the polymer which mobilizes the +oil by reducing the capillary pressure. However, to better understand the non-Newtonian effects of polymers +we are only focusing on shear thinning polymer flooding in this paper. +Prior research conducted in the field of Saffman-Taylor instabilities suggests that instabilities arise when +a less viscous fluid displaces a fluid with higher viscosity. It has been seen from theoretical studies [7, 6] +that shear thinning behavior of non-Newtonian displacing fluids may minimize the instability effects (finger +formation) on the interface separating the displaced and displacing fluids. The polymers that are currently +used in field operations are considered inelastic shear thinning fluids. There have been numerical studies +analyzing the inelastic behavior of polymers. The study by Durst et al. [15] shows that elastic behavior is +observed only above a critical Deborah (De) number. This number is very small when we consider velocity +of shear thinning fluid in porous medium. This has been shown experimentally by Marshall & Metzner [16]. +Therefore, the focus of efforts in the past has been to analyze the shear thinning behavior of non-Newtonian +fluids as the elastic behavior has less significance in this field of application. +After establishing the importance of shear thinning effects of polymer, the next big question is which +polymer is the best for oil recovery. While there is no one good answer to this question multiple studies +indicate a couple of good choices. The choice of polymer plays a crucial role in CEOR. The benefits of +polymer flood is contingent on the extent of polymer retention in the oil field with minimum retention +favorable for high recovery [17]. The three major mechanisms identified by Willhite et al. [18] for polymer +flow in porous media are polymer adsorption, mechanical entrapment and hydrodynamic retention. +Apart from the performance of polymer in oil recovery, the choice must also be made based on its +environmental impacts. Natural polymers such as Xanthane and Schizophyllan are less detrimental to the +environment when compared to hydrolyzed polyacrylamide (HPAM), a widely used industrial polymer which +can cause environmental problems (increases the difficulty in oil-water separation, degrades naturally to +produce toxic acrylamide and endanger local ecosystem). Agi et al. [19] reviewed different natural polymers +in reference to its application in enhanced oil recovery. Many previous studies show non-Newtonian behavior +of natural polymer. For example, Liu et al. [20] in their rheology study of Cassava starch finds that the +polymer exhibits shear thinning behavior i.e. viscosity decreases with increasing strain rate. Most of the +other natural polymers show similar behavior. These natural polymers in an eco-friendly way enhances oil +recovery with higher sweep efficiency. This sweep efficiency can be improved if these are used as nanofluids +[21]. Nanoparticles have advantages of being tolerant to high salinity, high temperature and retention in +highly permeable reservoir. However, there are concerns on the cost of full-scale field implementation and +toxicity of nanofluids. Agi et al [22] identified Cissus populnea(CP) as an important biopolymer and showed +an efficient process to synthesize Cissus populnea nanofluid (CPNF). They showed at same concentrations +3 + +the viscosity of CPNF is higher than CP which in turn is higher than commonly used natural polymer +– Xanthane. While Xanthane shows a decrease in viscosity with increasing temperatures, CP and CPNF +show an evident increase in viscosity. All these recent studies show promise on the possibility of improving +enhanced oil recovery if the properties of the polymers are understood and applied carefully. Introduction of +biopolymer nanoparticles is another emerging field where focus must be given to analyze the fluid properties +not just experimentally but through polymer flood simulations. +Extensive numerical and computational studies for tertiary oil recovery is available in the literature. +Daripa et al. [4] provids a comprehensive analytical study of instability control in tertiary oil recovery. +Afsharpoor et al. [23] uses upper convected Maxwell equation to model strong extensional flow effect to +relate the shear stress with the shear rate. Clemens et al. [24] performs a pore-scale evaluation of polymers +displacing oil using experiments and CFD simulation. The experiments show shear thinning behavior of +the polymer. CFD simulations show that viscosity is lower at pore throats than that in the pore. This +shear thinning behavior affects the displacement efficiency of the polymer flooding. However, the power law +model with fixed parameters (n, ϵ) is usually considered in CFD simulations when in relality these should be +functions of polymer concentration. This assumption of constant parameters becomes even more detrimental +in predicting recovery in real flooding simulations as the concentration of the polymer changes in space and +time. Nandwani et al. [25] models surfactant flooding process using ANSYS FLUENT (a commercial CFD +code) and shows that ultralow IFT, minimal fingering and low diffusion rate of the surfactant in oil are +responsible for higher oil recovery. For Newtonian fluids, Daripa [26] studies a system of three fluids where +the driving fluid drives the fluid in the middle which simultaneously drives the third fluid. He shows that +there exists a critical viscosity for the middle fluid which leads to the most stable setup i.e. least finger +growth. Recently, Manzoor et. al. [27] tested the effects of pressure variations on heavy oil recovery process +in a homogeneous glass beads-packed physical model. The study compares the simulation results with the +experiments. The main conclusions are that maximum pressure and periodic pressure variations have the +potential to enhance heavy-oil recovery. Moreover, they report that the numerical simulations are highly +sensitive to uncertainty in permeability, porosity, diffusion coefficient of polymer, number of grid points and +heavy-oil density. +Xie et al. [28] perform experiments on a microchip with heterogeneous porous structures where oil is +displaced by dispersed polymer. They show dispersion effect even when the polymer particles are smaller +than the pore throat. So the plugging effect is not the major mechanism for preferential flow control by +dispersed polymers. Simulations [28][29] for this type of flow show that dispersed polymers smartly controls +the preferential flow by inducing pressure fluctuations and thereby increasing efficiency. Xie et al. [30] uses +Herschel-Bulkley model to find viscosity of the polymer in a two phase (polymer aqueous solution and oil) +Lattice-Boltzman simulation. They show that apart from modelling shear rheological behavior of polymer +it is also important to model the three phases (polymer, water and oil) individually. Although the Herschel- +Bulkley model has been widely used in the literature to model pseudo-plastics, we use the power law model. +The major difference between the two is that the power law model has one less empirical parameter. The +reason for this choice is the widespread availability of the power-law coefficients for different polymers used +in this study. The different rheological models important for polymer flooding simulations are shown in +figure 1. +Surfactant-polymer flooding simulations helps us in understanding the complex phenomenon of multi- +component fluids moving across a porous region. The simulations get even more challenging as one of the +components (polymer) behaves as a non-Newtonian fluid i.e. shear rate is not a linear function of strain +rate. To account for this, we propose and numerically study a data driven approach to incorporate shear +thinning effect of polymer (shear thickening case is similar) in a hybrid numerical method developed by +Daripa & Dutta [2, 3]. The data driven approach proposed here is based on the recognition that (i) shear +thinning property can be modeled by an effective viscosity of polysolution based on a strain rate using a +power law model, (ii) values of the power law parameters in the power law model that fit for the given +shear rate and concentration of polymer vary and can be estimated from curve fit based on experimental +results, (iii) effective viscosity can be used in an otherwise Darcy’s law based Newtonian model of SP floods +without changing the fundamental equations yet implementing an accurate physical model which reduces +4 + +Figure 1: Rheological models +to the Newtonian rheology with a constant viscosity in regions where the polymer concentration is zero, +and (iv) shear thinning effect of any displacing fluid is easily incorporated in any existing Darcy’s law and +Newtonian fluid mechanics based code. This paper is laid out as follows. Data driven mathematical model +of SP flooding, which consists of the governing equations for SP flooding based on Newtonian fluid dynamics +and Ostwald de Waele power law model for shear thinning polysolution, is described in §2. +Numerical +method is described in §3. Numerical results in rectilinear and quarter five-spot geometries are presented in +§4. Finally we conclude in §5. The algorithm and the flow chart for the method are given in Appendix A.1. +The method for calculation of finger width is given in Apprndix A.2. +2 +Non-Newtonian Mathematical model +The non-Newtonian mathematical model is built from combining Newtonian model of porous media flow +for multi-phase multi-component porous media flow involving polymer as a component among others with +power law model for the shear thinning fluid. This is discussed in this section. Next section discuses the +numerical method. +2.1 +Newtonian model of SP flooding +The Newtonian model of Surfactant-Polymer (SP) flooding in porous media is used here as a prototype +model. +In this section, we recall from Daripa & Dutta [2] some relevant facts about this model. +It is +given by the following coupled nonlinear system of elliptic equation (1)2 for global pressure p and transport +equations (2), (3) and (4) for saturation s, concentration c of polymer, and concentration Γ of surfactant +respectively. +v = −K(x)λ(s, c, Γ)∇p, +−∇ · (K(x)λ(s, c, Γ)∇p) = qa + qo, +(1) +φ∂s +∂t + ∂fa +∂s v · ∇s + ∇ · +� +D∂pc +∂s ∇s +� += gs − ∂fa +∂c v · ∇c − ∂fa +∂Γ v · ∇Γ − ∇ · +� +D∂pc +∂Γ ∇Γ +� +, +(2) +φ∂c +∂t + +�fa +s v + D +s +∂pc +∂s ∇s + D +s +∂pc +∂Γ ∇Γ +� +· ∇c = gc, +(3) +5 + +1.0 +Newtonian Model +PowerLaw Model +Herschel-Bulkley Model +0.8 +Bingham Plastic Model +Stress +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +Non-Dimensional Shear Rate [-]φ∂Γ +∂t + +�fa +s v + D +s +∂pc +∂s ∇s + D +s +∂pc +∂Γ ∇Γ +� +· ∇Γ = gΓ. +(4) +The terms in the above equations are defined in Table-1 and after Table-1. +Term +Description +K(x) +Absolute Permeability Tensor +λ(s, c) +Total Mobility +qa, qo +Source/sink terms for global pressure equation +s, c, Γ +Saturation (volume fraction), Concentration of polymer, Concentration of surfactant +φ +Porosity of the medium +pc +Capillary pressure +krj +(j=o) Relative permeability of oil; (j=a) Relative permeability of aqueous phase +µj +(j=o) Viscosity of oil (constant); (j=a) Viscosity of aqueous phase +fj +(j=o) fractional flow function of oil; (j=a) fractional flow function of the aqueous phase +gs, gc, gΓ +Source terms for saturation, concentration and surfactant transport +Table 1: Nomenclature +The elliptic equation (1)2 solves for a new function called global pressure which is defined in terms of a +thermodynamic pressure and capillary phase pressures so that we have a canonical elliptic equation (1) for +the global pressure p. The following relation between this global pressure and phase pressures is taken from +Daripa & Dutta [2]. +p = 1 +2(po + pa) + 1 +2 +� s +sc +(fo − fa)(ζ, c, Γ)dpc +dζ (ζ, Γ)dζ + 1 +2 +� Γ +Γc +(fo − fa)(s, c, ξ)dpc +dξ (s, ξ)dξ +− 1 +2 +� +ηc(s, c, Γ) +� ∂c +∂xdx + ∂c +∂y dy +� +− 1 +2 +� +ηs(s, c, Γ) +� ∂s +∂xdx + ∂s +∂y dy +� +− 1 +2 +� +ηΓ(s, c, Γ) +�∂Γ +∂xdx + ∂Γ +∂y dy +� ++ C, +(5) +where C is a reference pressure which takes the place of an integration constant in the calculations. In the +above, +ηc(s, c, Γ) = +� s +sc +∂ +∂c (fo − fa) (ζ, c, Γ)dpc +dζ (ζ, Γ)dζ + +� Γ +Γc +∂ +∂c (fo − fa) (s, c, ξ)dpc +dξ (s, ξ)dξ, +(6a) +ηs(s, c, Γ) = +� Γ +Γc +∂ +∂s (fo − fa) (s, c, ξ)dpc +dξ (s, ξ)dξ, +(6b) +ηΓ(s, c, Γ) = +� s +sc +∂ +∂Γ (fo − fa) (ζ, c, Γ)dpc +dζ (ζ, Γ)dζ, +(6c) +where sc is the value of the aqueous phase saturation at which pc(sc, Γ) = 0 and Γc is the surfactant +concentration value defined similarly by pc(s, Γc) = 0. +The aqueous phase viscosity µa in this Newtonian model depends linearly on c: µa = µw + βc where µw +is the constant viscosity of water and β is a polymer specific constant. Thus, this system does not include +non-Newtonian effect of polymer. The phase relative permeability krj depends on saturation s and capillary +pressure dependent residual saturation sr (see [2]). Since capillary pressure depends on concentration Γ of +surfactant, the phase relative permeability krj depends on both s and Γ. The total mobility λ = λa + λo +where λj = krj/µj is the mobility of phase j. Therefore λa depends on the triplet (s, c, Γ) but λo depends +only on (s, Γ). However, both of these mobility we write generically as λj(s, c, Γ) for convenience. The +6 + +function fj(s, c, Γ) = λj(s, c, Γ)/λ(s, c, Γ) is the fractional flow function of the phase j and the coefficient +D(s, c, Γ) = K(x)λo(s, Γ)fa(s, c, Γ). +The source/sink terms in the elliptic equation (1)2 are given by (see Daripa & Dutta [2]) +qa = +� +� +� +� +� +Q +−(λa/λ) Q ; +0 +qo = +� +� +� +� +� +0 +−(λo/λ) Q +0 +at +� +� +� +� +� +xi = (0, 0) +(Source) +xp = (1, 1) +(Sink) +x ∈ Ω \ {(0, 0) ∪ (1, 1)} (Elsewhere) +, +(7) +and three source terms in the transport equations are defined as (see Daripa & Dutta [2]) +gs = +� +(1 − fa)Q +0 +gc = +� +(ci − c)Q/s +0 +gΓ = +� +(Γi − Γ)Q/s +0 +at x = +� +xi +(source) +Ω \ {xi} (elsewhere) +, +where Q is the volumetric injection/production rate. The following initial and boundary conditions are +prescribed. +s(x) = s0(x), +c(x) = c0(x), +Γ(x) = Γ0(x); +x ∈ Ω, +t = 0, +(8a) +∇s · ˆn = 0, +∇c · ˆn = 0, +∇Γ · ˆn = 0 +& +vj · ˆn = 0; +x ∈ ∂Ω, +t > 0, +(j = a, o), +(8b) +where ˆn is a unit vector normal to ∂Ω. +This SP-flooding model reduces to polymer flooding model if we set surfactant concentration Γ = 0 in +the above equations. This eliminates the transport equation (4) for surfactant, the source term gΓ, the terms +involving Γ from equations (2) and (3), and dependency of D(s, c, Γ), λ(s, c, Γ) and fractional flow functions +fj(s, c, Γ) on Γ. This completes the description of the Newtonian model of Surfactant-Polymer (SP) flooding. +Finer details can be found in Daripa & Dutta [2]. +2.2 +Model of shear thinning +To include shear thinning effect of polymers in the above model, following Ostwald de Waele power law +model (9) is used for the calculation of the effective viscosity µa of the aqueous phase. +µa = ρε˙γn−1, +(9) +where shear rate ˙γ is given by +˙γ = 2 +� +| ΠD |, +(10) +in terms of the second invariant ΠD of the strain rate tensor given by [31] +ΠD = −1 +4 +�� +∂u +∂y + ∂v +∂x +��2 ++ ∂u +∂x +∂v +∂y . +(11) +The power law (9) depends on density ρ, strain rate ˙γ, and two parameters (ε, n). The density ρ(x, t) is the +weighted sum of water and polymer densities based on the local in time and space value of concentration, +c(x, t), of polymer which evolves according to its transport equation (3). The strain rate ˙γ relates to the +second invariant of the strain rate tensor, ΠD, according to (10) [31]. +The strain rate ˙γ is calculated from the velocity field (u, v) = v which is obtained as part of the solution +of model (1) through (4) described in the next section §3. +The parameters (ε, n) depend on the type +and concentration of polymers. +This is exemplified in Fig. 2 in which experimental values of ε and n +versus concentration for two shear thinning polymers, namely Xanthane and Schizophyllan, are shown from +Hatscher [32] and Lindner et al. [7]. Values of these two parameters in the flow domain at any specific time +are found from curve fitting local values of c with experimental data similar to the ones shown in Fig. 2 for +the specific polymers in use. Local values of density ρ of polysolution is also found from local values of c, +7 + +concentration of polymer. Similarly, values of shear rate ˙γ in the flow domain at any specific time are found +from velocity field (u, v) = v which is obtained as part of the solution of the model (1) through (4) described +in §3. All these values are then used in the power law (9) to find the local effective viscosity µa(x, t). This +affects the model equations (1)-(4) through the functions λa, λ, fa and D appearing in these equations which +depend on µa(x, t). This procedure is a significant improvement and provides accurate values of interest +than conventional practices in which usually power law parameter values are taken at fixed values of polymer +concentration at the injection point. In §3, we show this by integrating shear thinning model in the numerical +method. +(a) +(b) +(c) +(d) +Figure 2: Top row: ε vs concentration for Xanthane (L) n Vs concentration for Xanthane (R)[7] +Bottom row: ε vs concentration for Schizophyllan (L) n vs concentration for Schizophyllan (R)[32] +This nonlinear coupling of effective viscosity µa(x, t) with the system (1)-(4) and influence of experimental +data make it increasingly hard to predict associated response of viscosity variation in the flow domain, flow +features, fingering instability and oil recovery performance measures to shear thinning. Therefore, accurate +physical interpretation of numerical solution of this data driven model problem is not going to be easy. +Regardless, various numerical results obtained using this data driven numerical method discussed in the +next section are analyzed and interpreted later in terms of physics. +8 + +600 +Data +Curve fit +500 +400 +lus +- +m +200 +100 +0 +250 +500 +75010001250150017502000 +Polymerconcentration[wppm]1.0 +Data +Curve fit +0.8 +0.6 +n +0.4 +0.2 +0.0 +0 +250 +500 +75010001250150017502000 +Polymerconcentration[wppm]600 +Data +Curve fit +500 +400 +snj +- +300 +m +200 +100 +0 +250 +500 +75010001250150017502000 +Polymerconcentration[wppm]1.0 +Data +Curve fit +0.8 +0.6 +n +0.4 +0.2 +0.0 +0 +250 +500 +75010001250150017502000 +Polymerconcentration[wppm]3 +Numerical Method +3.1 +A Brief Review of the Numerical Method for the Newtonian Model +We first briefly review from Daripa & Dutta [2] the hybrid numerical method for solving the mathematical +model of §2. In this method, global pressure equation (1) is solved using a Discontinuous Finite Element +Method (DFEM) [33]. The system of transport equations (2), (3), & (4) is solved by a time-implicit finite +difference method based on the Modified Method Of Characteristics (MMOC) [34, 35]. Two different types of +grid are used in the numerical method: the finite difference grid shown in Fig. 3(a) for the transport equations +and the finite element grid shown in Fig. 3(b) for the pressure equation. Thus, transfer of data between +these two grid system during numerical solution process is required in this numerical method because of the +coupling of elliptic and transport equations. Details about the computational grid, the numerical method, +and the entire computational algorithm for the above Newtonian SP flooding model are given in Daripa & +Dutta [?]. The convergence of the method has been proven in one-dimension in Daripa & Dutta [3]. +(0, 0)− Source +(1, 1)− Sink +Ω− +Ω+ +Σ +(xi, yj) +(a) Uniform FD grid for the transport equations +(0, 0)− Source +(1, 1)− Sink +Ω− +Ω+ +Σ +κ +(b) Uniform FE mesh for the pressure equation +Figure 3: Discretization of the computational domain for (a) the Transport Equations and (b) the Pressure +Equation. +3.2 +Numerical Method for the non-Newtonian Model +The numerical method discussed above in §3.1 is adapted for this shear thinning model by first calculating the +aqueous phase (polysolution) viscosity µa(x, t) using data driven power law model (9) as discussed in §2.2. +The effective viscosity and appropriate associated quantities which depend on this viscosity are calculated +at all grid points of both types of grid. For example, total mobility λ which the elliptic equation (1) depends +on is calculated at all finite element grid points at every time level since effective viscosity µa changes in +time which λ depends on as explained in §2.2. Similarly, λa, λ, fa and the function D which the transport +equations (2), (3), & (4) depend on are calculated at all finite difference grid points at every time level. All +this adds to the computational cost to study the shear thinning effect of polymer. In appendix A.1, the +entire algorithm that implements this numerical method and the flow chart are given. +4 +Results +Three sets with two simulations per set (for two polymers) are carried out for homogeneous and two different +heterogeneous permeability fields. Extensive comparisons have been made to validate this approach. Results +9 + +are summarised below. +4.1 +Shear thinning induced nonuniform viscosity and travelling waves +As discussed in §2.2, variable viscosity profile in a flow domain due to shear thinning effect is very hard +to predict a priori due to the influence of experimental data and permeability field of the porous media. +It is even harder to predict its development in time a priori as the aqueous phase viscosity µa, calculated +in space and time from the shear thinning power law model (9), is nonlinearly coupled with the elliptic +equation for pressure and the transport equations for saturation and polymer as discussed earlier. However, +a posteriori one can analyze such profiles to get an insight into the development of any phenomena and/or +distinct patterns discovered in the process. Here we exemplify this by analyzing the evolution of such profiles +in a quarter five spot simulation in heterogeneous porous media. +Figure 4: Logarithmic permeability plot from the SPE10 benchmark dataset on a 30 × 30 grid. +(a) +(b) +Figure 5: Viscosity profiles at three different time levels along the horizontal mid-section for polymers (a) +Xanthane and (b) Schizophyllan. +10 + +UpperNesstypeformation +30 +9 +8 +25 +20 +6 +5 +15 +4 +3 +10 +2 +1 +5 +0 +5 +10 +15 +20 +25 +30t=500 +t=1000 +t=1500 +20 +- +IViscosity +15 +Non-Dimensional +10 +5 +0 +F5 +10 +15 +20 +25 +30 +Non-Dimensional XDistance[-]t=500 +16 +t=1000 +t=1500 +14 +I Viscosity [-] +12 +Non-Dimensional +10 +8 +6 +4 +2 +5 +10 +15 +20 +25 +30 +Non-Dimensional X Distance [-]Simulations were carried out in a quarter five spot geometry with heterogeneity shown in Fig. 4. Fig. 5(a) +and Fig. 5(b) show plots of viscosity profile at the horizontal mid-section at three different times for the two +polymers. We see in these figures several localized travelling waves (peaks) in viscosity at each time level for +both the polymers. Development of these patterns appear to be a consequence of shear thinning property +of the displacing fluids since we notice in these two figures that these patterns and their speeds are different +for these two polymers even though simulations for both the polymers were carried with the same injected +polymer concentration (IPC) and same injection rate (IR). These patterns and trends also change as we +change the IPC and IR which will be discussed in the future when we have mathematically analyzed this +complex model and have a better quantitative understanding of the nonlinear complex dynamics hidden in +this model. +4.2 +Competing effects: viscosity ratio vs shear thinning +Shear thinning polymers during its passage through the pores undergo shear stresses which lead to a reduction +in viscosity. This viscosity reduction is proportional to the local velocity gradients which relates to the +injection rates. So, if the polymer were to behave as a Newtonian fluid, high injection rates would mean high +oil recovery. But due to the shear effect, a high injection rate may not necessarily correspond to high oil +recovery. Therefore, modelling polymer as a non-Newtonian fluid gives a very practical benefit of assessing +different injection rates and testing an ideal injection rate which helps in maintaining the viscosity ratio +while avoiding the viscosity reduction due to shear thinning. This competing effect can be seen in Fig. 6 +where the injection rates were varied and cumulative oil recovery (COR) recorded for time step t=100. +Figure 6: Cumulative oil recovered in heterogeneous rectilinear polymer simulations of Xanthane and Schizo- +phyllan for different injected polymer concentration (IPC) (300wppm and 1500wppm) and normalized injec- +tion rate (IR) (IR=0.084, 0.167, 0.250, 0.500, 1.00). at t=100 +It can be seen in the figure that at lower injection rates Xanthane has slightly higher COR as compared +to Schizophyllan at high IPC. But at these low injection rates, Schizophyllan has a higher COR at low IPC. +As we increase the injection rates, both the polymers are subject to higher shear rates. Due to this, the +advantage they provide by maintaining the viscosity ratio will start to diminish depending on the power +law parameters of the polymer. We see that reduction in viscosity is higher for Schizophyllan because of +diminishing returns in COR with increasing injection rate. For Xanthane, on the other hand, reduction in +viscosity is not at the same rate as for Schizophyllan. Another interesting observation is flipping trend in +COR for Schizophyllan. At low IR, higher IPC results in higher COR but at high IR the trend is flipped +11 + +1.0 +1500wppmXanthane +1500wppmSchizophyllan +300wppmXanthane +300wppmSchizophyllan +Oil Recovered +0.8 +0.6 +ICumulative +Normalized +0.4 +0.2 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Injection Rate [-]and lower IPC results in higher COR. This non-linearity is a clear indication of the two competing effects of +mobility ratio and viscosity reduction due to shear thinning. +This competitive behavior is also observed in Fig. 7 for simulations in rectilinear geometry with hetero- +geneity generated using the following permeability tensor K(x) taken from [36] and shown in Fig. 10. +K(x) = 50 +� +0.5(1 − 10−7)(sin (6π cos x) cos (4π sin (3y)) − 1) + 1 +� +1, +(12) +where 1 is the second order identity tensor. The shear rate is higher in regions of high permeability. This +directly relates to the ease with which the flow moves in these regions. +Due to high velocity gradient, +the flow experiences higher shear which eventually affects the viscosity of polysolution. +Even when the +concentration is same, viscosity varies for the four cases due to differences in the shear. Therefore, it can +be safely concluded that while high concentration leads to higher viscosity, it is the permeability field that +affects the flow movement through shear. Therefore, a shear thinning fluid will perform differently in different +permeability fields. +Figure 7: Contour plots of shear rate (top), polymer concentration (middle), viscosity (bottom) for rectilinear +heterogeneous polymer flood simulations comparing different injection conditions and polymers at t=100 +4.3 +Simulation in rectilinear geometry +4.3.1 +homogeneous porous media +First, simulations in rectilinear geometry are carried out for a homogeneous permeability field with polymers +Xanthane and Schizophyllan in separate experiments. +Fig. 8 shows the saturation fields obtained with +Xanthane and Schizophyllan at four time levels. This shows comparative development of the level sets of +saturation due to the effect of two polymers that respond differently to the shear in the field at flooding +conditions IR=120,000 and IPC=1500 wppm. It shows that the flow with polymer Xanthane moves faster +12 + +IPC=300wppm +IPC=300wppm +IPC=1500wppm +IPC=1500wppm +IR = 10,000 (Xanthane) +IR=10,000 (Schizophyllan) +IR = 120,000 (Xanthane) +IR = 120,000 (Schizophyllan) +Shearrate +Polymer +concentration +Viscositythan polymer Schizophyllan in this case as IR and IPC are set to their respective highest values in the range +of values tested. At such low concentration of polymer Xanthane, power law index n is closer to 1 and +hence the viscosity has very little shear rate dependency (see (9)). It behaves almost like a non-uniform +Newtonian fluid, non-uniform embodies the fact that aqueous phase viscosity will still vary in space and +time due to the data and time dependency of ρ and ε which the power law (9) depends on. On the other +hand, for Schizophyllan viscosity depends on shear rate with the index n around 0.7 and affects the viscosity +profoundly in comparison to Xanthane. This difference justifies the difference in the speed with which the +saturation fronts move in these two cases. Perhaps polymer Xanthane undergoes significant shear thinning +in comparison to Schizophyllan and hence moves faster due to reduction in viscosity. +Fig. 8 also shows narrow mixing regions of mild viscous fingers for both the polymers, though the mild +fingers in the case of Schizophyllan are somewhat more pronounced. These fingers are mild in comparison +to classical ones for displacements involving Newtonian flows because mobility ratio across the interfaces in +this figure is position dependent and can change from favorable to unfavorable constantly along an interface +because of data dependent shear thinning property of the fluids as discussed earlier. Finger widths in these +cases have been computed using the procedure described in appendix A.2. Fig. 9(a) shows finger width +growth for polymers Xanthane and Schizophyllan at IR=10,000 and IPC=300 wppm. Fig. 9(b) shows the +effect of finger width growth on cumulative oil recovery (COR). It is evident that Schizophyllan with higher +finger width shows higher recovery as compared to Xanthane. However, this trend is reversed for higher values +of IPC and IR. Plots in Fig. 9(c),(d) show Xanthane to have higher finger width and COR at IR=120,000 +and IPC=1500 wppm. This clearly shows the importance of choosing a polymer based on the IR and IPC +conditions in a given polymer flood. +Figure 8: Temporal evolution of saturation with polymers Xanthane (top) and Schizophyllan (bottom) at +IR=120000 and IPC=1500wppm in rectilinear polymer flood simulation with homogeneous permeability +field. +4.3.2 +heterogeneous porous media +Next, heterogeneous permeability field shown in Fig. 10 is used for simulation in core flood. The saturation +fields and COR are shown in Fig. 11 and Fig. 12 respectively. Although most of the flooding is now affected +by this heterogeneous permeability field, but for the same IR and IPC clear differences can be noticed on +the predicted COR for Schizophyllan and Xanthane. This is a clear indicator that choice of polymer also +depends on the permeability field. For the given case, Xanthane seems to outperform Schizophyllan in terms +13 + +500 +1000 +1500 +0.75 +0.4 +0.3 +0.2 +0.2 +0.4 +0.6 +0.8 +0.2 +0.4 +0.6 +0.8 +0.2 +0.4 +0.6 +0.8 +0.2 +0.8 +500 +1000 +1500 +0. +0.8 +0. +0.6 +0.5 +0.4 +0.3 +0.2 +0.2 +0.4 +0.6 +0.8 +0.2 +0.4 +0.6 +0.8 +0.2 +0.4 +0.6 +0.8 +0.2 +O. +0.6(a) +(b) +(c) +(d) +Figure 9: COR and MFW from rectilinear homogeneous polymer flood simulation. Top row: Comparison +of Xanthane and Schizohyllan at IR=10,000 and IPC=300wppm (a) Mean Finger Width (MFW) and (b) +Cumulative Oil Recovery (COR). Bottom row: Comparison of Xanthane and Schizohyllan at IR=120,000 +and IPC=1500wppm (c) Mean Finger Width (MFW) and (d) COR +Figure 10: The heterogeneous permeability field given by (12) and taken from [36] is plotted in a rectilinear +geometry with a 60 × 60 spatial resolution. Red regions represent higher permeability while the blue regions +represent lower permeability. +14 + +35 +300wppmXanthane +300wppmSchizophyllan +30 +25 +n Finger Width [-] +20 +15 +Mean +10 +5 +0 +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +Non-Dimensional Time [-]1e7 +1.0 +300wppmXanthane +300wppmSchizophyllan +0.8 +Cumulative Oil Recovered [ +0.6 +0.4 +0.2 +0.0 +0 +500 +1000 +1500 +2000 +2500 +Non-Dimensional Time [-]20.0 +1500wppmXanthane +1500wppmSchizophyllan +17.5 +15.0 +12.5 +Width +Finger +10.0 +Mean +7.5 +5.0 +2.5 +0.0 +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +Non-Dimensional Time [-]1e8 +1.2 +1.0 +I Recovered | +0.8 +Cumulative Oil +0.6 +0.4 +0.2 +COR:1500wppmXanthane +0.0 +COR:150oOwppmSchizophyllan +0 +500 +1000 +1500 +2000 +2500 +Non-Dimensional Time [-]60 +55 +45 +50 +40 +45 +35 +40 +30 +35 +30 +25 +25 +20 +20 +15 +15 +10 +10 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +55 +60of COR. But given a different set of IR, IPC and permeability field, the choice of polymer may change. In +practice, therefore permeability fields also influence the choice of polymer and other outcomes from shear +thinning polymer flooding. The fingering nature of the flow pattern seen in Fig. 11 is due to channeling +effect as there are regions of high permeability. +Figure 11: Temporal evolution of saturation with polymers Xanthane (top) and Schizophyllan (bottom) at +IR=120000 and IPC=1500 wppm in a rectilinear geometry with heterogeneous permeability field shown in +Fig.10. +(a) +(b) +Figure 12: (a) Comparison of COR for Xanthane and Schizohyllan at IR=10,000 and IPC=300wppm (a) and +(b) Comparison of COR for Xanthane and Schizohyllan at IR=120,000 and IPC=1500wppm in rectilinear +heterogeneous polymer flood simulation. +4.4 +Quarter five spot simulation +Finally, a quarter five spot geometry with a heterogeneous logarithmic permeability field shown in Fig. 4 is +used for simulation. Fig. 13 and Fig. 14 show the saturation field and COR for two different polymers. Here +again the influence of the permeability field is evident. Also the comparative simulations show significant +difference as the variable permeability field leads to higher shear and subsequent variation for different +polymers used in the flooding. In simulations with heterogeneous permeability fields, mean finger width +15 + +2000 +2500 +YYTYT +YYTTY +0.9 +0.8 +0.7 +0.7 +0.6 +0.6 +0.6 +0.5 +0.5 +0.5 +0.4 +0.3 +0.40.3 +0.4 +0.3 +0.3 +0.2 +0.2 +0.2 +0.2 +0.4 +0.8 + 0.2 +50 +0 +D. +0.5 +0.5 +0.5 +0.4 +0.5 +0.4 +0.4 +0.4 +0.2 +0.2 +0.2 +0. +0.2 +0.4 +0.61e6 +8 +Recovered +6 +4 +Cumulative +Xanthane +Schizophyllan +0 +1000 +2000 +3000 +4000 +Non-Dimensional Time [-]1e7 +Recovered +8 +6 +O +Cumulative +4 +2 +-Xanthane +Schizophyllan +0 +1000 +2000 +3000 +4000 +Non-Dimensional Time [-]does not really help in understanding the underlying physics as the flow becomes very complex due to the +interaction of heterogeneous permeability, time dependent variable viscosity field and travelling viscosity +waves generated by the shear thinning effect which has been alluded to earlier. The only metric that can +help understand the flow then becomes the COR shown in Fig. 14 for both the polymers. +Figure 13: Temporal evolution of saturation for Xanthane (top) and Schizophyllan (bottom) at IR=120000 +and IPC=1500 wppm in quarter-five spot heterogeneous polymer flood simulation +(a) +(b) +Figure 14: Comparison of cumulative oil recovered in quarter five-spot heterogeneous polymer flood simula- +tion with polymers Xanthane and Schizohyllan at (a) IR=10,000 and IPC=300 wppm and at (b) IR=120,000 +and IPC=1500 wppm. +4.5 +Effect of injection rate (IR) and injected polymer concentration (IPC) on +Cumulative Oil Recovered (COR) +The effect of IR and IPC is polymer specific and can be quantified on a common parameter namely COR. +Figures 15 and 16 show the COR for varying IR and IPC conditions for homogeneous rectilinear polymer +flooding simulation. It can be seen that both the polymers predict COR with a strong dependence on IR. +However, IPC seems to be a less important parameter and the degree to which it influences the COR differs +for the two polymers. Xanthane has a stronger dependence on IPC for predicting the COR as compared to +16 + +2000 +4000 +6000 +0.8 +0.7 +0.6 +0.3 +0.2 +0.1 +0.8 +0.2 +0.8 +0.2 +0.2 +0.8 +6000 +0.8 +0.8 +0.7 +0. +0.6 +0.6 +0.5 +0.3 +0.2 +0.2 +0. +0.2 +0.4 +0.6 +0.8 +0.2 +0.4 +0.6 +0.8 +0.8 +0.2 +0.4 +0.6 +0.8le7 +1.2 +0.8 +言 0.6 +0.4 +Cumulat +0.2 +Xanthane +Schizophyllan +0.0 +0 +2000 +4000 +6000 +Non-Dimensional Time [-]1e8 +Recovered +Xanthane +1.5 +Schizophyllan +Cumulative Oil +1.0 +0.5 +0.0 +0 +2000 +4000 +6000 +Non-Dimensional Time [-]Schizophyllan. This information enables correct estimation of polymer required for any flooding process and +also indicates the importance of IR that will be required for optimum recovery. +Figure 15: COR at water breakthrough for Xanthane +Figure 16: COR at water breakthrough for Schizophyllan +4.6 +Sensitivity of the power law parameters (ε and η) +Significance and implications of the way the power law viscosity model (9) is implemented here are exemplified +in this section. This model is not an ordinary power law model with fixed parameters, rather values of the +parameters n and ε are based on the type and concentration c(x, t) of polymer. Since the concentration of +polymer is evolving in space and time according to its own transport equation and the power law viscosity +model coefficients at every point in space and time are derived from experimental data, effectively this means +the classical shear thinning fluid, if thought to have constant values of the power law parameters, itself is +changing in space and time. In this way we treat the flow in the most physical way. To the best of the +authors’ knowledge, to-date this type of space and time dependent viscosity model has not been implemented +in a full scale shear thinning polymer flood simulations. +Fig. 17 shows plots of cumulative oil recovery (COR) versus time for constant values of n and ε (average +values over the entire range chosen) and also for experimental data driven values of these two parameters. +Interestingly, difference is observed right from the beginning and is very significant at breakthrough. The +graph has three distinct regions. +The first region from t=0 to t=300. +The second one from t=300 to +t=1600 and the third one from t=1600 to t=2500. We see in this figure that simulation with experimental +data dependent parameters results in significant variation in COR rate over all three regions as compared +17 + +120000 +×108 +100000 +1.0 +80000 +0.8 +OR +B +60000 +0.60 +40000 +0.4 +20000 +0.2 +0.0002 +0.0004 +0.0006 +0.0008 +0.0010 +IPC [-]120000 +X108 +1.0 +100000 +0.8 +80000 +COR +0.6 +B +60000 +0.4 +40000 +20000 +0.2 +0.0002 +0.0004 +0.0006 +0.0008 +0.0010 +IPC [-]to a more or less constant COR rate for constant parameter simulation. +In the first region, COR rate +is lower for the data dependent parameters than that for constant values of parameters indicating that +higher effective polymer viscosity makes the flow slow. +This is also seen in Fig. 18 where level sets of +saturation at four different time levels with constant power law coefficients (top) and with variable power +law coefficients (bottom) in rectilinear homogeneous polymer flooding are compared. The parameters chosen +are for high polymer concentration relating to high viscosity. In the second region, due to now mobilized +flow which corresponds to higher shear rates, the viscosity starts to decrease even if the concentration stays +the same leading to higher COR rate. Finally, in the third region the COR rate decreases due to reduction +in sweep efficiency as some aqueous phase reaches the production well. In this region, both constant and +data dependent parameter simulations show similar trend. The maximum variation at water breakthrough +is around 33%. This is a clear indication of the importance of data dependent shear thinning polymer flood +simulations. It is worth observing in Fig. 18 that saturation in the data dependent parameter case is more +diffused than in the constant parameter case. This can be attributed to the viscosity which is not only +varying in space but is also dependent on the local concentration of polymer and therefore smearing the +so-called interface. +Figure 17: Plots of cumulative oil recovered versus time for constant values of shear thinning parameters +and for data driven values of space-time varying parameters in rectilinear homogeneous polymer flooding for +IR=120000 and IPC=1500 wppm. +4.7 +Sensitivity to Kmax +Any given polymer flooding simulation is highly sensitive to the permeability field. In this section, effect of +varying the maximum value, Kmax, of permeability has been discussed. Multiple simulations with the two +polymers at different injection rates and Kmax values were run. Fig. 19 shows the cumulative oil recovered +at water breakthrough for different conditions. A clear trend is not seen which is expected given the highly +non-linear and data driven nature of the problem. However there are some trends that are seen across the +two polymers. For lower injection rates, increasing the Kmax value is seen to increase the COR. At some +particular cutoff this trend flips which is expected to be lower than IR=60000 for Xanthane but higher for +Schizophyllan. As seen from the IR=60000 the overall effect of increasing Kmax is different for Xanthane +and Schizophyllan, however the trend is decreasing for both the polymers at the maximum injection rate +(IR=120000). So to generalize, increasing the value of Kmax increases (decreases) COR for lower (higher) +injection rates. This can be explained from the relation of shear rate with viscosity for complex fluids. At +18 + +1e8 +Data driven parameters +1.4 +Constantparameters +Cumulative Oil Recovered +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +500 +1000 +1500 +2000 +2500 +Non-Dimensional Time [-]Figure 18: Temporal evolution of saturation in polymer flood simulation with constant power law coeffi- +cients (top) and variable power law coefficients (bottom) in rectilinear homogeneous polymer flooding at +IR=120000, IPC=1500wppm. +lower injection rates the flow experiences lower shear rate resulting in higher viscosity of the shear thinning +fluid. Effect of this higher viscosity is opposed by the lower resistance to the flow from the medium at higher +Kmax value. It appears that this second effect wins over the first one with the net effect of increasing overall +COR at water breakthrough. +A similar trend is seen for the homogeneous case. Fig. 18 shows the cumulative oil recovered for different +polymers and injection rate across Kmax values. However, in this case for the chosen range of Kmax values, the +cutoff value is not seen for Xanthane but is seen to be between IR=60,0000 and IR=120,000 for Schizophyllan. +5 +Conclusions +A dynamic empiric coefficient based shear thinning model of polymer flooding has been implemented in an +in-house code for modelling multi-component multi-phase fluid flow in porous media. Simulations with this +data driven model using this code have been performed for two shear thinning polymers, Xanthane and +Schizophyllan, at various values of the parameters of the problem. Numerical results which are qualitatively +consistent with physics show the merits of this easy-to-implement inexpensive and fast method. +Time +dependent viscosity profiles in the flow field show a train of travelling viscosity waves which are not yet fully +understood theoretically and opens the door for future research for a better theoretical understanding of +the complex PDE model for the shear tinning polymer flooding. The simulations also show (i) competing +effects of shear thinning and mobility ratio; (ii) injection conditions such as injection rate and injected +polymer concentration influence the choice of polymers to optimise cumulative oil recovery (these effects +are relatively much more significant in the case of Schizophyllan compared to Xanthane); (iii) permeability +field also affects the choice of polymer as polymers show varying movement for different shear rates that +are caused by heterogeneity; and (iv) dynamically evolving travelling wave patterns of viscosity profiles +which interact with the underlying flow and the heterogeneity field to generate narrow region of fingers +which need to be further probed in the future. The significance of this work is that it shows a simple way, +yet accurate enough, to incorporate shear thinning effect of polymer in an otherwise Newtonian model of +multiphase multicomponent porous media flow model and provides an effective yet easy approach to make +design choices of polymers for CEOR in any given flooding condition. We summarize below some of the +specific results. +1. Schizophyllan polymer viscosity decreases faster with increasing shear rate as compared to Xanthane. +2. Xanthane performs better in terms of COR for higher IPC and IR conditions while Schizophyllan shows +19 + +1 +500 +1000 +1300 +0.9 +0.9 +0.9 +0.9 +0.8 +0.8 +0.8 +0.8 +0.8 +0.8 +0.8 +0.8 +0.7 +0.7 +0.7 +0.6 +0.7 +0.6 +0.6 +0.6 +0.6 +0.6 +0.6 +0.6 +0.4 +0.4 +0.50.4 +0.50.4 +0.5 +0.5 +0.4 +0.4 +0.4 +0.4 +0.2 +0.2 +0.2 +0.2 +0.3 +0.3 +0.3 +0.3 +0.2 +0.8 +0 +0.20.40.6 +0.8 +0 +0.4 +0.6 +1 +1 +0 +0.2 +0.40.60.8 +1 +0 +0.2 +0.4 +0.60.8 +1 +1 +500 +1000 +1300 +0.9 +0.9 +0.9 +0.9 +0.8 +0.8 +0.8 +0.8 +0.8 +0.8 +0.8 +0.8 +0.7 +0.7 +0.6 +0.7 +0.6 +0.6 +0.7 +0.6 +0.6 +0.4 +0.50.4 +0.4 +0.50.4 +0.6 +0.4 +0.4 +0.2 +0.2 +0.2 +0.40.2 +0.5 +0.3 +0.3 +0.3 +0.4 +0 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.4 +0.6 +0.8Figure 19: Cumulative Oil Recovered (at water breakthrough) showing sensitivity to Kmax for different +injection rates and polymers (Comparison made for heterogeneous rectilinear geometry) +higher COR for lower IPC and IR. This fact is reflected from the MFW which in turn shows that the +mean finger width is higher for the polymer producing higher COR for a given condition. +3. Competing effects of viscosity ratio and shear thinning behavior is seen in Schizophyllan polymer for +the range of IR and IPC tested. COR is higher for higher IPC at lower injection rates which reduces to +a COR less than the case of lower IPC at higher injection rates. This indicates that lower IPC might +be preferable for some polymers at higher injection rates. +4. Permeability field directly affects the shear rates and therefore the viscosity of the aqueous solution. +Polymers might perform differently in different permeability fields when shear thinning behavior is +accounted for. +5. The percentage difference for COR is at 33% for shear thinning model when comparing varying pa- +rameters and constant power law parameters. This shows the importance of these models to predict +the overall recovery in a given polymer flood simulation. These percentages may vary depending on +the IPC, IR and permeability. +20 + +1e7 +Xanthane;IR-10000 +le6 Schizophyllan;IR-10000 +1.0095 +7.80 +1.0090 +7.78 +1.0085 +1.0080 +7.76 +40 +60 +80 +100 +40 +60 +80 +100 +Cumulative Oil Recovered +le7 +Xanthane:iR-60000 +le7 Schizophyllan;IR-60000 +3.810 +4.650 +4.648 +3.805 +4.646 +3.800 +40 +60 +80 +100 +40 +60 +80 +100 +1e7 +Xanthane;IR-120000 +le7 Schizophyllan;IR-120000 +8.460 +9.510 +8.458 +9.508 +8.456 +8.454 +9.506 +40 +60 +80 +100 +40 +60 +80 +100 +Kmax [-]Figure 20: Cumulative Oil Recovered (at water breakthrough) showing sensitivity to Kmax for different +injection rates and polymers (Comparison made for homogeneous rectilinear geometry) +A +Appendix +A.1 +Algorithm +Here we outline the algorithm for the SP-flood simulation that includes shear thinning effect of polymer. +The algorithm for the polymer flood is essentially a special case of the same with zero concentration for the +surfactant. The step-by-step algorithm built on the one described in Daripa & Dutta [2] is given below. To +accomplish some of the steps below, one needs to refer Daripa & Dutta [2]. +1. Define the Cartesian grid in the domain using equal, uniform grid sizes in both the axes. Generate the +finite element mesh. +2. Generate a heterogeneity field on this grid. +3. Choose an initial interface separating the injected fluid from the resident fluid. +4. Set the model parameters: µo, µw, sσ0 +ro , sσ0 +ra. +5. Initialize the state variables s, c and Γ as +s0 = +� +1 +x ∈ Ω+ +sσ0 +0 +x ∈ Ω− , +c0 = +� +0.01 +x ∈ Ω+ +0 +x ∈ Ω− , +Γ0 = +� +0.005 +x ∈ Ω+ +0 +x ∈ Ω− . +21 + +1e7 +Xanthane;R-10000 +le7 Schizophyllan;IR-10000 +1.134 +1.1950 +1.133 +1.1925 +1.132 +1.1900 +1.131 +1.1875 +1.130 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +IRecovered +1e7 +Xanthane;IR-60000 +le7 +Schizophyllan;IR-60000 +5.590 +6.319 +5.585 +Oil +6.318 +Cumulative +6.317 +5.580 +6.316 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 ++1.2e8Xanthane;IR-120000 +le8 Schizophyllan;IR-120000 +80000 +1.022 +60000 +40000 +1.020 +20000 +1.018 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +Kmax [-]6. Calculate σ(sn, Γn), µa(cn, ∇vn−1), sra(sn, Γn),sro(sn, Γn), λa(sn, cn, Γn), λo(sn, cn, Γn), λ(sn, cn, Γn) +using sn, cn, Γn which are values of s, c and Γ respectively at the nth time level. +7. Solve the global pressure equation to get pn and subsequently compute vn. +8. Use vn, sn, cn, Γn and the quantities calculated in Step 6, to solve for sn+1, cn+1 and Γn+1, thus +completing a full time step. +9. If breakthrough is achieved: then stop; else update n = n + 1 and repeat from Step 6 above. +In order to reduce computational cost, a few iterations of Steps 6 and 8 are done before updating the pressure +in Step 7. The pseudocode (see Algorithm 1) and flow-chart (see Fig. 21) for the procedure are given here. +Algorithm 1: SP flooding simulation +/* Set up Cartesian grid, FE Mesh, permeability field and model parameters +*/ +1 Set i, j = 0, . . . , M; xij = +� i +M , j +M +� +; +/* (M × M is the grid size ) */ +2 Set Σ = Initial interface; +/* Σ = ∂Ω+ ∩ ∂Ω− */ +3 Set K(x) +4 Set µo, µw, sσ0 +ro , sσ0 +ra, ˜q = values from 1; +/* Initialization +*/ +5 Set t = 0; ∆t = 1 +N ; Tstop = N∆t; +/* N chosen for accuracy */ +6 for i = 0, . . . , M do +7 +for j = 0, . . . , M do +8 +Set (s, c, Γ)(xij, 0) = +� +(1, 0, 0) +xij ∈ Ω+ +(sσ0 +0 , c0, Γ0) +xij ∈ Ω− ; +9 +end +10 end +/* Computation loop +*/ +11 while +� +s(xM,M, t) ≤ 1 − sσ0 +0 +&& t < Tstop +� +do +12 +Compute {σ, µa, sra, sro, λa, λo, λ, pc} using (sn, cn, Γn, vn−1); +13 +Use data driven power law model to calculate µa; +14 +Update λa, λ, fa, D(s, c, Γ) using µa; +15 +Solve global pressure equation for pn, vn; +16 +Recompute {sra, sro, λa, λo, λ} using (sn, cn, Γn, vn); +17 +Solve transport equations for sn+1, cn+1 and Γn+1; +18 +Set t = t + ∆t; +19 end +A.2 +Post processing for mean finger width calculation +This appendix has bearing for section 4.3.1. Following procedure is used to calculate the mean finger width +(flowchart shown in Fig. 22) which is based on identifying the mixing layer, fingers and numerical interface +shown in Fig. 23. +1. Identify the mixing layer: The 2D domain is scanned at each time step to locate a reduction in +saturation. For each y level, let x0 denote the start of mixing layer and x1 denote the end of mixing +layer. +22 + +Setup grid +Load K(x) +Set: +µo, µw, +sσ0 +ro , sσ0 +ra +Initialize: +s0, c0, Γ0 +Compute +σ, µa, sra, +sro, λa, λo +Calculate +u, v +Compute +σ, pc +Solve for +s, c, Γ +update +model +Water +break- +through? +stop +no +yes +Figure 21: Flow-chart for SP flooding simulation +Figure 22: Flowchart for mean finger width calculation +2. Find the mean saturation for the mixing layer: The mean saturation is found from the mixing region +by taking a mean of saturation at each y level. All the means are then averaged to find a level set +based on that cutoff concentration. As the mean concentration goes below this set value, the x value +corresponding to it is defined as xMFW (shown in green in Fig. 23). +23 + +Start +a. Scan the domain for saturation +c. Find mean saturation for the +between set range (Umin and Umax) +mixing layeridentified in step-1 +b. Xo identified when average +d. Going from left to right identify X +saturation goes below Umax from +locationforagivenylevelwhere +step-a; Xi identified when average +the saturation goes below the +saturation goes below Umin from +saturation found from step-c +step-a +Step-1: Find the mixing layer +Step-2:Findnumericalinterface +h. Count the number of fingers by +e. Find mean saturation (Uo) for all +couting number of switches from 1 +the x locations identified from +to O and vice versa, Calculate +step-d +g. Assign binary values to all the +strings of O's and 1's indicating +cells in the XMFw based on: +finger width for each finger. +f. Find location of XmFw going from +U > Uo (0) +U 0 (i.e., no marked points). Let mH be the Masur-Veech measure on H, let +Z ∼= Rk−1 be the corresponding action given by translation along the leaves of the +real Rel foliation, and let Z0 ⊂ Z be any one-dimensional connected subgroup of Z. +Then the Z0-flow on (H, mH) is mixing of all orders (and in particular, ergodic). +The definition of mixing of all orders is given in §3.3. +For purposes of this +introduction it is enough to note that it implies ergodicity of any nontrivial element. +Note that when H has marked points, there will be subgroups Z0 which only move +the marked points on surfaces without otherwise changing the geometry, and the +conclusion of Theorem 1.1 will not hold. This is the only obstruction to generalizing +our results to strata with marked points, see Theorem 8.1. +1 + +2 +JON CHAIKA AND BARAK WEISS +The proof of Theorem 1.1, as well as the other results of this paper, relies crucially +on measure-rigidity results of Eskin, Mirzakhani and Mohammadi [EM, EMM], and +further forthcoming work extending these results, which we will describe in §5. +Theorem 1.1 improves on the results of several authors. In those results, ergod- +icity for the full Rel foliation was studied. The full Rel foliation (also referred to +as the ‘kernel foliation’, ‘isoperiodic foliation’, or ‘absolute period foliation’) will +also be defined in §2.2. Its leaves are of dimension 2(k − 1), that is, twice the di- +mension of the real Rel leaves. Loosely speaking, two surfaces are in the same leaf +for this foliation if one can be obtained from the other by moving the singularities +(without otherwise affecting the geometry of the surface). That is, we relax the +hypothesis that points can only be moved horizontally. The first ergodicity results +for the full Rel foliation were obtained by McMullen [McM], who proved ergodic- +ity in the two strata H(1, 1) and H(1, 1, 1, 1). Subsequently, Calsamiglia, Deroin +and Francaviglia [CDF] proved ergodicity in all principal strata, and Hamenst¨adt +[Ham] reproved their result by a simpler argument. Recently, Winsor [Wi1] proved +ergodicity for most of the additional strata, and in [Wi2], showed that there are +dense orbits for the Z0-flow, for any Z0 as in Theorem 1.1. Note that ergodicity for +a foliation is implied by ergodicity for any of its subfoliations, and that ergodicity +implies the existence of dense leaves. and thus Theorem 1.1 generalizes all of these +results. Also note that full Rel is a foliation which is not given by a group action, +and the notions of mixing and multiple mixing do not make sense in this case. +The papers [McM, CDF] go beyond ergodicity and obtain classifications of full +Rel leaves in their settings. We suspect that there is not a reasonable classification +of real Rel leaf-closures, indeed it is already known (see [HW]) that there are real +rel trajectories that leave every compact set never to return. +The strata H support other interesting measures for which similar questions +could be asked. +Namely, by work of Eskin, Mirzakhani and Mohammadi [EM, +EMM], for any q ∈ H, the orbit-closure L +def += Gq is the support of a unique smooth +G-invariant measure which we denote by mL. Let ZL be the subgroup of Z leaving L +invariant. Then ZL also preserves mL and for many choices of L, we have dim ZL > +0. In these cases, for any closed connected Z1 ⊂ ZL, there is a complexification R1, +which gives a foliation of L whose leaves R1(q) have dimension 2 dim Z1 (see §2.2). +The leaves R1(q) have a natural translation structure, and this induces a natural +locally finite translation-invariant measure on each leaf. With this terminology we +can now state the main result of this paper: +Theorem 1.2. Let L be a G-orbit closure, and let mL, ZL, RL be as above, where +dim ZL > 0. Let z0 be a nontrivial element of ZL and let Z0 = spanR(z0). Then +either +(1) Z0 is mL mixing of all orders (and in particular, z0 acts ergodically); or +(2) there is an intermediate closed connected subgroup Z1 so that Z0 ⊂ Z1 ⊂ +ZL, and the complexification R1 of Z1 satisfies +• for every q ∈ L, the leaf R1(q) is closed, and +• for mL-a.e. q, R1(q) is of finite volume with respect to its translation- +invariant measure, and Z0q = R1(q). +Thus, in order to establish ergodicity of real Rel subfoliations on G-orbit-closures, +it is enough to rule out Case (2). We will prove Proposition 7.1, which provides a + +ERGODICITY OF REL +3 +simple way to achieve this, using cylinder circumferences of surfaces in L. Theorems +1.1 and 8.1 are deduced from Theorem 1.2 using Proposition 7.1. +The following statement is an immediate consequence of Theorem 1.2. +Corollary 1.3. Let L be a G-orbit-closure, let mL, ZL be as above, and let Z1 ⊂ ZL +be one-dimensional. Suppose that the foliation induced by the complexification R1 +has a dense leaf. +Then the Z1-flow on (L, mL) is mixing of all orders (and in +particular, ergodic). +The density of certain leaves of the full Rel foliation in G-orbit-closures of rank +one was obtained by Ygouf in [Y]. Using these results we obtain ergodicity of one- +dimensional subgroups of the real Rel foliation in many cases. For instance, using +[Y, Thm. A & Prop. 5.1] we have: +Corollary 1.4. The real Rel foliation is mixing of all orders (and in particular, +ergodic) in any eigenform locus in H(1, 1) with a non-square discriminant. +Recall that in [Wi2] Winsor proved the existence of dense real Rel leaves, and +dense leaves for one-dimensional flows Z0, in all strata. +Using these results in +conjunction with Corollary 1.3, one can obtain an alternative proof of Theorem 1.1 +that avoids the use of Proposition 7.1. +1.1. Outline. In §2 we give background material on translation surfaces, their +moduli spaces, and the Rel foliation. In §3 we use standard facts about joinings to +build a measure θ on the product of two strata (see (3.1)), depending on a real- +Rel flow Z0, such that if θ is the product measure, then Z0 is ergodic. In §3.3 +we discuss a technique of Mozes that makes it possible to upgrade ergodicity to +mixing of all orders. In §4 we show that θ is ergodic for the diagonal action of +the upper triangular group P ⊂ G on the product of two strata. In §5 we state a +far-reaching measure rigidity result of Brown, Eskin, Filip and Rodriguez-Hertz for +P-ergodic measures on products of two strata. In §6 we use this measure rigidity +result, as well as prior results for the action on one stratum due to Wright, in order +to characterize the situations in which θ is not a product measure, thus proving +Theorem 1.2. +1.2. Acknowledgements. We are very grateful to Alex Eskin for crucial contri- +butions to this project. We also thank Simion Filip and Alex Wright for useful dis- +cussions, and acknowledge the support of ISF grants 2919/19, BSF grant 2016256, +NSFC-ISF grant 3739/21, a Warnock chair, a Simons Fellowship, and NSF grants +DMS-1452762 and DMS-2055354. +2. Preliminaries about translation surfaces +2.1. Strata, period coordinates. In this section we collect standard facts about +translation surfaces, and fix our notation. For more details, we refer to reader to +[Zo, Wr1, BSW]. Below we briefly summarize the treatment in [BSW, §2]. +Let S be a compact oriented surface of genus g, Σ = {ξ1, . . . , ξk} ⊂ S a finite set, +a1, . . . , ak non-negative integers with � ai = 2g−2, and H = H(a1, . . . , ak) the cor- +responding stratum of unit-area translation surfaces. We let Hm = Hm(a1, . . . , ak) +denote the stratum of unit-area marked translation surfaces and π : Hm → H the + +4 +JON CHAIKA AND BARAK WEISS +forgetful mapping. Our convention is that singular points are labeled, or equiv- +alently, H = Hm/ Mod(S, Σ), where Mod(S, Σ) is the group of isotopy classes of +orientation-preserving homeomorphisms of S fixing Σ, up to an isotopy fixing Σ. +There is an R>0-action that dilates the atlas of a translation surface by c ∈ R>0. +For a stratum H and marked stratum Hm, we denote the collection of surfaces +of arbitrary area, obtained by applying such dilations, by ¯H, ¯Hm. The marked +stratum ¯Hm is a linear manifold modeled on the vector space H1(S, Σ; R2). It has +a developing map dev : ¯Hm → H1(S, Σ; R2), sending an element of ¯Hm represented +by f : S → M, where M is a translation surface, to f ∗(hol(M, ·)), where for +an oriented path γ in M which is either closed or has endpoints at singularities, +hol(M, γ) = +�� +γ dx +� +γ dy +� +, and dx, dy are the 1-forms on M inherited from the plane. +Furthermore, there is an open cover {Uτ} of Hm, indexed by triangulations τ of +S with triangles whose vertices are in Σ, and maps dev|Uτ : Uτ → H1(S, Σ; R2), +which are homeomorphisms onto their image, and such that the transition maps on +overlaps for this atlas are restrictions of linear automorphisms of H1(S, Σ; R2). +This atlas of charts {(Uτ, dev|Uτ )} is known as period coordinates. +Since +each Uτ is identified via period coordinates with an open subset of the vector +space H1(S, Σ; R2), the tangent space at each Uτ is identified canonically with +H1(S, Σ; R2), and thus the tangent bundle of Hm is locally constant. A sub-bundle +of the tangent bundle is called locally constant or flat if it is constant in the charts +afforded by period coordinates. The Mod(S, Σ)-action on Hm is properly discontin- +uous, and hence H is an orbifold, and the map π : Hm → H is an orbifold covering +map. +The group G acts on translation surfaces in H by modifying planar charts, and +acts on H1(S, Σ; R2) via its action on R2, thus inducing a G-action on Hm. The G- +action commutes with the Mod(S, Σ)-action, and thus the map π is G-equivariant +for these actions. The G-action on Hm is free, since dev(gq) ̸= dev(q) for any +nontrivial g ∈ G. We will use the following subgroups of G: +gt = +� +et +0 +0 +e−t +� +, +us = +� +1 +s +0 +1 +� +U = {us : s ∈ R}, +P = +��a +b +0 +a−1 +� +: a > 0, b ∈ R +� +. +2.2. Rel foliation and real Rel foliation. We define and list some important +properties of the Rel foliation, the real-Rel foliation, and the corresponding action +on the space of surfaces without horizontal saddle connections. See [MW2, BSW] +for more details. See also [Zo, McM], and references therein. +We have a canonical splitting R2 = R ⊕ R and we write R2 = Rx ⊕ Ry to +distinguish the two summands in this splitting. There is a corresponding splitting +(2.1) +H1(S, Σ; R2) = H1(S, Σ; Rx) ⊕ H1(S, Σ; Ry). +We also have a canonical restriction map Res : H1(S, Σ; R2) → H1(S; R2) (given +by restricting a cochain to absolute periods). Since Res is topologically defined, +its kernel ker(Res) is Mod(S, Σ)-invariant. +Moreover, from our convention that +singular points are marked, the Mod(S, Σ)-action on ker(Res) is trivial. + +ERGODICITY OF REL +5 +Let +(2.2) +R +def += ker(Res) +and +Z +def += R ∩ H1(S, Σ; Rx). +Since H1(S, Σ; Rx) and H1(S, Σ; Ry) are naturally identified with each other via +their identification with H1(S, Σ; R), for each Z1 ⊂ Z we can define the space R1 +spanned by the two copies of Z1 in H1(S, Σ; Rx) and H1(S, Σ; Ry) respectively. +The space R1 is the complexification of Z1. This terminology arises from view- +ing H1(S, Σ; R2) as H1(S, Σ; C), a vector space over C, viewing H1(S, Σ; Rx) and +H1(S, Σ; Ry) as the real and imaginary subspace of this complex vector space. With +this viewpoint, R1 is the C-span of Z1. +For any subspace Z1 ⊂ R, we can foliate the vector space H1(S, Σ; R2) by affine +subspaces parallel to Z1. Pulling back this foliation using the period coordinate +charts gives rise to a foliation of ¯Hm. Since monodromy acts trivially on R, this +foliation descends to a well-defined foliation on ¯H. It is known (see e.g. [BSW, +Prop. 4.1]) that the area of a surface is constant on leaves of the Rel foliation, +and thus the Rel foliation and any of its subfoliations descends to a foliation of H. +The foliation corresponding to R (respectively, to Z) is known as the Rel foliation +(respectively, the real Rel foliation). +Because the Mod(S, Σ)-monodromy action fixes all points of R, the leaves of +the Rel foliation, and any of its sub-foliations, acquire a translation structure. In +particular, they are equipped with a natural measure. +For any v ∈ Z we have a constant vector field, well-defined on Hm and on H, +everywhere equal to v. Integrating this vector field we get a partially defined real +REL flow (corresponding to v) (t, q) �→ Reltv(q); the flow may not be defined for +all time due to possible ‘collide of zeroes’. For every q ∈ H it is defined for t ∈ Iq, +where the domain of definition Iq = Iq(v) is an open subset of R which contains 0. +The sets Iq(v), are explicitly described in [BSW, Thm. 6.1]. Let ˆH denote the set +of surfaces in H with no horizontal saddle connections. Then Iq = R for all q ∈ ˆH. +If q ∈ H, s ∈ R and τ ∈ Iq then τ ∈ Iusq, and Relτv(usq) = usRelτv(q). +Similarly, if q ∈ H, t ∈ R and τ ∈ Iq then τ ′ def += etτ ∈ Igtq and Relτ ′v(gtq) = +gtRelτv(q). In particular, since P preserves ˆH and P = {gtus : t, s ∈ R}, there is +an action of P ⋉ Z on ˆH, given by (p, z).q = pRelz(q). +3. Preliminaries from ergodic theory +3.1. Ergodic decomposition. We will use the notation G ⟳ (X, µ) to indicate +that G is a locally compact second countable group, (X, B) is a standard Borel +space, and µ is a probability measure on B preserved by the G-action. We say that +G ⟳ (Y, ν) is a factor of (X, µ) if there is a measurable G-invariant conull subset +X0 ⊂ X, and a measurable map T : X0 → Y such that T ◦ g = g ◦ T for all g ∈ G, +and ν = T∗µ. In this situation we refer to T as the factor map. Given a factor +map, there is a (unique up to nullsets) measure disintegration µ = +� +µy dν(y), for +a Borel mapping y �→ µy from Y to the space of Borel probability measures on X, +such that µy(T −1(y)) = 1 for ν-a.e. y. Equivalently we can write µ = +� +x µ′ +x dµ(x), +where µ′ +x +def += µT (x). +For a closed subgroup H ⊂ G, we say that µ is H-ergodic +if any invariant set is null or conull. We have the following well-known ergodic +decomposition theorem: + +6 +JON CHAIKA AND BARAK WEISS +Proposition 3.1. Suppose G ⟳ (X, µ), and H is a closed subgroup of G. Then +there is a factor of H ⟳ (X, µ), called the space of ergodic components and denoted +by X//H, with the following properties: +(i) For ν-a.e. y ∈ X//H, µy is H-invariant and H-ergodic. +(ii) H acts trivially on X//H. +(iii) H ⟳ (X, µ) is ergodic if and only if X//H = {pt.}. +(iv) The properties (i)–(iii) uniquely determine the factor X//H up to measur- +able isomorphism. +(v) If H ✁ G then G ⟳ (X//H, ν). +Proof. For (i) and (ii) see [Va, Thm. 4.4] (in the notation of [Va], these assertions +follow from the fact that β is a map on points and is H-invariant). Assertion (iii) +is immediate from definitions and (iv) follows from [Va, Lemma 4.4]. For (v), one +can argue using the uniqueness property (iv), and the fact that the image of an +H-invariant ergodic measures under any element g ∈ G is also H-invariant and +ergodic. +□ +Remark 3.2. An action is called prime if it has no factors (besides the action itself, +and the trivial action on a point). The construction above shows that if H ✁ G, G′ +is a subgroup of G so that G′ ⟳ (X, µ) is prime and H ⟳ (X, µ) is not isomorphic +to the trivial action, then H ⟳ (X, µ) is ergodic. This is not the approach we will +take for proving Theorem 1.1. +3.2. Joinings. We recall some well-known facts about joinings, see [dlR] and ref- +erences therein. Let G ⟳ (Xi, µi) for i = 1, 2. A joining is a measure θ on X1 × X2, +invariant under the diagonal action of G, such that πi∗θ = µi. +A self-joining +is a joining in case X1 = X2. +If (Xi, µi) → (Z, ν) is a joint factor then the +relatively independent joining over Z is the joining � +Z(µ1)z × (µ2)z dν(z), where +µi = +� +Z(µi)z dν(z) is the disintegration of µi. In case X1 = X2 = X, and Z = X//H +is the space of ergodic components of the action of H on (X, µ) as in Proposition 3.1, +we obtain the relatively independent self-joining over X//H. This joining satisfies: +Proposition 3.3. The following are equivalent: +• H ⟳ (X, µ) is ergodic. +• The relatively independent self-joining over X//H is µ × µ. +We note two properties of this self-joining. +We fix a topology on X which +generates the σ-algebra, and denote by supp µ the topological support of µ, i.e., +the smallest closed set of full measure. +Proposition 3.4. Let θ be the measure on X×X which is the relatively independent +self-joining over X//H, for some H, and let T : X → X//H be the factor map. Then +the following hold: +• We have +(3.1) +θ = +� +X +µT (x) × µT (x) dµ(x). +• If X = supp µ then supp θ contains the diagonal ∆X +def += {(x, x) : x ∈ X}. + +ERGODICITY OF REL +7 +Proof. Formula (3.1) is immediate from the definition of the relatively independent +self-joining over X//H. +Since each µ′ +x = µT (x) is H-invariant and ergodic, and +µ′ +x(T −1(T (x))) = 1, the set {x ∈ X : x /∈ supp µ′ +x} is a nullset. From this, and +from (3.1) we obtain the second assertion. +□ +3.3. Ergodicity, mixing, and mixing of all orders. For G ⟳ (X, µ), let L2 +0(µ) +denote the Hilbert space of L2-functions on (X, µ) of integral zero, and let k ≥ 2. +The action is called k-mixing if for any f1, . . . , fk ∈ L2 +0(µ) and for any k−1 sequences +� +g(i) +n +� +n∈N ∈ G, i = 1, . . . , k − 1, for which all of the sequences +� +g(i) +n +� +n∈N +(1 ≤ i ≤ k − 1) +and +� +g(i) +n (g(j) +n )−1� +n∈N +(1 ≤ i < j ≤ k − 1) +eventually leave every compact subset of G, we have +� +X +f1 +� +g(1) +n x +� +· · · fk−1 +� +g(k−1) +n +x +� +fk(x) dµ(x) +n→∞ +−→ +k +� +i=1 +� +X +fi dµ. +We say that the action is mixing if it is 2-mixing, and mixing of all orders if it is +mixing of order k for any k ≥ 2. It is easy to check that mixing implies ergodicity +of any unbounded subgroup of G. We have the following: +Proposition 3.5. Let Z0 ∼= R and let {gt} be a one-parameter group acting on Z0 +by dilations, i.e., for all v ∈ Z0 and t ∈ R we have gtv = eλtv for some λ ̸= 0. +Let F = {gt} ⋉ Z0 and let F ⟳ (X, µ) be a probability space. The following are +equivalent: +(a) the restricted flow Z0 ⟳ (X, µ) is ergodic; +(b) the restricted flow Z0 ⟳ (X, µ) is mixing of all orders; +(c) the restricted flow Z0 ⟳ (X, µ) is mixing; +(d) any nontrivial element of Z0 acts ergodically. +Remark 3.6. The group F appearing in Proposition 3.5 is isomorphic as a Lie +group to the subgroup P of upper triangular matrices in G, but in our application +we will use it for the group generated by a one-parameter real Rel flow Z0 and the +diagonal flow {gt}. +Proof. Clearly (b) +=⇒ +(c) +=⇒ +(d) +=⇒ +(a). We assume that the Z0-flow +is ergodic. To see that it is mixing, it is enough by [P, Chap. 2, Prop. 5.9] to +prove that it has countable Lebesgue spectrum, and for this, use [KT, Prop. 1.23 +& Prop. 2.2]. The proof of mixing of all orders follows verbatim from an argument +of Mozes [Mo], for mixing actions of Lie groups which are ‘Ad-proper’. Since our +group F is not Ad-proper, we cannot cite [Mo] directly, so we sketch the proof. +For notational convenience we deduce 3-fold mixing from mixing (the proof that +‘k-fold mixing +=⇒ k + 1-fold mixing’, for k ≥ 3, is identical but requires more +cumbersome notation). +We use additive notation in the group Z0, and denote the action of Z0 on X by +(z, x) �→ z.x. Let (bn)n∈N and (cn)n∈N be sequences in Z0 such that each of the +sequences (bn)n∈N , (cn)n∈N , (bn + cn)n∈N eventually leaves every compact set, and +let f1, f2, f3 be in L2 +0(µ). We need to prove that +� +X +f1(x)f2(bn.x)f3((bn + cn).x) dµ(x) +n→∞ +−→ +� +X +f1 dµ +� +X +f2 dµ +� +X +f3 dµ. + +8 +JON CHAIKA AND BARAK WEISS +For each n, define a measure µn on X3 def += X × X × X by +� +X3 f dµn +def += +� +X +f(x, bn.x, (bn + cn).x) dµ(x), +∀f ∈ Cc(X3). +That is, µn is the pushforward of the diagonal measure on X3 by the sequence +(0, bn, bn + cn). It is easy to see that 3-mixing is equivalent to the fact that the +weak-* limit of µn is the measure µ3 def += µ×µ×µ. The group F 3 def += F ×F ×F acts on +X3 by acting separately on each component, and as in [Mo], since Z0 is mixing, it +suffices to show that any measure ν on X3 which is a weak-* limit of a subsequence +of (µn)n∈N, is invariant under (0, u, v) ∈ R3 ⊂ F 3, for some (u, v) ∈ R2 ∖ (0, 0). We +claim that for any s ∈ R the measure µn is invariant under +hn(s) +def += (gs, bn · gs · (−bn), (bn + cn) · gs · (−bn − cn)) , +where the multiplication is in the group F 3. Indeed, since µ is {gs}-invariant, +� +X3 f dµn = +� +X +f (gsx, bn.(gsx), (bn + cn).(gsx)) dµ(x), +and +hn(s) · (idF , bn, bn + cn) = (gs, bn · gs, (bn + cn) · gs). +That is, applying hn(s) changes one description of µn to another. +We embed F as a multiplicative group of matrices in GL2(R) and let dF be the +metric on F induced by some norm on GL2(R). By a straightforward computation +we have +hn(s) = +� +gs, (1 − eλs)bn · gs, (1 − eλs)(bn + cn) · gs +� +, +and dF (idF , hn(sn)) is a continuous function of s which goes to 0 as s → 0 and for +any fixed s > 0, increases to infinity as n → ∞. Therefore we can choose sn → 0 +so that dF (idF , hn(sn)) = 1 for all large enough n. +As in [Mo], ν is invariant +under some subsequential limit of hn(sn) which is of the form (0, u, v) for some +(u, v) ∈ R2 ∖ (0, 0). This establishes our sufficient condition. +□ +4. The relatively independent self-joining for a Rel flow +Recall that ˆL ⊂ L is the set of surfaces without horizontal saddle connections, +and this is a P-invariant set of full measure with respect to mL. We can combine +the product action of ZL × ZL on ˆL × ˆL with the diagonal action of P to obtain an +action of the semi-direct product P ⋉ (ZL × ZL) on ˆL × ˆL. Since ˆL ⊂ L is of full +measure, and the arguments of this section involve passing to sets of full measure, +in the remainder of this section we will ignore the distinction between L and ˆL. +Proposition 4.1. Let Z ⊂ ZL be a closed connected subgroup. If θ is an invariant +probability measure for an action of the semidirect product P ⋉ (Z × Z) on L × L, +then any f ∈ L2(θ) which is {gt}-invariant is also Z × Z-invariant. +Proof. For any z ∈ Z ×Z, gtzg−t →t→−∞ 0. So the claim follows from the Mautner +phenomenon, see e.g. [EW, Prop 11.18]. +□ +Proposition 4.2. Let (L, mL) be a G-orbit-closure with a fully supported P- +invariant ergodic measure, let Z ⊂ ZL be a connected closed subgroup, and let +θ on L × L be the relatively independent joining over L//Z. Then θ is P-invariant +and {gt}-ergodic (and hence P-ergodic). Also ∆L ⊂ supp θ. + +ERGODICITY OF REL +9 +As we will see in §5, under the conditions of the Proposition, mL is the so-called +‘flat measure’ on L. +Proof. Let π : L × L → L be the projection onto the first factor, and let ν = π∗θ. +For each x ∈ L, let Ωx +def += π−1(x) = {x} × L be the fiber, and let θx be the fiber +measure appearing in the disintegration θ = +� +L θx dν(x). Then Z acts on Ωx via +the second factor in Z × Z, and θx is Z-invariant and ergodic by the definition of +the ergodic decomposition. +It follows from Proposition 3.1(v) that θ is P-invariant. To prove ergodicity, let +f ∈ L2(L×L, θ) be a P-invariant function. By Proposition 4.1, f is Z×Z-invariant. +For each x ∈ L, let fx +def += f|Ωx. There is L0 ⊂ L such that mL(L0) = 1 and for +every x ∈ L0, fx belongs to L2(Ωx, θx) and is Z-invariant. Hence, by ergodicity, +there is ¯f : L0 → R such that for every x ∈ L0, ¯f(x) is the θx-almost-sure value +of fx. Since f is P-invariant for the diagonal action of P, ¯f is P-invariant for the +action of P on L. By ergodicity of P ⟳ (L, mL), ¯f is ν-a.e. constant, and thus f +is θ-a.e. constant. +The last assertion follows from Proposition 3.4. +□ +5. An upgraded magic wand theorem +The celebrated ‘magic wand’ Theorem of Eskin and Mirzakhani [EM], and en- +suing work of Eskin, Mirzakhani and Mohammadi [EMM], classified P- and G- +invariant probability measures and orbit-closures on strata of translation surfaces. +These results can be summarized as follows (see [EM, Defs. 1.1 & 1.2, Thms. 1.4 +& 1.5]): +Theorem 5.1. Let H, Hm, ¯H, ¯Hm be as in §2.1. Any P-invariant ergodic proba- +bility measure m has the following properties: +(i) It is G-invariant. +(ii) There is a complex-affine manifold N and a proper immersion ϕ : N → ¯H +such that +L +def += supp m = H ∩ ϕ(N). +(iii) There is an open G-invariant subset U ⊂ ¯H satisfying m(U) = 1, and for +any x ∈ U ∩ L there is an open set V containing x such that V is evenly +covered by V ⊂ Hm under the map π : ¯Hm → ¯H, and ψ +def += dev ◦ (π|V)−1 ◦ ϕ +coincides on its domain with a C-linear map, with real coefficients. +(iv) The subspace W +def += Im(ψ) is symplectic, and the measure m is obtained via +the cone construction from the Lebesgue measure on W. +(v) The complement L ∖ U is a finite union of supports of measures satisfying +properties (i)–(iv), for which the manifolds N ′ appearing in (ii) satisfy +dim N ′ < dim N. +Any orbit-closure for the P-action is a set L as above. +We will refer to L as an orbit-closure and to m = mL as a flat measure on L. +Orbit-closures are referred to as affine invariant manifolds and also as invariant +subvarieties. The use of an evenly covered neighborhood in item (iii) is a standard +approach for defining period coordinates (see e.g. [MS]). We refer to [Wr1] for a +survey containing more information on orbit-closures. + +10 +JON CHAIKA AND BARAK WEISS +In a forthcoming work of Brown, Eskin, Filip and Rodriguez-Hertz, the same +conclusion is obtained for the diagonal actions of P and G on a product of strata +H × H′. Namely, the following is shown: +Theorem 5.2. Let H, H′ be strata of translation surfaces, and let P and G act +on H × H′ via their diagonal embeddings in G × G. Then all of the conclusions of +Theorem 5.1 hold for this action (with ¯H × ¯H′ replacing ¯H). +6. Proof of main result +Using Theorem 5.2 and further work of Wright [Wr2], we can prove our main +result. +Proof of Theorem 1.2. Let Z0 = spanR(z0) be a one-dimensional connected real +Rel subgroup. Assume that (1) fails, so that the action of Z0 on (L, mL) is not +mixing of all orders. +Then, by Proposition 3.5 it is not ergodic. +Let θ be the +relatively independent self-joining over L//Z0. Applying Propositions 3.3 and 3.4 +we have that θ ̸= mL × mL and ∆L ⊂ supp θ. +Applying Proposition 4.2 and +Theorem 5.2, we have that there is a G-invariant open subset U of full θ-measure +such that U ∩ supp θ is the isomorphic image of an affine complex-linear manifold +whose dimension is strictly smaller than 2 dim ¯H, and θ is obtained from Lebesgue +measure on this complex-linear manifold by the cone construction. +We claim that the set +U1 +def += {q ∈ H : (q, q) ∈ U} +is of full measure for (π1)∗θ, where π1 : L × L → L is the projection onto the +first factor. Indeed, the measure θ is invariant under Z0 × {Id}, and hence so is +its support. Since Z0 acts by homeomorphisms where defined, and using property +(v) in Theorems 5.1 and 5.2, we have that the set U is also Z0 × {Id}-invariant. +Thus for any Z0-ergodic measure, it is either null or conull. Thus if q /∈ U1 and q is +generic for the measure µT (q) appearing in (3.1), µT (q) assigns measure zero to U. +If this were to happen for a positive measure of q it would follow from (3.1) that U +does not have full measure for θ. +For q ∈ U1, let Nq denote the connected component of U ∩ π−1 +1 (q) ∩ supp θ +containing (q, q). Since the fibers π−1 +1 (q) are also affine submanifolds of L × L, +we have that the Nq are affine submanifolds contained in π−1 +1 (q) ∼= L, so we can +identify them with invariant submanifolds in L (which we continue to denote by +Nq). With this notation we have q ∈ Nq. +The mapping q �→ T (Nq) is locally constant; that is, letting V ⊂ ¯H and V ⊂ ¯Hm +be open sets such that π|V : V → V is a homeomorphism and q ∈ V , the map +q �→ dev ◦ π|−1 +V (q) sends a neighborhood of q in Nq to an affine subspace Wq of +H1(S, Σ; R2), and the corresponding linear spaces Wq − Wq are the same for all +q ∈ V . Since mL × mL is the unique P-invariant ergodic measure on L × L of full +support, we have dim Nq < dim L for every q ∈ U1. +Let ¯Nq denote the set of surfaces (not necessarily of area one) which are obtained +by rescaling surfaces in Nq, and let +Nq +def += Tq( ¯Nq) + +ERGODICITY OF REL +11 +(the tangent space to ¯Nq at q, thought of as a subset of the tangent space Tq( ¯L)). +The assignment q �→ Nq defines a proper flat sub-bundle of the tangent bundle +T ( ¯L). Flat sub-bundles of T ( ¯L) were classified in [Wr2]. According to [Wr2, Thm. +5.1], Nq ⊂ RL for each q, and Nq is a complex linear subspace which is locally +constant. Since RL is acted on trivially by monodromy, we in fact have that Nq is +independent of q, and we denote it by R. The leaves R(q) are contained in ¯Nq for +each q, and of the same dimension. That is, R(q) is the connected component of +¯Nq containing q. Since Rel deformations do not affect the area of the surface, we +see that ¯Nq = Nq. In particular R(q) is closed for each q. +By Proposition 3.4, for a.e. q, Nq is the support of the ergodic component (mL)q, +and in particular +(mL)q(Nq) < ∞, +for a.e. q. +Since Z0 acts ergodically with respect to (mL)q, we have that almost surely Nq = +R(q). Since the measure (mL)q is affine in charts, it is a scalar multiple of the +translation-invariant measure on R(q), and thus the volume Vq of R(q) (with respect +to its translation-invariant measure) is almost surely finite. +It is clear that the +function q �→ Vq is U-invariant, and by ergodicity, it is constant almost surely. +□ +Remark 6.1. We note that the above argument works under much weaker conclu- +sions than those given in Theorem 5.2. Indeed, in the first step of the argument, +Theorem 5.2 was used simply to extract a G-invariant assignment q �→ Nq, where +Nq is a subspace of Tq(L), which is proper if θ is not the product joining. A fun- +damental fact about such G-invariant assignments is that they are very restricted +– besides [Wr2], see [EFW] and [Fi]. In particular, [Fi] gives strong restrictions on +assignments that are only assumed to be defined almost everywhere and measurable. +7. A topological condition for Rel ergodicity +Let Z0 ⊂ Z be a subspace. We say that a translation surface x is Z0-stably +periodic if it can be presented as a finite union of horizontal cylinders and the +Z0-orbit of x is well defined. Recall that a horizontal separatrix is a horizontal +leaf whose closure contains at least one singularity, and it is a horizontal saddle +connection if its closure contains two singularities. Then the condition of being Z0- +stably periodic is equivalent to requiring that all horizontal separatrices starting at +singular points are on horizontal saddle connections, and Z0 preserves the holonomy +of every horizontal saddle connection on x. +In case Z = Z0 is the full real rel +group, we say that x is fully stably periodic. This is equivalent to saying that all +horizontal separatrices starting at singular points are on saddle connections, and all +horizontal saddle connections start and end at the same singularity. In particular, +for any cylinder C on a fully stably periodic surface, each boundary component of +C is made of saddle connections starting and ending at the same singular point ξ; +we say that the boundary component only sees singularity ξ. For more information +on the real Rel action on surfaces which are horizontally completely periodic, see +[HW, §6.1]. +Proposition 7.1. Suppose x is a surface which is Z0-stably periodic, and v ∈ Z0 +moves two singularities p and q with respect to each other. Suppose that x contains +two cylinders C1 and C2 that both only see singularity p on one boundary component +and only see singularity q on another boundary component. Finally suppose the + +12 +JON CHAIKA AND BARAK WEISS +circumferences c1, c2 of these cylinders satisfy c1 +c2 /∈ Q. Then Case (2) of Theorem +1.1 does not hold for x. +Proof. Since c1 +c2 /∈ Q, the trajectory {Reltv(x) : t ∈ R} is not closed, let L denote +its closure. We claim that the tangent space to L is not contained in Z. Let σ1 +denote a saddle connection from p to q in C1 and let σ2 denote a saddle connection +from q to p in C2. Let σ be the concatenation. Then σ represents an absolute +homology class because it goes from p back to p, and it is nontrivial because the +vertical component of its holonomy on x is nonzero. If we consider the restriction +of the rel-action to C1 ∪ C2 then it only affects the twist parameters, which is a +2-dimensional space. This space can be generated by the horizontal holonomy of σ1 +and the horizontal holonomy of σ2. Since c1 +c2 /∈ Q, this restricted action does not give +a closed orbit. So the tangent space to L contains directions, which continuously +affect the holonomy of σ. Since σ is an absolute period, we see that the tangent +space to L is not contained in Z. +□ +8. Checking the condition for strata +Let H = H(a1, . . . , ak) and for i, j ∈ {1, . . . , k}, let ξi, ξj be the corresponding +singular points of a surface in H. +Let z ∈ R be a Rel cohomology class. +We +say that z moves ξi, ξj with respect to each other if for some (equivalently, every) +α ∈ H1(S, Σ) represented by a path starting at ξi and ending at ξj, we have +z(α) ̸= 0. Below when we discuss a stratum H(a1, . . . , ak) we allow ai = 0, that is +we allow marked points. We call points with cone angle 2π (that is, with a = 0) +removable singularities, and otherwise we call them non-removable. The following +result, which clearly implies Theorem 1.1, allows strata with removable singularities. +Theorem 8.1. Let H be a connected component of a stratum H(a1, . . . , ak). Let +mH be the Masur-Veech measure on H, let Z be the corresponding real Rel foliation, +and let Z0 ⊂ Z be a one-dimensional connected subgroup of Z. Suppose that there +are 1 ≤ i < j ≤ k with corresponding singular points ξi, ξj, such that ai > 0, aj > 0 +and such that some element of Z0 moves ξi, ξj with respect to each other. Then the +Z0-flow on (H, mH) is mixing of all orders (and in particular, ergodic). +Clearly, Theorem 8.1 follows from Theorem 1.2, Proposition 7.1, and the follow- +ing result. +Proposition 8.2. Let H ⊂ H(a1, . . . , ak) be a connected component of a stratum of +translation surfaces with at least two non-removable singular points. If p ̸= q is any +pair of non-removable singularities then there exists M ∈ H, which has cylinders +C1, C2 with circumferences c1, c2 so that +(1) M is fully stably periodic. +(2) +c1 +c2 /∈ Q. +(3) Both C1 and C2 only see singularity p on one boundary component and only +see singularity q on the other boundary component. +For the proof of Proposition 8.2 we will also need the following: +Proposition 8.3. Let H = H(a1, . . . , ak) be a stratum of translation surfaces with +at least two singular points (that is k ≥ 2). If p ̸= q is any pair of distinct sin- +gularities (possibly removable), then there exists M ∈ H, so that M is fully stably + +ERGODICITY OF REL +13 +△ +△ +c +Figure 1. The surface M has a cylinder of circumference c, and its +boundary components see only the singularities ξi and ξj (denoted +by ◦ and •). The edges not labeled by △ are connected to M ∖ C. +periodic and there exists a cylinder on M that only sees singularity p on one bound- +ary component, and only sees singularity q on the other boundary component. +Propositions 8.2 and 8.3 will both be proved by induction, after some prepara- +tions. +Lemma 8.4 (The basic surgery – gluing in a torus). Let H = H(b1, . . . , bℓ) be +a stratum of translation surfaces, and let M ∈ H, with singularities labeled by +ξ1, . . . , ξℓ, so that the order of ξi is bi. Suppose M has a horizontal cylinder C, with +circumference c, where one boundary component is made of saddle connections that +begin and end at ξi, and the other is made of saddle connections that begin and end +at ξj, where bi ≥ 0 and bj ≥ 0 (so that ξi, ξj might be removable). Then for all +w > 0 there exists M ′ ∈ H(b1, . . . , bi+1, . . . , bj+1, . . . , bℓ), with singularities labeled +ξ′ +1, . . . , ξ′ +ℓ, which has two horizontal cylinders C′ +1, C′ +2, where C′ +1 has circumference +c + w and C′ +2 has circumference w. The complements M ∖ C and M ′ ∖ (C1 ∪ C2) +are isometric, by an isometry mapping ξ′ +i to ξi for all i. The cylinders C1 and C2 +only see singularity ξ′ +i on one boundary component, and ξ′ +j on another. Moreover, +if M is fully stably periodic then so is M ′. +Proof. It will be easier to follow the proof while consulting Figures 1 (before) and +2 (after). +Given a polygonal presentation for M, we give a polygonal presentation +for M ′. +Let M be a polygon representation for M in which the cylinder C is +represented by a parallelogram P (in Figure 1, the large rectangle in the center +of the presentation), with two horizontal sides of length c, non-horizontal sides +identified to each other, and the singular points ξi, ξj on adjacent corners of P. +Thus the non-horizontal sides of P represent a saddle connection σ on M connecting +ξi to ξj. We consider the two non-horizontal sides of P as distinct and label them +by σ1, σ2. Let P ′ be a parallelogram with sides parallel to those of P, where the +horizontal sides have length w and the nonhorizontal sides are longer than the ones +on P (in Figure 2, P ′ is to the right of P). +Label the two horizontal sides of P ′ by h′ +1 and h′ +2, and identify them by a trans- +lation. Partition the non-horizontal sides of P ′ into two segments. The segments + +14 +JON CHAIKA AND BARAK WEISS +c +w +/ +/ +△ +△ +□ +□ +Figure 2. To obtain M ′ from M, glue in a torus (rectangle on the +right). This transforms C into a cylinder C′ +1 of circumference c+w, +and adds a horizontal cylinder C′ +2 of circumference w. Edges not +labeled by △, □, / or the color green are attached to M ′∖(C′ +1∪C′ +2). +σ′ +1, σ′ +2 are parallel to each other and have the same length as σ1, σ2, and start at a +corner of P. The segments γ′ +1, γ′ +2 comprise the remainder of the non-horizontal sides +of P ′ (and in particular, have the same length). Identify γ′ +1 to γ′ +2 by a translation, +and identify σ′ +1, σ′ +2 to σ1, σ2 by a translation so that each σ′ +i is attached to the σj +with the opposite orientation. Let M ′ be the translation surface corresponding to +this presentation. It is clear that M ′ has the required properties. +□ +Proof of Proposition 8.3. The proof is by induction on � ai. +Base of induction: The base case is the stratum H(a1, 0s), that is, one sin- +gular point (removable or non-removable) of order a1, and some number s ≥ 1 of +removable singular points. In this case we take a surface in H(a1) which is made +of one horizontal cylinder. We label the singular point by ξ1 and place additional +removable singular points ξ2, . . . , ξs+1 in the interior of the cylinder, at different +heights (so that the resulting surface has no horizontal saddle connections between +distinct singularities) and so that ξi and ξj are on opposite sides of a cylinder. +Inductive step: Suppose H′ = H(a1, . . . , ak) is our stratum, where at least +two of the singularities are non-removable. Let p′, q′ be labels of singular points +for surfaces in H′, corresponding to indices i ̸= j. To simplify notation assume +i = 1, j = 2. There are three cases to consider: ai = aj = 0, or one of ai, aj are +positive, or both are positive. +If ai = aj = 0 then by assumption k ≥ 4. We take a cylinder C on a fully +stably completely periodic surface M in H = H(a1, . . . , ˆai, . . . , ˆaj, . . . , ak). +The +notation ˆai means that the symbol should be ignored; that is on a stratum of the +same genus with k − 2 ≥ 2 singular points obtained by removing two removable +singular points. We place two singular points marked p′, q′ in the interior of C at +different heights. If ai > 0 and aj = 0 is zero we take a fully stably periodic surface +M in H(a1, . . . , ai − 1, . . . , ˆaj, . . . , ak), find a cylinder C on M whose boundary + +ERGODICITY OF REL +15 +c +w′ +w +/ +/ +// +// +△ +△ +□ +□ +∇ +∇ +Figure 3. First option for M ′ in Lemma 8.5. Attaching the sub- +surface on the right increases the genus by 2. Unlabeled edges are +attached to M ′ ∖ (C1 ∪ C2 ∪ C3). +component is made of saddle connections starting and ending at ξi, place a marked +point labeled ξj in the interior of C. If ai and aj are both positive we use the +induction hypothesis to find a surface M ∈ H(a1, . . . , ai − 1, . . . , aj − 1, . . . , ak) +with a cylinder whose boundary components see ξi and ξj, and we perform the +surgery in Lemma 8.4 to this cylinder. +□ +Lemma 8.5 (Two surgeries involving genus two surfaces). Let H = H(b1, . . . , bk) +be a stratum of translation surfaces and let M ∈ H have a horizontal cylinder C, +with circumference c. Let p and q be singular points with order bi, bj respectively, +such that one boundary component of C only sees singularity p and the other only +sees singularity q. Then for any w1, w2 > 0 there exists M ′ ∈ H′ = H(b1, . . . , bi + +2, . . . , bj + 2, . . . , bk) which has three cylinders C1, C2, C3 with circumferences c + +w1 + w2, w1 and w2 respectively. The complements M ∖ C and M ′ ∖ (C1 ∪C2 ∪C3) +are isometric by an isometry preserving the labels of singular points, and C1, C2, C3 +all have one boundary component that sees only p, and another that sees only q. +Thus, if M is fully stably periodic so is M ′. Moreover, if the bi are all even, so that +H′ has even and odd spin components, we can choose M ′ to be in either the even +or odd connected component. +Proof. Once again we encourage the reader to consult Figures 3 and 4. +In Lemma 8.4 we made a slit in M, running through P from top to bottom, +and glued in a torus with a slit. In this case we make an identical slit, this time +gluing in a genus two surface with a slit. This surface is presented in Figures 3 and +4 as made up of three rectangles. It is straightforward to check that M ′ ∈ H′ and +that it has cylinders satisfying the desired properties. It remains to check the final +assertion about the parity of the spin structure. +Recall from [KZ, eqn. (4)] that where defined, the spin structure of a surface +M of genus g can be computed as follows. Let αi, βj (where 1 ≤ i, j ≤ g) be a +symplectic basis for H1(M), realized explicitly as smooth curves on M. This means + +16 +JON CHAIKA AND BARAK WEISS +c +w′ +w +/ +/ +// +// +△ +△ +∇ +∇ +□ +□ +Figure 4. Second option for M ′, with a different spin. +that all of these curves are disjoint, except for αi and βi which intersect once. For +each curve γ, let ind(γ) be the turning index, that is the total number of circles +made by the tangent vector to γ, as one goes around γ. The parity of M is then the +parity of the integer �g +i=1(1 + ind(αi))(1 + ind(βi)). It is shown in [KZ] that this +number is well-defined (independent of the choice of the symplectic basis) when all +the singular points have even order. +Suppose M has genus g and is equipped with a symplectic basis. Since any +non-separating simple closed curve can be completed to a symplectic basis, we can +assume that α1 is the core curve of C, and the other curves in the basis do not +intersect the saddle connection from p to q passing through C. We construct a sym- +plectic basis for M ′ in both cases, by modifying α1, keeping α2, . . . , αg, β1, . . . , βg, +and adding new curves αg+1, αg+2, βg+1, βg+2. The modified curves are shown in +Figures 5, 6, and the reader can easily check that these new curves still form a +symplectic basis, and that these two choices add two numbers of different parities +to the spin structure. +□ +Note that in Proposition 8.2 we care about all connected components of strata. +We need to record some information about the classification of connected compo- +nents of strata, due to Kontsevich and Zorich. A translation surface is hyperelliptic +if it admits an involution which acts on absolute homology as −Id (see [FM] or +[KZ, §2.1] for more details). A connected component of a stratum is hyperelliptic +if all surfaces in the component are hyperelliptic. +Proposition 8.6 ([KZ], Theorems 1 & 5 and Corollary 5 of Appendix B). Let +H(a1, . . . , ak) be a stratum with ai. > 0 for all i. The following holds: +• H has three connected components in the following cases: +– k = 1, a1 = 2g − 2, g ≥ 4. +– k = 2, a1 = a2 = g − 1, g ≥ 5 is odd. One is hyperelliptic, and the two +non-hyperelliptic strata are distinguished by the spin invariant. +• H has two connected components in the following cases: + +ERGODICITY OF REL +17 +αg+1 +βg+1 +βg+2 +αg+2 +α1 +Figure 5. Modifying the symplectic basis. Gluings as in Figure 3. +α1 +αg+1 +βg+1 +βg+2 +αg+2 +Figure 6. Modifying the symplectic basis, second case. Gluings +as in Figure 4. Note the change in the rotation number of βg+2. +– All of the ai are even, g ≥ 4, and either k ≥ 3 or a1 > a2. The +components are distinguished by their spin. +– a1 = a2 and g is either 3 or is even. One of the components is hyper- +elliptic and the other is not. When g = 3 the hyperelliptic component +is even, and the other one is odd. +• H is connected in all other cases. +Proof of Proposition 8.2. The proof will be case-by-case. Here are the cases: +(i) H(1, 1). +(ii) All the ai are nonzero and H is connected. +(iii) All the ai are nonzero and H has two connected components distinguished +by spin. + +18 +JON CHAIKA AND BARAK WEISS +(iv) All the ai are nonzero and H has two connected components distinguished +by hyperellipticity. +(v) All the ai are nonzero and H has three connected components. +(vi) Some of the ai are zero. +Case (i). There is just one connected component and the desired surface is +a Z-shaped surface, with three horizontal cylinders C1, C2, C3 of circumferences +c1, c1 + c3, c3, where C1, C3 are simple. We put all of the removable singular points +in the interior of C3, and choose choose c1, c3 so that c1/(c1 + c3) /∈ Q. It is clear +that with these choices the conditions are satisfied. +Case (ii). The stratum H is connected, and we have at least two singularities +of positive order. So with no loss of generality that they are labelled 1 and 2. The +result follows from Lemma 8.4, applied to a surface in H(a1 − 1, a2 − 1, a3, . . . , ak), +and taking w /∈ cQ, so that w/(c + w) /∈ Q. +Case (iii). We apply the surgery in Lemma 8.5, with w1/w2 /∈ Q. Namely if p +and q are labelled i, j, we let bi = ai − 2, bj = aj − 2 and bℓ = aℓ for ℓ ̸= i, j. +Case (iv). There are two connected components. One is hyperelliptic, one is +not. This means that a1 = a2 and either g = 3 (in which case a1 = a2 = 2) or +g ≥ 4 is even (in which case a1 = a2 = g − 1). In this case we give explicit surfaces, +one in each connected component. The first surface (the H(2, 2) case is shown in +Figure 7) is a ‘staircase’ surface made of gluing 2g rectangles to each other. The +rectangles are labelled (k, B) and (k, T ) for k = 1, . . . , g. The top (respectively, +bottom) of (k, B) is glued to the bottom (resp., top) of (k, T ) for k = 1, . . . , g, and +the left (resp., right) of (k, T ) is glued to the right (resp., left) of (k + 1, B) for +k = 1, . . . , g − 1. The horizontal sides of (1, B) are glued to each other, as are the +horizontal sides of (g, T ). This surface is hyperelliptic since it has a hyperelliptic +involution rotating each rectangle around its midpoint, and this involution swaps +the singularities (see [KZ, Remark 3]). The second surface is obtained as follows. +We first construct a hyperelliptic surface in H(a1 − 2, a2 − 2) as in the previous +paragraph. Then we perform the surgery described in Lemma 8.5. The resulting +surface has a horizontal cylinder intersecting three vertical cylinders, and thus, by +[Li, Lemma 2.1], is not hyperelliptic. See Figure 8 for an example in H(2, 2). In both +of these constructions there are no restrictions on the sidelengths of the rectangles, +and we can easily arrange that two of the circumferences are incommensurable. +Case (v). In this case a1 = a2 = g − 1 for g ≥ 5 odd. Applying the argument +in Case (iii), we obtain the required surfaces in the odd and even connected com- +ponents. To obtain the required surface in the hyperelliptic component we use the +‘staircase surface’ describe in Case (iv). +Case (vi). +Assume with no loss of generality that the removable singularities +are labelled k′+1, . . ., k for some k′ ≥ 2, and let H′ = H(a1, . . . , ak′). Note that the +singularities p and q have label in {1, . . . , k′}. Apply the preceding considerations +to obtain a surface in H′ with the required cylinders. By examining the proof in +all preceding case one sees that the number of horizontal cylinders on this surface +is at least three, that is there is at least one cylinder C3 which is distinct from +the cylinders C1, C2, and we modify M by adding k − k′ in general position in the +interior of C3, to obtain the desired surface. +□ + +ERGODICITY OF REL +19 +Figure 7. A surface in Hhyp(2, 2). +References +[BSW] M. Bainbridge, J. Smillie and B. Weiss, Horocycle dynamics: new invariants and eigen- +form loci in the stratum H(1, 1), preprint (2016) to appear in Mem. AMS. +[CDF] G. Calsamiglia, B. Deroin and S. Francaviglia, A transfer principle: from periods to isope- +riodic foliations, preprint (2015) https://arxiv.org/pdf/1511.07635.pdf +[EW] M. Einsiedler and T. Ward, Ergodic theory with a view towards number theory, +Grad. Texts in Math. 259, Springer (2010). +[EFW] A. Eskin, S. Filip and A. Wright, The algebraic hull of the Kontsevich-Zorich cocycle, +Ann. of Math. (2) 188 (2018), no. 1, 281—313. +[EM] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL2(R)-action on +moduli space, Publ. Math. Inst. Hautes ´Etudes Sci. 127 (2018), 95-–324. +[EMM] A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit clo- +sures for the SL2(R)-action on moduli space, Ann. Math. (2) 182 (2015), no. 2, 673–721. +[Ham] U. Hamenst¨adt, Ergodicity of the absolute period foliation, Israel J. Math. 225 (2018), no. +2, 661–680. +[FM] B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical +Series, Vol. 49 (2012), Princeton University Press. +[Fi] S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich Cocycle, Invent. Math. 205 +(2016), no. 3, 617-–670 +[HW] P. Hooper and B. Weiss, Rel leaves of the Arnoux-Yoccoz surfaces, with an appendix by L. +Bary-Soroker, M. Shusterman and U. Zannier, Selecta Math., (2018) 24, no. 2, 875–934. +[KT] A. B. Katok and J.-P., Thouvenot, Spectral properties and combinatorial constructions in +ergodic theory, in Handbook of Dynamical Systems, Vol. 1B, 649–753, Elsevier (2006). + +20 +JON CHAIKA AND BARAK WEISS +Figure 8. A surface in Hnonhyp(2, 2). +[KZ] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differ- +entials with prescribed singularities, Invent. Math. 153 (2003), no. 3, 631–678. +[Li] K. Lindsey, Counting invariant components of hyperelliptic translation surfaces, Israel J. +Math. 210 (2015), p. 125-146. +[MS] H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations, +Ann. Math. 134 (1991) 455–543. +[MaTa] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of +dynamical systems, Enc. Math. Sci. Ser. (2001). +[McM] C. McMullen, Moduli spaces of isoperiodic forms on Riemann surfaces, Duke Math. J. +163 (2014), no. 12, 2271?2323. +[MW2] Y. Minsky and B. Weiss, Cohomology classes represented by measured foliations, and +Mahler’s question for interval exchanges, Annales Sci. de L’ENS 2 (2014) 245–284. +[Mo] S. Mozes, Mixing of all orders of Lie groups actions, Invent. Math. 107 (1992), no. 2, +235-–241. +[P] +K. Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, 2. Cambridge +University Press (1983). +[dlR] T. de la Rue, An introduction to joinings in ergodic theory, +Disc. Cont. Dyn. Sys. 15 1 +121–142 (2006). +[Va] V. S. Varadarajan, Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 109 +(1963), 191-–220. +[Wi1] K. Winsor, Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms, +preprint (2021) https://arxiv.org/abs/2109.12669 +[Wi2] K. Winsor, Dense real Rel flow orbits and absolute period leaves, preprint (2022), +https://people.math.harvard.edu/~kwinsor/preprints/Winsor DenseLeaves 060622.pdf +[Wr1] A. Wright, Translation surfaces and their orbit closures: an introduction for a broad au- +dience, EMS Surv. Math. Sci. 2 (2015), no. 1, 63–108. +[Wr2] A. Wright, The field of definition of affine invariant submanifolds of the moduli space of +abelian differentials, Geom. Topol. 18 (2014), no. 3, 1323-–1341. + +ERGODICITY OF REL +21 +[Y] F. Ygouf, A criterion for density of the isoperiodic leaves in rank 1 affine invariant suborb- +ifolds, J. Top. 16 (2023) 1–19. +[Zo] A. Zorich, Flat surfaces, in Frontiers in number theory, physics and geometry, P. +Cartier, B. Julia, P. Moussa and P. Vanhove (eds), Springer (2006). +University of Utah chaika@math.utah.edu +Tel Aviv University barakw@tauex.tau.ac.il + diff --git a/TtE0T4oBgHgl3EQflQFO/content/tmp_files/load_file.txt b/TtE0T4oBgHgl3EQflQFO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..723f8bdb2cbd4d8ab0311d78d2f8f100cfcb1086 --- /dev/null +++ b/TtE0T4oBgHgl3EQflQFO/content/tmp_files/load_file.txt @@ -0,0 +1,970 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf,len=969 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='02483v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='DS] 6 Jan 2023 ON THE ERGODICITY OF THE REL FOLIATION JON CHAIKA AND BARAK WEISS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H be a stratum of translation surfaces with at least two sin- gularities, let mH denote the Masur-Veech measure on H, and let Z0 be a flow on (H, mH) obtained by integrating a Rel vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We prove that Z0 is mixing of all orders, and in particular is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We also characterize the ergodicity of flows defined by Rel vector field, for more general spaces (L, mL), where L ⊂ H is an orbit-closure for the action of G = SL2(R) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=', an affine invariant subvariety) and mL is the natural measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Our results are condi- tional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Introduction Let H be a stratum of area-one translation surfaces and let G def = SL2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' There is a G-invariant finite measure mH on H known as the Masur-Veech measure, and the dynamics of the G-action on (H, mH) have been intensively studied in recent years and are intimately connected to many problems in geometry and ergodic theory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' [MaTa, Zo].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Suppose that surfaces in H have k singularities, where k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then there is a k − 1-dimensional foliation of H, known as the real Rel foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A precise definition of the foliation and some of its properties will be given below in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Loosely speaking, two surfaces are in the same real Rel leaf if one can be obtained from the other by a surgery in which singular points are moved with respect to each other in the horizontal direction, without otherwise changing the geometry of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A natural question, which we address here, is the ergodic properties of this foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' As we review in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2, by labeling the singularities and removing a set of leaves of measure zero, we can think of the real Rel leaves as being the orbits of an action of a group Z on H, where Z ∼= Rk−1, and the restriction of this action to any one- dimensional subgroup of Z defines a flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Our first main result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H be a connected component of a stratum H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) with all ai > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=', no marked points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let mH be the Masur-Veech measure on H, let Z ∼= Rk−1 be the corresponding action given by translation along the leaves of the real Rel foliation, and let Z0 ⊂ Z be any one-dimensional connected subgroup of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then the Z0-flow on (H, mH) is mixing of all orders (and in particular, ergodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The definition of mixing of all orders is given in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For purposes of this introduction it is enough to note that it implies ergodicity of any nontrivial element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Note that when H has marked points, there will be subgroups Z0 which only move the marked points on surfaces without otherwise changing the geometry, and the conclusion of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 will not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This is the only obstruction to generalizing our results to strata with marked points, see Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 1 2 JON CHAIKA AND BARAK WEISS The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1, as well as the other results of this paper, relies crucially on measure-rigidity results of Eskin, Mirzakhani and Mohammadi [EM, EMM], and further forthcoming work extending these results, which we will describe in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 improves on the results of several authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In those results, ergod- icity for the full Rel foliation was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The full Rel foliation (also referred to as the ‘kernel foliation’, ‘isoperiodic foliation’, or ‘absolute period foliation’) will also be defined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Its leaves are of dimension 2(k − 1), that is, twice the di- mension of the real Rel leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Loosely speaking, two surfaces are in the same leaf for this foliation if one can be obtained from the other by moving the singularities (without otherwise affecting the geometry of the surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' That is, we relax the hypothesis that points can only be moved horizontally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The first ergodicity results for the full Rel foliation were obtained by McMullen [McM], who proved ergodic- ity in the two strata H(1, 1) and H(1, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Subsequently, Calsamiglia, Deroin and Francaviglia [CDF] proved ergodicity in all principal strata, and Hamenst¨adt [Ham] reproved their result by a simpler argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Recently, Winsor [Wi1] proved ergodicity for most of the additional strata, and in [Wi2], showed that there are dense orbits for the Z0-flow, for any Z0 as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Note that ergodicity for a foliation is implied by ergodicity for any of its subfoliations, and that ergodicity implies the existence of dense leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' and thus Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 generalizes all of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Also note that full Rel is a foliation which is not given by a group action, and the notions of mixing and multiple mixing do not make sense in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The papers [McM, CDF] go beyond ergodicity and obtain classifications of full Rel leaves in their settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We suspect that there is not a reasonable classification of real Rel leaf-closures, indeed it is already known (see [HW]) that there are real rel trajectories that leave every compact set never to return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The strata H support other interesting measures for which similar questions could be asked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Namely, by work of Eskin, Mirzakhani and Mohammadi [EM, EMM], for any q ∈ H, the orbit-closure L def = Gq is the support of a unique smooth G-invariant measure which we denote by mL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let ZL be the subgroup of Z leaving L invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then ZL also preserves mL and for many choices of L, we have dim ZL > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In these cases, for any closed connected Z1 ⊂ ZL, there is a complexification R1, which gives a foliation of L whose leaves R1(q) have dimension 2 dim Z1 (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The leaves R1(q) have a natural translation structure, and this induces a natural locally finite translation-invariant measure on each leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' With this terminology we can now state the main result of this paper: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let L be a G-orbit closure, and let mL, ZL, RL be as above, where dim ZL > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let z0 be a nontrivial element of ZL and let Z0 = spanR(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then either (1) Z0 is mL mixing of all orders (and in particular, z0 acts ergodically);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' or (2) there is an intermediate closed connected subgroup Z1 so that Z0 ⊂ Z1 ⊂ ZL, and the complexification R1 of Z1 satisfies for every q ∈ L, the leaf R1(q) is closed, and for mL-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' q, R1(q) is of finite volume with respect to its translation- invariant measure, and Z0q = R1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Thus, in order to establish ergodicity of real Rel subfoliations on G-orbit-closures, it is enough to rule out Case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We will prove Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1, which provides a ERGODICITY OF REL 3 simple way to achieve this, using cylinder circumferences of surfaces in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 are deduced from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2 using Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The following statement is an immediate consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let L be a G-orbit-closure, let mL, ZL be as above, and let Z1 ⊂ ZL be one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Suppose that the foliation induced by the complexification R1 has a dense leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then the Z1-flow on (L, mL) is mixing of all orders (and in particular, ergodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The density of certain leaves of the full Rel foliation in G-orbit-closures of rank one was obtained by Ygouf in [Y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Using these results we obtain ergodicity of one- dimensional subgroups of the real Rel foliation in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For instance, using [Y, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A & Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1] we have: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The real Rel foliation is mixing of all orders (and in particular, ergodic) in any eigenform locus in H(1, 1) with a non-square discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Recall that in [Wi2] Winsor proved the existence of dense real Rel leaves, and dense leaves for one-dimensional flows Z0, in all strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Using these results in conjunction with Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3, one can obtain an alternative proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 that avoids the use of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In §2 we give background material on translation surfaces, their moduli spaces, and the Rel foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In §3 we use standard facts about joinings to build a measure θ on the product of two strata (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1)), depending on a real- Rel flow Z0, such that if θ is the product measure, then Z0 is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3 we discuss a technique of Mozes that makes it possible to upgrade ergodicity to mixing of all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In §4 we show that θ is ergodic for the diagonal action of the upper triangular group P ⊂ G on the product of two strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In §5 we state a far-reaching measure rigidity result of Brown, Eskin, Filip and Rodriguez-Hertz for P-ergodic measures on products of two strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In §6 we use this measure rigidity result, as well as prior results for the action on one stratum due to Wright, in order to characterize the situations in which θ is not a product measure, thus proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We are very grateful to Alex Eskin for crucial contri- butions to this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We also thank Simion Filip and Alex Wright for useful dis- cussions, and acknowledge the support of ISF grants 2919/19, BSF grant 2016256, NSFC-ISF grant 3739/21, a Warnock chair, a Simons Fellowship, and NSF grants DMS-1452762 and DMS-2055354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Preliminaries about translation surfaces 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Strata, period coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In this section we collect standard facts about translation surfaces, and fix our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For more details, we refer to reader to [Zo, Wr1, BSW].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Below we briefly summarize the treatment in [BSW, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let S be a compact oriented surface of genus g, Σ = {ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ξk} ⊂ S a finite set, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak non-negative integers with � ai = 2g−2, and H = H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) the cor- responding stratum of unit-area translation surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We let Hm = Hm(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) denote the stratum of unit-area marked translation surfaces and π : Hm → H the 4 JON CHAIKA AND BARAK WEISS forgetful mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Our convention is that singular points are labeled, or equiv- alently, H = Hm/ Mod(S, Σ), where Mod(S, Σ) is the group of isotopy classes of orientation-preserving homeomorphisms of S fixing Σ, up to an isotopy fixing Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' There is an R>0-action that dilates the atlas of a translation surface by c ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For a stratum H and marked stratum Hm, we denote the collection of surfaces of arbitrary area, obtained by applying such dilations, by ¯H, ¯Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The marked stratum ¯Hm is a linear manifold modeled on the vector space H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It has a developing map dev : ¯Hm → H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2), sending an element of ¯Hm represented by f : S → M, where M is a translation surface, to f ∗(hol(M, ·)), where for an oriented path γ in M which is either closed or has endpoints at singularities, hol(M, γ) = �� γ dx � γ dy � , and dx, dy are the 1-forms on M inherited from the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Furthermore, there is an open cover {Uτ} of Hm, indexed by triangulations τ of S with triangles whose vertices are in Σ, and maps dev|Uτ : Uτ → H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2), which are homeomorphisms onto their image, and such that the transition maps on overlaps for this atlas are restrictions of linear automorphisms of H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This atlas of charts {(Uτ, dev|Uτ )} is known as period coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since each Uτ is identified via period coordinates with an open subset of the vector space H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2), the tangent space at each Uτ is identified canonically with H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2), and thus the tangent bundle of Hm is locally constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A sub-bundle of the tangent bundle is called locally constant or flat if it is constant in the charts afforded by period coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The Mod(S, Σ)-action on Hm is properly discontin- uous, and hence H is an orbifold, and the map π : Hm → H is an orbifold covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The group G acts on translation surfaces in H by modifying planar charts, and acts on H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2) via its action on R2, thus inducing a G-action on Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The G- action commutes with the Mod(S, Σ)-action, and thus the map π is G-equivariant for these actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The G-action on Hm is free, since dev(gq) ̸= dev(q) for any nontrivial g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We will use the following subgroups of G: gt = � et 0 0 e−t � , us = � 1 s 0 1 � U = {us : s ∈ R}, P = ��a b 0 a−1 � : a > 0, b ∈ R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Rel foliation and real Rel foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We define and list some important properties of the Rel foliation, the real-Rel foliation, and the corresponding action on the space of surfaces without horizontal saddle connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' See [MW2, BSW] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' See also [Zo, McM], and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We have a canonical splitting R2 = R ⊕ R and we write R2 = Rx ⊕ Ry to distinguish the two summands in this splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' There is a corresponding splitting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1) H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2) = H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Rx) ⊕ H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Ry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We also have a canonical restriction map Res : H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2) → H1(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2) (given by restricting a cochain to absolute periods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since Res is topologically defined, its kernel ker(Res) is Mod(S, Σ)-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Moreover, from our convention that singular points are marked, the Mod(S, Σ)-action on ker(Res) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' ERGODICITY OF REL 5 Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2) R def = ker(Res) and Z def = R ∩ H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Rx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Rx) and H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Ry) are naturally identified with each other via their identification with H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R), for each Z1 ⊂ Z we can define the space R1 spanned by the two copies of Z1 in H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Rx) and H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Ry) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The space R1 is the complexification of Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This terminology arises from view- ing H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2) as H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' C), a vector space over C, viewing H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Rx) and H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Ry) as the real and imaginary subspace of this complex vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' With this viewpoint, R1 is the C-span of Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For any subspace Z1 ⊂ R, we can foliate the vector space H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2) by affine subspaces parallel to Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Pulling back this foliation using the period coordinate charts gives rise to a foliation of ¯Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since monodromy acts trivially on R, this foliation descends to a well-defined foliation on ¯H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It is known (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' [BSW, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1]) that the area of a surface is constant on leaves of the Rel foliation, and thus the Rel foliation and any of its subfoliations descends to a foliation of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The foliation corresponding to R (respectively, to Z) is known as the Rel foliation (respectively, the real Rel foliation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Because the Mod(S, Σ)-monodromy action fixes all points of R, the leaves of the Rel foliation, and any of its sub-foliations, acquire a translation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In particular, they are equipped with a natural measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For any v ∈ Z we have a constant vector field, well-defined on Hm and on H, everywhere equal to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Integrating this vector field we get a partially defined real REL flow (corresponding to v) (t, q) �→ Reltv(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' the flow may not be defined for all time due to possible ‘collide of zeroes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For every q ∈ H it is defined for t ∈ Iq, where the domain of definition Iq = Iq(v) is an open subset of R which contains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The sets Iq(v), are explicitly described in [BSW, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let ˆH denote the set of surfaces in H with no horizontal saddle connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then Iq = R for all q ∈ ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If q ∈ H, s ∈ R and τ ∈ Iq then τ ∈ Iusq, and Relτv(usq) = usRelτv(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Similarly, if q ∈ H, t ∈ R and τ ∈ Iq then τ ′ def = etτ ∈ Igtq and Relτ ′v(gtq) = gtRelτv(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In particular, since P preserves ˆH and P = {gtus : t, s ∈ R}, there is an action of P ⋉ Z on ˆH, given by (p, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='q = pRelz(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Preliminaries from ergodic theory 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Ergodic decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We will use the notation G ⟳ (X, µ) to indicate that G is a locally compact second countable group, (X, B) is a standard Borel space, and µ is a probability measure on B preserved by the G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We say that G ⟳ (Y, ν) is a factor of (X, µ) if there is a measurable G-invariant conull subset X0 ⊂ X, and a measurable map T : X0 → Y such that T ◦ g = g ◦ T for all g ∈ G, and ν = T∗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In this situation we refer to T as the factor map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Given a factor map, there is a (unique up to nullsets) measure disintegration µ = � µy dν(y), for a Borel mapping y �→ µy from Y to the space of Borel probability measures on X, such that µy(T −1(y)) = 1 for ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Equivalently we can write µ = � x µ′ x dµ(x), where µ′ x def = µT (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For a closed subgroup H ⊂ G, we say that µ is H-ergodic if any invariant set is null or conull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We have the following well-known ergodic decomposition theorem: 6 JON CHAIKA AND BARAK WEISS Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Suppose G ⟳ (X, µ), and H is a closed subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then there is a factor of H ⟳ (X, µ), called the space of ergodic components and denoted by X//H, with the following properties: (i) For ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' y ∈ X//H, µy is H-invariant and H-ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (ii) H acts trivially on X//H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (iii) H ⟳ (X, µ) is ergodic if and only if X//H = {pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (iv) The properties (i)–(iii) uniquely determine the factor X//H up to measur- able isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (v) If H ✁ G then G ⟳ (X//H, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For (i) and (ii) see [Va, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4] (in the notation of [Va], these assertions follow from the fact that β is a map on points and is H-invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Assertion (iii) is immediate from definitions and (iv) follows from [Va, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For (v), one can argue using the uniqueness property (iv), and the fact that the image of an H-invariant ergodic measures under any element g ∈ G is also H-invariant and ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' An action is called prime if it has no factors (besides the action itself, and the trivial action on a point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The construction above shows that if H ✁ G, G′ is a subgroup of G so that G′ ⟳ (X, µ) is prime and H ⟳ (X, µ) is not isomorphic to the trivial action, then H ⟳ (X, µ) is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This is not the approach we will take for proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Joinings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We recall some well-known facts about joinings, see [dlR] and ref- erences therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let G ⟳ (Xi, µi) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A joining is a measure θ on X1 × X2, invariant under the diagonal action of G, such that πi∗θ = µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A self-joining is a joining in case X1 = X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If (Xi, µi) → (Z, ν) is a joint factor then the relatively independent joining over Z is the joining � Z(µ1)z × (µ2)z dν(z), where µi = � Z(µi)z dν(z) is the disintegration of µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In case X1 = X2 = X, and Z = X//H is the space of ergodic components of the action of H on (X, µ) as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1, we obtain the relatively independent self-joining over X//H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This joining satisfies: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The following are equivalent: H ⟳ (X, µ) is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The relatively independent self-joining over X//H is µ × µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We note two properties of this self-joining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We fix a topology on X which generates the σ-algebra, and denote by supp µ the topological support of µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=', the smallest closed set of full measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let θ be the measure on X×X which is the relatively independent self-joining over X//H, for some H, and let T : X → X//H be the factor map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then the following hold: We have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1) θ = � X µT (x) × µT (x) dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If X = supp µ then supp θ contains the diagonal ∆X def = {(x, x) : x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' ERGODICITY OF REL 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1) is immediate from the definition of the relatively independent self-joining over X//H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since each µ′ x = µT (x) is H-invariant and ergodic, and µ′ x(T −1(T (x))) = 1, the set {x ∈ X : x /∈ supp µ′ x} is a nullset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' From this, and from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1) we obtain the second assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Ergodicity, mixing, and mixing of all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For G ⟳ (X, µ), let L2 0(µ) denote the Hilbert space of L2-functions on (X, µ) of integral zero, and let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The action is called k-mixing if for any f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , fk ∈ L2 0(µ) and for any k−1 sequences � g(i) n � n∈N ∈ G, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , k − 1, for which all of the sequences � g(i) n � n∈N (1 ≤ i ≤ k − 1) and � g(i) n (g(j) n )−1� n∈N (1 ≤ i < j ≤ k − 1) eventually leave every compact subset of G, we have � X f1 � g(1) n x � · · fk−1 � g(k−1) n x � fk(x) dµ(x) n→∞ −→ k � i=1 � X fi dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We say that the action is mixing if it is 2-mixing, and mixing of all orders if it is mixing of order k for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It is easy to check that mixing implies ergodicity of any unbounded subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We have the following: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let Z0 ∼= R and let {gt} be a one-parameter group acting on Z0 by dilations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=', for all v ∈ Z0 and t ∈ R we have gtv = eλtv for some λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let F = {gt} ⋉ Z0 and let F ⟳ (X, µ) be a probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The following are equivalent: (a) the restricted flow Z0 ⟳ (X, µ) is ergodic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (b) the restricted flow Z0 ⟳ (X, µ) is mixing of all orders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (c) the restricted flow Z0 ⟳ (X, µ) is mixing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (d) any nontrivial element of Z0 acts ergodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The group F appearing in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='5 is isomorphic as a Lie group to the subgroup P of upper triangular matrices in G, but in our application we will use it for the group generated by a one-parameter real Rel flow Z0 and the diagonal flow {gt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Clearly (b) =⇒ (c) =⇒ (d) =⇒ (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We assume that the Z0-flow is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' To see that it is mixing, it is enough by [P, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 2, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='9] to prove that it has countable Lebesgue spectrum, and for this, use [KT, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='23 & Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The proof of mixing of all orders follows verbatim from an argument of Mozes [Mo], for mixing actions of Lie groups which are ‘Ad-proper’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since our group F is not Ad-proper, we cannot cite [Mo] directly, so we sketch the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For notational convenience we deduce 3-fold mixing from mixing (the proof that ‘k-fold mixing =⇒ k + 1-fold mixing’, for k ≥ 3, is identical but requires more cumbersome notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We use additive notation in the group Z0, and denote the action of Z0 on X by (z, x) �→ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let (bn)n∈N and (cn)n∈N be sequences in Z0 such that each of the sequences (bn)n∈N , (cn)n∈N , (bn + cn)n∈N eventually leaves every compact set, and let f1, f2, f3 be in L2 0(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We need to prove that � X f1(x)f2(bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='x)f3((bn + cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='x) dµ(x) n→∞ −→ � X f1 dµ � X f2 dµ � X f3 dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 8 JON CHAIKA AND BARAK WEISS For each n, define a measure µn on X3 def = X × X × X by � X3 f dµn def = � X f(x, bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='x, (bn + cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='x) dµ(x), ∀f ∈ Cc(X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' That is, µn is the pushforward of the diagonal measure on X3 by the sequence (0, bn, bn + cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It is easy to see that 3-mixing is equivalent to the fact that the weak-* limit of µn is the measure µ3 def = µ×µ×µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The group F 3 def = F ×F ×F acts on X3 by acting separately on each component, and as in [Mo], since Z0 is mixing, it suffices to show that any measure ν on X3 which is a weak-* limit of a subsequence of (µn)n∈N, is invariant under (0, u, v) ∈ R3 ⊂ F 3, for some (u, v) ∈ R2 ∖ (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We claim that for any s ∈ R the measure µn is invariant under hn(s) def = (gs, bn · gs · (−bn), (bn + cn) · gs · (−bn − cn)) , where the multiplication is in the group F 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Indeed, since µ is {gs}-invariant, � X3 f dµn = � X f (gsx, bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (gsx), (bn + cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (gsx)) dµ(x), and hn(s) · (idF , bn, bn + cn) = (gs, bn · gs, (bn + cn) · gs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' That is, applying hn(s) changes one description of µn to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We embed F as a multiplicative group of matrices in GL2(R) and let dF be the metric on F induced by some norm on GL2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' By a straightforward computation we have hn(s) = � gs, (1 − eλs)bn · gs, (1 − eλs)(bn + cn) · gs � , and dF (idF , hn(sn)) is a continuous function of s which goes to 0 as s → 0 and for any fixed s > 0, increases to infinity as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Therefore we can choose sn → 0 so that dF (idF , hn(sn)) = 1 for all large enough n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' As in [Mo], ν is invariant under some subsequential limit of hn(sn) which is of the form (0, u, v) for some (u, v) ∈ R2 ∖ (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This establishes our sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The relatively independent self-joining for a Rel flow Recall that ˆL ⊂ L is the set of surfaces without horizontal saddle connections, and this is a P-invariant set of full measure with respect to mL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We can combine the product action of ZL × ZL on ˆL × ˆL with the diagonal action of P to obtain an action of the semi-direct product P ⋉ (ZL × ZL) on ˆL × ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since ˆL ⊂ L is of full measure, and the arguments of this section involve passing to sets of full measure, in the remainder of this section we will ignore the distinction between L and ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let Z ⊂ ZL be a closed connected subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If θ is an invariant probability measure for an action of the semidirect product P ⋉ (Z × Z) on L × L, then any f ∈ L2(θ) which is {gt}-invariant is also Z × Z-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For any z ∈ Z ×Z, gtzg−t →t→−∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' So the claim follows from the Mautner phenomenon, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' [EW, Prop 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let (L, mL) be a G-orbit-closure with a fully supported P- invariant ergodic measure, let Z ⊂ ZL be a connected closed subgroup, and let θ on L × L be the relatively independent joining over L//Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then θ is P-invariant and {gt}-ergodic (and hence P-ergodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Also ∆L ⊂ supp θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' ERGODICITY OF REL 9 As we will see in §5, under the conditions of the Proposition, mL is the so-called ‘flat measure’ on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let π : L × L → L be the projection onto the first factor, and let ν = π∗θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For each x ∈ L, let Ωx def = π−1(x) = {x} × L be the fiber, and let θx be the fiber measure appearing in the disintegration θ = � L θx dν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then Z acts on Ωx via the second factor in Z × Z, and θx is Z-invariant and ergodic by the definition of the ergodic decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1(v) that θ is P-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' To prove ergodicity, let f ∈ L2(L×L, θ) be a P-invariant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1, f is Z×Z-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For each x ∈ L, let fx def = f|Ωx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' There is L0 ⊂ L such that mL(L0) = 1 and for every x ∈ L0, fx belongs to L2(Ωx, θx) and is Z-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Hence, by ergodicity, there is ¯f : L0 → R such that for every x ∈ L0, ¯f(x) is the θx-almost-sure value of fx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since f is P-invariant for the diagonal action of P, ¯f is P-invariant for the action of P on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' By ergodicity of P ⟳ (L, mL), ¯f is ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' constant, and thus f is θ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The last assertion follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' An upgraded magic wand theorem The celebrated ‘magic wand’ Theorem of Eskin and Mirzakhani [EM], and en- suing work of Eskin, Mirzakhani and Mohammadi [EMM], classified P- and G- invariant probability measures and orbit-closures on strata of translation surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' These results can be summarized as follows (see [EM, Defs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2, Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4 & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='5]): Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H, Hm, ¯H, ¯Hm be as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Any P-invariant ergodic proba- bility measure m has the following properties: (i) It is G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (ii) There is a complex-affine manifold N and a proper immersion ϕ : N → ¯H such that L def = supp m = H ∩ ϕ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (iii) There is an open G-invariant subset U ⊂ ¯H satisfying m(U) = 1, and for any x ∈ U ∩ L there is an open set V containing x such that V is evenly covered by V ⊂ Hm under the map π : ¯Hm → ¯H, and ψ def = dev ◦ (π|V)−1 ◦ ϕ coincides on its domain with a C-linear map, with real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (iv) The subspace W def = Im(ψ) is symplectic, and the measure m is obtained via the cone construction from the Lebesgue measure on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (v) The complement L ∖ U is a finite union of supports of measures satisfying properties (i)–(iv), for which the manifolds N ′ appearing in (ii) satisfy dim N ′ < dim N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Any orbit-closure for the P-action is a set L as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We will refer to L as an orbit-closure and to m = mL as a flat measure on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Orbit-closures are referred to as affine invariant manifolds and also as invariant subvarieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The use of an evenly covered neighborhood in item (iii) is a standard approach for defining period coordinates (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' [MS]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We refer to [Wr1] for a survey containing more information on orbit-closures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 10 JON CHAIKA AND BARAK WEISS In a forthcoming work of Brown, Eskin, Filip and Rodriguez-Hertz, the same conclusion is obtained for the diagonal actions of P and G on a product of strata H × H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Namely, the following is shown: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H, H′ be strata of translation surfaces, and let P and G act on H × H′ via their diagonal embeddings in G × G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then all of the conclusions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 hold for this action (with ¯H × ¯H′ replacing ¯H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof of main result Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2 and further work of Wright [Wr2], we can prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let Z0 = spanR(z0) be a one-dimensional connected real Rel subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Assume that (1) fails, so that the action of Z0 on (L, mL) is not mixing of all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='5 it is not ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let θ be the relatively independent self-joining over L//Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Applying Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4 we have that θ ̸= mL × mL and ∆L ⊂ supp θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2, we have that there is a G-invariant open subset U of full θ-measure such that U ∩ supp θ is the isomorphic image of an affine complex-linear manifold whose dimension is strictly smaller than 2 dim ¯H, and θ is obtained from Lebesgue measure on this complex-linear manifold by the cone construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We claim that the set U1 def = {q ∈ H : (q, q) ∈ U} is of full measure for (π1)∗θ, where π1 : L × L → L is the projection onto the first factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Indeed, the measure θ is invariant under Z0 × {Id}, and hence so is its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since Z0 acts by homeomorphisms where defined, and using property (v) in Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2, we have that the set U is also Z0 × {Id}-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Thus for any Z0-ergodic measure, it is either null or conull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Thus if q /∈ U1 and q is generic for the measure µT (q) appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1), µT (q) assigns measure zero to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If this were to happen for a positive measure of q it would follow from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1) that U does not have full measure for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For q ∈ U1, let Nq denote the connected component of U ∩ π−1 1 (q) ∩ supp θ containing (q, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since the fibers π−1 1 (q) are also affine submanifolds of L × L, we have that the Nq are affine submanifolds contained in π−1 1 (q) ∼= L, so we can identify them with invariant submanifolds in L (which we continue to denote by Nq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' With this notation we have q ∈ Nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The mapping q �→ T (Nq) is locally constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' that is, letting V ⊂ ¯H and V ⊂ ¯Hm be open sets such that π|V : V → V is a homeomorphism and q ∈ V , the map q �→ dev ◦ π|−1 V (q) sends a neighborhood of q in Nq to an affine subspace Wq of H1(S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' R2), and the corresponding linear spaces Wq − Wq are the same for all q ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since mL × mL is the unique P-invariant ergodic measure on L × L of full support, we have dim Nq < dim L for every q ∈ U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let ¯Nq denote the set of surfaces (not necessarily of area one) which are obtained by rescaling surfaces in Nq, and let Nq def = Tq( ¯Nq) ERGODICITY OF REL 11 (the tangent space to ¯Nq at q, thought of as a subset of the tangent space Tq( ¯L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The assignment q �→ Nq defines a proper flat sub-bundle of the tangent bundle T ( ¯L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Flat sub-bundles of T ( ¯L) were classified in [Wr2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' According to [Wr2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1], Nq ⊂ RL for each q, and Nq is a complex linear subspace which is locally constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since RL is acted on trivially by monodromy, we in fact have that Nq is independent of q, and we denote it by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The leaves R(q) are contained in ¯Nq for each q, and of the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' That is, R(q) is the connected component of ¯Nq containing q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since Rel deformations do not affect the area of the surface, we see that ¯Nq = Nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In particular R(q) is closed for each q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' q, Nq is the support of the ergodic component (mL)q, and in particular (mL)q(Nq) < ∞, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since Z0 acts ergodically with respect to (mL)q, we have that almost surely Nq = R(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since the measure (mL)q is affine in charts, it is a scalar multiple of the translation-invariant measure on R(q), and thus the volume Vq of R(q) (with respect to its translation-invariant measure) is almost surely finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It is clear that the function q �→ Vq is U-invariant, and by ergodicity, it is constant almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We note that the above argument works under much weaker conclu- sions than those given in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Indeed, in the first step of the argument, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2 was used simply to extract a G-invariant assignment q �→ Nq, where Nq is a subspace of Tq(L), which is proper if θ is not the product joining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A fun- damental fact about such G-invariant assignments is that they are very restricted – besides [Wr2], see [EFW] and [Fi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In particular, [Fi] gives strong restrictions on assignments that are only assumed to be defined almost everywhere and measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A topological condition for Rel ergodicity Let Z0 ⊂ Z be a subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We say that a translation surface x is Z0-stably periodic if it can be presented as a finite union of horizontal cylinders and the Z0-orbit of x is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Recall that a horizontal separatrix is a horizontal leaf whose closure contains at least one singularity, and it is a horizontal saddle connection if its closure contains two singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then the condition of being Z0- stably periodic is equivalent to requiring that all horizontal separatrices starting at singular points are on horizontal saddle connections, and Z0 preserves the holonomy of every horizontal saddle connection on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In case Z = Z0 is the full real rel group, we say that x is fully stably periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This is equivalent to saying that all horizontal separatrices starting at singular points are on saddle connections, and all horizontal saddle connections start and end at the same singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In particular, for any cylinder C on a fully stably periodic surface, each boundary component of C is made of saddle connections starting and ending at the same singular point ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' we say that the boundary component only sees singularity ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For more information on the real Rel action on surfaces which are horizontally completely periodic, see [HW, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Suppose x is a surface which is Z0-stably periodic, and v ∈ Z0 moves two singularities p and q with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Suppose that x contains two cylinders C1 and C2 that both only see singularity p on one boundary component and only see singularity q on another boundary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Finally suppose the 12 JON CHAIKA AND BARAK WEISS circumferences c1, c2 of these cylinders satisfy c1 c2 /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then Case (2) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 does not hold for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since c1 c2 /∈ Q, the trajectory {Reltv(x) : t ∈ R} is not closed, let L denote its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We claim that the tangent space to L is not contained in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let σ1 denote a saddle connection from p to q in C1 and let σ2 denote a saddle connection from q to p in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let σ be the concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then σ represents an absolute homology class because it goes from p back to p, and it is nontrivial because the vertical component of its holonomy on x is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If we consider the restriction of the rel-action to C1 ∪ C2 then it only affects the twist parameters, which is a 2-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This space can be generated by the horizontal holonomy of σ1 and the horizontal holonomy of σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since c1 c2 /∈ Q, this restricted action does not give a closed orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' So the tangent space to L contains directions, which continuously affect the holonomy of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since σ is an absolute period, we see that the tangent space to L is not contained in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Checking the condition for strata Let H = H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) and for i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , k}, let ξi, ξj be the corresponding singular points of a surface in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let z ∈ R be a Rel cohomology class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We say that z moves ξi, ξj with respect to each other if for some (equivalently, every) α ∈ H1(S, Σ) represented by a path starting at ξi and ending at ξj, we have z(α) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Below when we discuss a stratum H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) we allow ai = 0, that is we allow marked points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We call points with cone angle 2π (that is, with a = 0) removable singularities, and otherwise we call them non-removable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The following result, which clearly implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1, allows strata with removable singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H be a connected component of a stratum H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let mH be the Masur-Veech measure on H, let Z be the corresponding real Rel foliation, and let Z0 ⊂ Z be a one-dimensional connected subgroup of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Suppose that there are 1 ≤ i < j ≤ k with corresponding singular points ξi, ξj, such that ai > 0, aj > 0 and such that some element of Z0 moves ξi, ξj with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then the Z0-flow on (H, mH) is mixing of all orders (and in particular, ergodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Clearly, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1 follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1, and the follow- ing result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H ⊂ H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) be a connected component of a stratum of translation surfaces with at least two non-removable singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If p ̸= q is any pair of non-removable singularities then there exists M ∈ H, which has cylinders C1, C2 with circumferences c1, c2 so that (1) M is fully stably periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (2) c1 c2 /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (3) Both C1 and C2 only see singularity p on one boundary component and only see singularity q on the other boundary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For the proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2 we will also need the following: Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H = H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) be a stratum of translation surfaces with at least two singular points (that is k ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If p ̸= q is any pair of distinct sin- gularities (possibly removable), then there exists M ∈ H, so that M is fully stably ERGODICITY OF REL 13 △ △ c Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The surface M has a cylinder of circumference c, and its boundary components see only the singularities ξi and ξj (denoted by ◦ and •).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The edges not labeled by △ are connected to M ∖ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' periodic and there exists a cylinder on M that only sees singularity p on one bound- ary component, and only sees singularity q on the other boundary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3 will both be proved by induction, after some prepara- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4 (The basic surgery – gluing in a torus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H = H(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , bℓ) be a stratum of translation surfaces, and let M ∈ H, with singularities labeled by ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ξℓ, so that the order of ξi is bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Suppose M has a horizontal cylinder C, with circumference c, where one boundary component is made of saddle connections that begin and end at ξi, and the other is made of saddle connections that begin and end at ξj, where bi ≥ 0 and bj ≥ 0 (so that ξi, ξj might be removable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then for all w > 0 there exists M ′ ∈ H(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , bi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , bj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , bℓ), with singularities labeled ξ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ξ′ ℓ, which has two horizontal cylinders C′ 1, C′ 2, where C′ 1 has circumference c + w and C′ 2 has circumference w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The complements M ∖ C and M ′ ∖ (C1 ∪ C2) are isometric, by an isometry mapping ξ′ i to ξi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The cylinders C1 and C2 only see singularity ξ′ i on one boundary component, and ξ′ j on another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Moreover, if M is fully stably periodic then so is M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It will be easier to follow the proof while consulting Figures 1 (before) and 2 (after).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Given a polygonal presentation for M, we give a polygonal presentation for M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let M be a polygon representation for M in which the cylinder C is represented by a parallelogram P (in Figure 1, the large rectangle in the center of the presentation), with two horizontal sides of length c, non-horizontal sides identified to each other, and the singular points ξi, ξj on adjacent corners of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Thus the non-horizontal sides of P represent a saddle connection σ on M connecting ξi to ξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We consider the two non-horizontal sides of P as distinct and label them by σ1, σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let P ′ be a parallelogram with sides parallel to those of P, where the horizontal sides have length w and the nonhorizontal sides are longer than the ones on P (in Figure 2, P ′ is to the right of P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Label the two horizontal sides of P ′ by h′ 1 and h′ 2, and identify them by a trans- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Partition the non-horizontal sides of P ′ into two segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The segments 14 JON CHAIKA AND BARAK WEISS c w / / △ △ □ □ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' To obtain M ′ from M, glue in a torus (rectangle on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This transforms C into a cylinder C′ 1 of circumference c+w, and adds a horizontal cylinder C′ 2 of circumference w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Edges not labeled by △, □, / or the color green are attached to M ′∖(C′ 1∪C′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' σ′ 1, σ′ 2 are parallel to each other and have the same length as σ1, σ2, and start at a corner of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The segments γ′ 1, γ′ 2 comprise the remainder of the non-horizontal sides of P ′ (and in particular, have the same length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Identify γ′ 1 to γ′ 2 by a translation, and identify σ′ 1, σ′ 2 to σ1, σ2 by a translation so that each σ′ i is attached to the σj with the opposite orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let M ′ be the translation surface corresponding to this presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It is clear that M ′ has the required properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ Proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The proof is by induction on � ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Base of induction: The base case is the stratum H(a1, 0s), that is, one sin- gular point (removable or non-removable) of order a1, and some number s ≥ 1 of removable singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In this case we take a surface in H(a1) which is made of one horizontal cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We label the singular point by ξ1 and place additional removable singular points ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ξs+1 in the interior of the cylinder, at different heights (so that the resulting surface has no horizontal saddle connections between distinct singularities) and so that ξi and ξj are on opposite sides of a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Inductive step: Suppose H′ = H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) is our stratum, where at least two of the singularities are non-removable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let p′, q′ be labels of singular points for surfaces in H′, corresponding to indices i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' To simplify notation assume i = 1, j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' There are three cases to consider: ai = aj = 0, or one of ai, aj are positive, or both are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If ai = aj = 0 then by assumption k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We take a cylinder C on a fully stably completely periodic surface M in H = H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ˆai, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ˆaj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The notation ˆai means that the symbol should be ignored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' that is on a stratum of the same genus with k − 2 ≥ 2 singular points obtained by removing two removable singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We place two singular points marked p′, q′ in the interior of C at different heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If ai > 0 and aj = 0 is zero we take a fully stably periodic surface M in H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ai − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ˆaj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak), find a cylinder C on M whose boundary ERGODICITY OF REL 15 c w′ w / / // // △ △ □ □ ∇ ∇ Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' First option for M ′ in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Attaching the sub- surface on the right increases the genus by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Unlabeled edges are attached to M ′ ∖ (C1 ∪ C2 ∪ C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' component is made of saddle connections starting and ending at ξi, place a marked point labeled ξj in the interior of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' If ai and aj are both positive we use the induction hypothesis to find a surface M ∈ H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ai − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , aj − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) with a cylinder whose boundary components see ξi and ξj, and we perform the surgery in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4 to this cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='5 (Two surgeries involving genus two surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H = H(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , bk) be a stratum of translation surfaces and let M ∈ H have a horizontal cylinder C, with circumference c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let p and q be singular points with order bi, bj respectively, such that one boundary component of C only sees singularity p and the other only sees singularity q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then for any w1, w2 > 0 there exists M ′ ∈ H′ = H(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , bi + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , bj + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , bk) which has three cylinders C1, C2, C3 with circumferences c + w1 + w2, w1 and w2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The complements M ∖ C and M ′ ∖ (C1 ∪C2 ∪C3) are isometric by an isometry preserving the labels of singular points, and C1, C2, C3 all have one boundary component that sees only p, and another that sees only q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Thus, if M is fully stably periodic so is M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Moreover, if the bi are all even, so that H′ has even and odd spin components, we can choose M ′ to be in either the even or odd connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Once again we encourage the reader to consult Figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4 we made a slit in M, running through P from top to bottom, and glued in a torus with a slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In this case we make an identical slit, this time gluing in a genus two surface with a slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This surface is presented in Figures 3 and 4 as made up of three rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It is straightforward to check that M ′ ∈ H′ and that it has cylinders satisfying the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It remains to check the final assertion about the parity of the spin structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Recall from [KZ, eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (4)] that where defined, the spin structure of a surface M of genus g can be computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let αi, βj (where 1 ≤ i, j ≤ g) be a symplectic basis for H1(M), realized explicitly as smooth curves on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This means 16 JON CHAIKA AND BARAK WEISS c w′ w / / // // △ △ ∇ ∇ □ □ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Second option for M ′, with a different spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' that all of these curves are disjoint, except for αi and βi which intersect once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' For each curve γ, let ind(γ) be the turning index, that is the total number of circles made by the tangent vector to γ, as one goes around γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The parity of M is then the parity of the integer �g i=1(1 + ind(αi))(1 + ind(βi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It is shown in [KZ] that this number is well-defined (independent of the choice of the symplectic basis) when all the singular points have even order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Suppose M has genus g and is equipped with a symplectic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Since any non-separating simple closed curve can be completed to a symplectic basis, we can assume that α1 is the core curve of C, and the other curves in the basis do not intersect the saddle connection from p to q passing through C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We construct a sym- plectic basis for M ′ in both cases, by modifying α1, keeping α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , αg, β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , βg, and adding new curves αg+1, αg+2, βg+1, βg+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The modified curves are shown in Figures 5, 6, and the reader can easily check that these new curves still form a symplectic basis, and that these two choices add two numbers of different parities to the spin structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ Note that in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2 we care about all connected components of strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We need to record some information about the classification of connected compo- nents of strata, due to Kontsevich and Zorich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A translation surface is hyperelliptic if it admits an involution which acts on absolute homology as −Id (see [FM] or [KZ, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A connected component of a stratum is hyperelliptic if all surfaces in the component are hyperelliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='6 ([KZ], Theorems 1 & 5 and Corollary 5 of Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Let H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak) be a stratum with ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The following holds: H has three connected components in the following cases: – k = 1, a1 = 2g − 2, g ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' – k = 2, a1 = a2 = g − 1, g ≥ 5 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' One is hyperelliptic, and the two non-hyperelliptic strata are distinguished by the spin invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' H has two connected components in the following cases: ERGODICITY OF REL 17 αg+1 βg+1 βg+2 αg+2 α1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Modifying the symplectic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Gluings as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' α1 αg+1 βg+1 βg+2 αg+2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Modifying the symplectic basis, second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Gluings as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Note the change in the rotation number of βg+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' – All of the ai are even, g ≥ 4, and either k ≥ 3 or a1 > a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The components are distinguished by their spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' – a1 = a2 and g is either 3 or is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' One of the components is hyper- elliptic and the other is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' When g = 3 the hyperelliptic component is even, and the other one is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' H is connected in all other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The proof will be case-by-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Here are the cases: (i) H(1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (ii) All the ai are nonzero and H is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (iii) All the ai are nonzero and H has two connected components distinguished by spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' 18 JON CHAIKA AND BARAK WEISS (iv) All the ai are nonzero and H has two connected components distinguished by hyperellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (v) All the ai are nonzero and H has three connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' (vi) Some of the ai are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' There is just one connected component and the desired surface is a Z-shaped surface, with three horizontal cylinders C1, C2, C3 of circumferences c1, c1 + c3, c3, where C1, C3 are simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We put all of the removable singular points in the interior of C3, and choose choose c1, c3 so that c1/(c1 + c3) /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' It is clear that with these choices the conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The stratum H is connected, and we have at least two singularities of positive order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' So with no loss of generality that they are labelled 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The result follows from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='4, applied to a surface in H(a1 − 1, a2 − 1, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak), and taking w /∈ cQ, so that w/(c + w) /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We apply the surgery in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='5, with w1/w2 /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Namely if p and q are labelled i, j, we let bi = ai − 2, bj = aj − 2 and bℓ = aℓ for ℓ ̸= i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Case (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' There are two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' One is hyperelliptic, one is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This means that a1 = a2 and either g = 3 (in which case a1 = a2 = 2) or g ≥ 4 is even (in which case a1 = a2 = g − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In this case we give explicit surfaces, one in each connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The first surface (the H(2, 2) case is shown in Figure 7) is a ‘staircase’ surface made of gluing 2g rectangles to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The rectangles are labelled (k, B) and (k, T ) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The top (respectively, bottom) of (k, B) is glued to the bottom (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=', top) of (k, T ) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , g, and the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=', right) of (k, T ) is glued to the right (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=', left) of (k + 1, B) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , g − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The horizontal sides of (1, B) are glued to each other, as are the horizontal sides of (g, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' This surface is hyperelliptic since it has a hyperelliptic involution rotating each rectangle around its midpoint, and this involution swaps the singularities (see [KZ, Remark 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The second surface is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' We first construct a hyperelliptic surface in H(a1 − 2, a2 − 2) as in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Then we perform the surgery described in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' The resulting surface has a horizontal cylinder intersecting three vertical cylinders, and thus, by [Li, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content='1], is not hyperelliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' See Figure 8 for an example in H(2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In both of these constructions there are no restrictions on the sidelengths of the rectangles, and we can easily arrange that two of the circumferences are incommensurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Case (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' In this case a1 = a2 = g − 1 for g ≥ 5 odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Applying the argument in Case (iii), we obtain the required surfaces in the odd and even connected com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' To obtain the required surface in the hyperelliptic component we use the ‘staircase surface’ describe in Case (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Case (vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Assume with no loss of generality that the removable singularities are labelled k′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=', k for some k′ ≥ 2, and let H′ = H(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , ak′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Note that the singularities p and q have label in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' , k′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' Apply the preceding considerations to obtain a surface in H′ with the required cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' By examining the proof in all preceding case one sees that the number of horizontal cylinders on this surface is at least three, that is there is at least one cylinder C3 which is distinct from the cylinders C1, C2, and we modify M by adding k − k′ in general position in the interior of C3, to obtain the desired surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' □ ERGODICITY OF REL 19 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQflQFO/content/2301.02483v1.pdf'} +page_content=' A surface in 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100644 index 0000000000000000000000000000000000000000..dab0cd5e76920227a2d3e1b24a503c541394801e --- /dev/null +++ b/VNAzT4oBgHgl3EQf1P5q/content/tmp_files/2301.01796v1.pdf.txt @@ -0,0 +1,1658 @@ +Recursive classification of satellite imaging time-series: +An application to water and land cover mapping +Helena Calatravaa,1,˚, Bhavya Duvvuria,1, Haoqing Lia, Ricardo Borsoib, Edward Beighleya, Deniz +Erdo˘gmu¸sa, Pau Closasa, Tales Imbiribaa +aNortheastern University, Boston, 02215, MA, USA +bCRAN, University of Lorraine, CNRS, Vandoeuvre-les-Nancy, F-54000, France +Abstract +A wide variety of applications of fundamental importance for security, environmental protection and +urban development need access to accurate land cover monitoring and water mapping, for which the +analysis of optical remote sensing imagery is key. Classification of time-series images, particularly with +recursive methods, is of increasing interest in the current literature. Nevertheless, existing recursive ap- +proaches typically require large amounts of training data. This paper introduces a recursive classification +framework that provides high accuracy while requiring low computational cost and minimal supervision. +The proposed approach transforms a static classifier into a recursive one using a probabilistic framework +that is robust to non-informative image variations. A water mapping and a land cover experiment are +conducted analyzing Sentinel-2 satellite data covering two areas in the United States. The performance +of three static classification algorithms and their recursive versions is compared, including a Gaussian +Mixture Model (GMM), Logistic Regression (LR) and Spectral Index Classifiers (SICs). SICs consist in +a new approach that we introduce to convert the Modified Normalized Difference Water Index (MNDWI) +and the Normalized Difference Vegetation Index (NDVI) into probabilistic classification results. Two +state-of-the-art deep learning-based classifiers are also used as benchmark models. Results show that +the proposed method significantly increases the robustness of existing static classifiers in multitemporal +settings. Our method also improves the performance of deep learning-based classifiers without the need +of additional training data. +Keywords: +Recursive Bayesian classification, spectral indices, water mapping, land cover mapping, land +cover change detection and time-series analysis, unsupervised classification +PACS: 0000, 1111 +2000 MSC: 0000, 1111 +1. Introduction +Given the vast amount of high resolution remotely sensed data available today, there exists a consid- +erable body of literature focused on remote sensing (RS) applications involving land cover mapping and +change detection (Anderson, 1976; Karan and Samadder, 2018; Rwanga et al., 2017; Srivastava et al., +2012). Some application examples are studies on land conservation, sustainable development, landscape +planning and management of resources such as, e.g., water. Changes in water dynamics can be studied +by surface water mapping, with the aim of monitoring floods (Farhadi et al., 2022; Koukoula et al., 2020; +Proud et al., 2011) and assessing the quality of water (Jiang et al., 2020; Son and Wang, 2020; Tong et al., +2010). Water mapping can also be used for coastline extraction and change assessment (Ekercin, 2007; +Hannv et al., 2013; Pardo-Pascual et al., 2012; Zhang et al., 2013), while land cover mapping is of fun- +damental importance when identifying the distribution of different types of crops (Griffiths et al., 2019; +Tewabe and Fentahun, 2020) or the dynamical evolution of land use in urban environments (Gadrani +et al., 2018; Liu et al., 2018). +Several sources of remotely sensed data are currently available, presenting different characteristics +when it comes to spatial, spectral, radiometric and temporal resolution (Satir and Berberoglu, 2012). +Spatial resolution typically varies from centimeters, in the case of very high resolution sensors, such +˚Corresponding author. +Email address: calatrava.h@northeastern.edu +1Indicates shared first authorship. +Preprint submitted to Remote Sensing of Environment +November 2022 +arXiv:2301.01796v1 [eess.IV] 4 Jan 2023 + +as the ones used by the GeoEye and QuickBird-2 satellites, to a few meters, in the case of sensors +used by the Landsat 8 and Sentinel-2 satellites. Such satellites can acquire images of the same scene +with a weekly temporal resolution. On the other hand, satellites equipped with the Moderate Resolution +Imaging Spectroradiometer (MODIS) and the Visible Infrared Imaging Radiometer Suite (VIIRS) sensors +achieve a higher temporal resolution, with daily image acquisitions. However, their spatial resolution is +significantly low, being in the order of hundreds of meters. Considering this and in order to combine +images with different spatial and temporal resolutions, multimodal image fusion techniques have been +developed to generate high spatio-temporal image sequences, contributing to generate a wealth of RS +data (Li et al., 2022; Zhu et al., 2010, 2018). +Spectral indices are one of the main land cover mapping tools given their simplicity and required low +computational cost. They compute scalar-valued features as a function of specific spectral bands, whose +value can be used to distinguish between different land cover classes contained in a pixel. Besides, spectral +index values can be easily interpreted or explained, as they minimize the effect of illumination in satellite +imagery while enhancing different spectral features present in the scene under study. For instance, the +Normalized Difference Vegetation Index (NDVI) enhances the presence of trees, bushes, and others. This +is due to the reflectance given by the spectral response of vegetation increasing for the wavelengths +defined by the NDVI. Water indices are used for water extraction at pixel level, given the difference in +spectral reflectance of land and water in the near and middle infrared wavelengths (Xie et al., 2016). The +most widely used water indices are the Normalized Difference Water Index (NDWI), the Modified NDWI +(MNDWI) and the Automated Water Extraction Index (AWEI) (Acharya et al., 2018). It has been shown +that challenging weather conditions create problems in the extraction of water bodies with spectral index +methods. Nevertheless, this can be solved with modified methods like the one proposed in Gao et al. +(2016), which effectively mitigates mountain shadows and allows the extraction of particularly challenging +small water bodies. Also, in Khalid et al. (2021), they suggest that the Land surface temperature Based +Water Extraction Index (LBWEI) provides high accuracy under all tested weather conditions. Although +spectral indices can achieve good results using only a subset of the spectral bands provided by the sensor, +it has been shown that using all Sentinel-2A bands can improve classification accuracy (Zhang et al., +2019). +Aside from spectral index methods, there is a wide choice of land cover classification approaches based +on machine learning methods available in the literature, whose main advantage is an increased flexibility. +Some of the explored machine learning methods used in RS include maximum likelihood classifiers (Frazier +and Page, 2000), decision trees, support vector machines (Hannv et al., 2013), logistic regression (LR) +(Mueller et al., 2015), random forests (Pelletier et al., 2016, 2017), naive Bayes and clustering methods +like the widely used K-means algorithm. The taxonomy of RS image classification methods proposed +in Satir and Berberoglu (2012) groups them into supervised/unsupervised, parametric/non-parametric +and hard/soft classifiers, among others. Results in Qiu et al. (2019) and Skakun (2010) suggest that deep +learning methods like artificial neural networks provide high accuracy results in land cover classification, +even when compared to other machine learning classifiers such as support vector machines. +The analysis of multitemporal or time-series data is of increasing interest for RS applications (Johnson +and Iizuka, 2016; Kuenzer et al., 2015). Exploiting multitemporal data makes it possible to improve the +performance of tasks such as classification (Deng et al., 2019; G´omez et al., 2016) or spectral mixture +analysis (Borsoi et al., 2021a; Halabisky et al., 2016) due to its temporal correlation, while at the same +time supplying the end-user with a more complete product that shows the spatial as well as the temporal +distribution of land classes or their proportions. The simplest approach to perform multitemporal land +cover mapping is to apply a static classifier to each image in the sequence, being spectral indices such as +the NDVI a popular choice (Jeevalakshmi et al., 2016; Sun et al., 2018). However, this does not exploit the +temporal information available in the data. Significant effort has been dedicated to developing techniques +specifically suited to process multitemporal image sequences. For instance, the authors in Hoberg et al. +(2015) and Kenduiywo et al. (2017) proposed classification methods based on conditional random field +models, which represent the interactions between class labels in both time and space. +Transfer and +active learning were combined in Demir et al. (2013) to adapt a pre-trained classifier to new images +acquired at other time instants. Time-series classification accounting for missing pixels using Gaussian +process regression was proposed in Constantin et al. (2022), while other works considered 1D temporal +covolutional neural networks (CNNs) (Pelletier et al., 2019), and 3D spatio-temporal CNNs (Ji et al., +2018). +The aforementioned techniques require a full image time-series as input in order to produce classifica- +tion maps, and are thus often referred to as batch or offline time-series classification methods. However, +in practice, instruments such as Sentinel-2 or Landsat 8 are continuously acquiring images. Generat- +2 + +ing classification maps online using batch algorithms require re-processing the whole image time-series +every time a new image is acquired, which can be computationally expensive. Thus, recursive meth- +ods are of particular interest, as they can iteratively update multitemporal classification maps as new +images are acquired by leveraging previously computed results. These methods can operate online and +be more efficient. The earliest recursive RS classification techniques were based on Bayesian filtering +ideas, by recursively updating the probabilities of each class given the measurements after each datum is +acquired (Strahler, 1980; Swain, 1978). These techniques are based on a statistical model that represents +the pixel spectra given its class, called a generative model, which is non-trivial to obtain. More recent +Bayesian approaches have proposed classification strategies that are recursive both in time, and across +multiple spatial scales (multiresolution) (Hedhli et al., 2016), using computationally expensive algorithms +such as the expectation maximization method to learn parameters and a generative model for the pix- +els. Other recent works have leveraged deep learning strategies, in particular different instantiations of +recurrent neural networks, such as long short-term memory (LSTM) networks. These have been applied +to predict flood susceptibility (Fang et al., 2021), for crop identification (Rubwurm and Korner, 2017), +and for land cover classification (Sharma et al., 2018). However, these models need large amounts of data +and long training times. +Despite their widespread use in land cover mapping, the previously mentioned classification algo- +rithms suffer from several limitations. First, they are highly sensitive to illumination and atmospheric +interferences (e.g., different aerosol concentrations or viewing angles), which can significantly impact the +spectra of pixels from a given material class (Borsoi et al., 2021b; Theiler et al., 2019). The lack of +robustness to such non-informative spectral variations is a significant limitation of spectral indices such +as the MNDWI (Yang et al., 2018). Moreover, due to the high sensor-to-target distances involved in RS +applications, many image pixels do not belong to a single class, but are instead composed of a mixture +of different material classes (Quintano et al., 2012). Although this can be addressed by spectral mixture +analysis techniques (Keshava and Mustard, 2002) or by assigning a pixel to more than one class (Mertens +et al., 2006), it poses a significant challenge to traditional classification algorithms. Furthermore, while +some deep learning-based algorithms such as artificial neural networks have sufficient flexibility to learn a +classification function in the presence of these interferences, such methods require large amounts of labeled +training data and have a high computational cost. Although spectral index classifiers have more limited +performance, they are widely used in RS applications because they do not rely on training data (Khalid +et al., 2021). Also, supervised time-series classification approaches require labeled training images of the +same region at multiple time instants, adding another layer of difficulty for their training. +t +time index +time index +Data +Classifier or +Likelihood Model + Bayesian +Recursion +Recursive Bayesian Classification (RBC) Framework +Image time-series +Time-series classification maps +Recursion +Generative version Eq. (3) +Discriminative version Eq. (5) +Discriminative +or Generative +Model +Figure 1: Overview of the proposed recursive Bayesian classifier. This method is able to convert a static generative model +(i.e., which models the observation of a pixel given its class label with the likelihood function p pzt|Ctq) or a discriminative +model (i.e., which models the observation of a class label given a corresponding pixel with p pCt|ztq) into a recursive Bayesian +classifier that exploits the temporal relationship of time-series data. Knowledge about the state transition matrix A and +the class prior probabilities p pCtq is assumed. The hat operator in pCt denotes the decision from the classifier. Finally, C +is an experiment-dependent set containing K different labels. +This paper addresses the need for a multitemporal, multispectral land cover classification method that +provides high accuracy while requiring low computational cost and minimal supervision. The proposed +approach, termed recursive Bayesian classification (RBC), is able to convert a static generative model +(i.e., which models the observation of a pixel given its class label) or discriminative classifier (i.e., which +models the observation of a class label given a corresponding pixel) into a recursive classifier using a +probabilistic framework. The method is based on a Markov assumption on the transition of the class +labels, which states that the probabilities of a change in a class label can be determined based only on the +information of labels at the most recent time instants. This simplifies the model, and allows to capture the +3 + +dynamical aspect of the classifier with a single set of class transition probabilities. Opposed to previous +approaches based on generative models (Hedhli et al., 2016; Strahler, 1980; Swain, 1978), the proposed +method can integrate existing state-of-the-art static classification algorithms in a recursive framework. +Moreover, unlike deep learning methods, it does not need huge training datasets. An overview of the +method introduced in this paper can be found in Fig. 1. +In our model, class transition probabilities can be tuned. This provides a transparent way to balance +the capability of the classifier to adapt to abrupt class changes with the increase of its robustness to +non-informative spectral variations originating from, e.g., atmospheric and illumination variations. Fur- +thermore, a recursive version of spectral index classifiers such as the MNDWI and the NDVI is proposed. +For this matter, spectral index values are converted into probabilistic classification results, which contain +class uncertainty information and can be integrated into the recursive discriminative Bayesian classifica- +tion framework. This results in an unsupervised, recursive classification method that is more robust in +multitemporal settings when compared to the spectral indices, addressing one of their main limitations. +The performance of the proposed approach is demonstrated through two different experiments ana- +lyzing 10 meter spatial resolution Sentinel-2 satellite data. Due to the lack of ground truth classification +maps for the selected study regions, classification methods are trained and evaluated based only on the +observed Sentinel-2 images. The first experiment consisted in water mapping of a reservoir and its down- +stream river, using the MNDWI as a benchmark spectral index method. The second experiment consisted +in land cover classification using the NDVI as a spectral index benchmark method. Two terrain areas +located in Oroville Dam, an embankment dam in California, have been evaluated in the first experi- +ment. These two areas pose several challenges to water mapping algorithms, such as ripples caused by +changes in the water flow, variations in illumination and abrupt changes in the water level of the reser- +voir. The terrain area evaluated in the second experiment is the Charles river basin, in Boston, which is +also challenging given the appearance of algal blooms in the summer, variations in illumination and in +atmospheric conditions, and the presence of highly reflective surfaces. In the case of both experiments, +we compared the performance of several static classification algorithms (namely, a Gaussian mixture +model (GMM), LR, and spectral index classifiers) and their recursive versions obtained with the RBC +approach proposed in this paper. The GMM belongs to the group of generative models, while the last +two belong to the group of discriminative models. For the water mapping experiment, we also included as +benchmark models two pre-trained state-of-the-art deep learning-based discriminative classifiers. These +are the DeepWaterMap (Isikdogan et al., 2017) and WatNet (Luo et al., 2021). +Experimental results show that the proposed RBC approach can significantly increase the robustness +of existing classification algorithms in a multitemporal setting. Moreover, for the water mapping exper- +iment, the proposed framework substantially improved the performance of pre-trained state-of-the-art +classification methods using deep artificial neural networks in the multitemporal setting, without requir- +ing additional training data. Provided results are fully reproducible. A Python implementation of the +proposed algorithms can be found at https://github.com/neu-spiral/RBC-SatImg. The dataset is +also available at (Calatrava et al., 2022b). +The remainder of this paper is structured as follows. The data that has been used and the area +under study are described in Section 2. +In Section 3, the proposed recursive Bayesian classification +technique based on both generative and discriminative models and also the proposed spectral index +classification model are derived mathematically and described in detail. The experimental setup, results +and a comparison with state-of-the-art approaches are presented in Section 4. +A discussion on the +implications and impact of our results is provided in Section 5. Finally, Section 6 concludes this paper. +2. Area of Study and Satellite Data +2.1. Area of Study +Our research focuses on three areas of study located in the United States (US), which are listed +in Table 1. Study areas A and B are located in Oroville Dam, an embankment dam on the east side of +the city of Oroville, in the state of California. Being 235 meters high, it is the tallest dam in the United +States and it is mostly used for water supply and flood control. Study areas A and B are analyzed in +the water mapping experiment using the MNDWI as one of the benchmark methods. Study area A is +located in the dam downstream and it contains a small water stream. Study area B is located in the dam +upstream. The Sentinel-2 RGB composite and ESA WorldCover Map images of the Oroville Dam region +can be found in Fig. 2. The sizes of study areas A and B are 200 ˆ 500 and 150 ˆ 110 pixels. +4 + +Table 1: Description of the three evaluation areas and the two training regions used in this research. Data size is given in +pixels. +Study Area (Evaluation) +Location +Data Size (px) +A (Fig. 2) +Oroville Dam (Downstream) in California, US +200 ˆ 500 +B (Fig. 2) +Oroville Dam (Upstream) in California, US +150 ˆ 110 +C (Fig. 3) +Charles river in Boston, Massachusetts, US +700 ˆ 1241 +Study Region (Training) +Location +Data Size (px) +1 (Fig. 2) +Oroville Dam in California, US +2229 ˆ 3341 +2 (Fig. 3) +Charles river in Boston, Massachusetts, US +927 ˆ 2041 +1 +Figure 2: Sentinel-2 RGB composite (upper figures) and ESA WorldCover Map (Zanaga et al., 2021) (lower figure) images +of study areas A and B, located in the downstream and upstream of the Oroville Dam and with approximate coordinates +39˝36102N 121˝27102W. Models are trained with pixel data covering study region 1 and evaluated with pixel data covering +study areas A and B. +Study area C extends over the Charles and Mystic rivers and the Boston harbor. To demonstrate +the performance of the proposed approach for land cover mapping, images of the Charles river basin are +analyzed using the NDVI as one of the benchmark methods. This site contains a big permanent water +body, urban vegetation and built-up area, which are interesting for land cover detection and classification +problems. The Sentinel-2 RGB composite and ESA WorldCover Map images of the Charles river area +5 + +esa2 km +1 +C +Study Area (Evaluation) +2 +Study Region (Training) +Figure 3: Sentinel-2 RGB composite (upper figure) and ESA WorldCover Map (Zanaga et al., 2021) (lower figure) images +of study area C, located in the Charles river basin and with approximate coordinates 71˝5102W 42˝22102N. Models are +trained with pixel data covering study region 2 and evaluated with pixel data covering study area C. +can be found in Fig. 3. The size of study area C is 700 ˆ 1241 pixels. +Study areas A, B and C pose significant challenges to multitemporal classification algorithms due to +seasonal changes in the land cover and variations in illumination and atmosphere. Study area A, as shown +in Fig. 4 (first row), suffers from the effect of illumination factors given by varying solar incidence angles +and the time of day at which images are captured. Moreover, ripples and other artifacts caused by the +high flows in the river stream make the classification of water pixels challenging, particularly to spectral +indices such as the MDNWI. Study area B is located in a reservoir where the water level changes seasonally +based on reservoir storage data obtained from the NWIS USGS website (https://waterdata.usgs.gov/ +nwis/). In October 2020, the recorded water storage was 200,485.8 hc-m, decreasing to 160,783.2 hc-m +in December 2020. This was followed by changes that resulted in recorded water storage of 183,223.8 and +97,542.6 hc-m in May and September of 2021. Fig. 4 shows Sentinel-2 RGB composite images of study +area B that clearly illustrate the significant changes caused by the varying water storage of this area. +Challenges in study area C include the seasonal cyanobacterial bloom in the Charles river and in the +Boston harbor waters. The presence of reflective surfaces from buildings in the area are also difficult to +classify, as they are easily mistaken for water. Algal blooms mostly occur during summer (Rome et al., +2021), as shown by the image captured on 2021-07-31 from Fig. 4. +6 + +esaesaesaestesaScattered clouds Algal blooms Pixel discrepancies +2020-10-16 +2021-01-19 2021-05-04 +2021-09-01 +Challenges in Area B: Abrupt water level +changes in the reservoir. +Challenges in Area A: Ripples are caused by +the water flow. See the challenging variations +in illumination. +Challenges in Area C: Appearance of algal +blooms. See the challenging variations in +illumination and atmosphere. +2021-03-23 +2021-07-31 +2020-10-21 +2021-06-23 +Figure 4: Sentinel-2 RGB composite images obtained with the Google Earth Engine (GEE) (Gorelick et al., 2017) showing +the challenges posed by the selected evaluation areas A, B and C. +The evaluation areas are selected to be smaller than the training regions in order to facilitate the +analysis of results since the temporal behavior of the images in these regions can be more easily interpreted. +Also, the overall computational cost decreases when lowering the number of pixels to be evaluated. The +GMM and LR models need to be trained. Training is performed using a weakly supervised approach. +First, pseudo-labels for a small set of images are obtained as classification maps from spectral index +classifiers, which are carefully checked so as to accurately represent the study areas. Then, these pseudo- +labels are used to train the models. +In the case of the Oroville Dam scenario, images covering the +training region shown in Fig. 2 are used, with size 2229 ˆ 3341 in pixels. Alternatively, for the Charles +river scenario, images with a size of 927ˆ2041 pixels are used, covering the training region shown in Fig. 3. +2.2. Satellite Data +In this research, Sentinel-2 images at the Coastal Aerosol, Blue, Green, Red, Near-Infrared (NIR, band +8 for Sentinel-2), Narrow NIR (band 8A for Sentinel-2), Shortwave Infrared (SWIR) 1 and SWIR 2 bands +are used to evaluate the proposed recursive Bayesian classification technique. The Sentinel-2 mission is +designed to give a high revisit frequency of 5 days alternating between two twin satellites, where each +twin satellite systematically acquires optical imagery from the target scene every 10 days. Besides, the +high spatial resolution of the data (10, 20 or 60 m) aids in observing the changing patterns in the land +more accurately than compared to lower resolution satellites such as the ones using the MODIS or the +VIIRS sensors, which provide resolutions of hundreds of meters. Taking this into account, the Sentinel-2 +data is of our interest due to its high temporal and spatial resolution, in addition to its availability via +the Google Earth Engine (GEE). +The resolution and central wavelength of the considered bands can be found in Table 2. Regarding the +RGB bands, band 4 (Red) is useful for identifying soil, water and many urban features, band 3 (Green) +gives excellent contrast between clear and turbid waters, and band 2 (Blue) is useful for identifying +vegetation and also human-made features (Maciej Huk, 2020). When it comes to the SWIR bands, they +are useful for measuring vegetation, water and soil moisture. +Table 2: Resolution (in meters) and central wavelength (in nanometers) of the Sentinel-2 spectral bands used in this research. +Band +Description +Resolution (m) +Central wavelength (nm) +1 +Coastal Aerosol +60 +443 +2 +Blue +10 +490 +3 +Green +10 +560 +4 +Red +10 +665 +8 +NIR +10 +842 +8A +Narrow NIR Edge +20 +865 +11 +SWIR 1 +20 +1610 +12 +SWIR 2 +20 +2190 +7 + +The Sentinel-2 level-2A (surface reflectance) data was downloaded with the GEE from the COPERNI- +CUS/S2 SR collection. GEE atmospherically corrects the images using the standard SEN2COR software +package. Only images with at most 10% cloud cover as indicated by the GEE are considered. In the +post-processing stage, the resolution of bands 8A, 11, 12 (20 meters) and band 1 (60 meters) is increased +to 10 meters by nearest-neighbor interpolation. +The pixel values are scaled by 0.0001 such that the +surface reflectance values are between 0 and 1. We consider images acquired between dates 2020-09-01 +and 2021-09-26 for the Oroville Dam scenario, and images between dates 2020-09-04 and 2021-09-29 for +the Charles river region. Images of the Charles river area with significant snow cover are removed. The +preprocessed data used in our paper was made available at (Calatrava et al., 2022b). +3. Methods +3.1. Recursive Bayesian Classification +In this paper, we propose a recursive Bayesian classification technique for water mapping and land +cover classification using multispectral and multitemporal data. The proposed technique is considered +to be unsupervised because it does not need labeled training data for multiple time instants. It may be +applied on top of other classifiers regardless of their supervised, semi-supervised or unsupervised nature. +Therefore, it can be stated that the recursion technique is agnostic to the classifier that is used. When +applied on top of an unsupervised classifier, the resulting technique can be considered to be completely +unsupervised. +Let us denote by Zt P RBˆN an image with B bands and N pixels observed at time instant t “ +t1, . . . , Tu. +The images at the different time instants are supposed to be coregistered, that is, they +constitute observations of the same geographical scene. For each pixel zt,n P RB, being n “ t1, . . . , Nu, +we associate a label Ct,n P C , where C is an experiment-dependent set containg the possible K labels. +For a set of images Zt over time, the most likely label Ct,n for each pixel zt,n (i.e., the n-th column of +Zt) can be estimated based on all the previously observed data tZt, Zt´1, . . . , Z1u by maximizing the +posterior probability ppCt,n|Zt, Zt´1, . . . , Z1q as +pCt,n “ arg max +Ct,nPC ppCt,n|Zt, Zt´1, . . . , Z1q , +(1) +where the hat operator denotes the decision from the classifier. This expression is powerful, as it considers +both temporal and spatial information. However, learning the posterior Probability Mass Function (PMF) +in Eq. (1) can be hard, specially with high dimensional images. A spatial independence assumption can +be applied to reduce the computational cost when calculating the conditional PMF. We propose to treat +the label of every pixel as independent of the data from other pixels, meaning that Ct,n only depends +on zt,n, zt´1,n, . . . , z1,n, or, equivalently, on z1:t,n fi tzt,n, zt´1,n, . . . , z1,nu. +This is without loss of +generality, as the proposed approach can be directly extended to consider spatial information (i.e., from +multiple pixels). Thus, the posterior in Eq. (1) becomes ppCt,n|z1:t,nq, disregarding spatial information, +and leading to +pCt “ arg max +CtPC ppCt|z1:tq, +(2) +where the pixel index n is omitted for simplicity. The classifier proposed in Eq. (2) still considers a +temporal dependence on previous data, meaning that the labels and images at previous time instants +influence the results of the current time t. +The posterior PMF in Eq. (2) can be computed recursively using Bayes theorem under conditional +independence assumptions and assuming knowledge about the prior ppC0q, the sate transition ppCt|Ct´1q +8 + +and the likelihood distribution ppzt|Ctq. Thus, the posterior PMF can be computed as +ppCt|z1:tq “ +ÿ +Ct´1PC +ppCt, Ct´1|z1:tq +paq +“ +ÿ +Ct´1PC +ppCt|Ct´1, ztqppCt´1|z1:t´1q +pbq +“ +ÿ +Ct´1PC +ppzt|CtqppCt|Ct´1q +ppzt|Ct´1q +ppCt´1|z1:t´1q +“ +ÿ +Ct´1PC +ppzt|CtqppCt|Ct´1q +ř +C1 +t ppzt|C1 +tqppC1 +t|Ct´1qppCt´1|z1:t´1q +“ ppzt|Ctq +ÿ +Ct´1PC +ppCt|Ct´1q +ř +C1 +t ppzt|C1 +tqppC1 +t|Ct´1qppCt´1|z1:t´1q. +(3) +where in equality paq we assumed a first-order Markov model, that is, given Ct´1 and zt, the class +labels Ct are independent of z1:t´1; in equality pbq we applied the Bayes theorem followed by the same +conditional independence assumption. We refer to Eq. (3) as the Recursive Bayesian Generative Model +(RBGM) due to its dependence on the likelihood function ppzt|Ctq. The term ppCt´1|z1:t´1q denotes the +posterior PMF of the previous time step. This term shows in the equation because states are assumed +to be independent of future measurements, meaning that Ct´1 depends on z1:t´1 but not on zt. When +t “ 1, ppCt´1|z1:t´1q becomes equivalent to the class prior probabilities since ppC0|z0q “ ppC0q. +Eq. (3) uses the Bayes theorem, where the posterior probability is given by the product of the likelihood +and the prior divided by the marginal probability. The denominator in this expression considers the +probability of all the possible transitions that a pixel can go through. The transition PMF ppCt|Ct´1q is +described later in this section. +The posterior probability can also be computed as a function of the probability of the labels given the +pixel values, which allows existing classification algorithms to be used in the RBC framework. In this case, +the posterior probability is computed by following Eq. (5) and this is referred to as Recursive Bayesian +Discriminative Model (RBDM). Applying the Bayes theorem to the likelihood ppzt|Ctq we obtain +ppzt|Ctq “ ppCt|ztqppztq +ppCtq +, +(4) +where ppCt|ztq is the prediction of the classifier to which the RBC framework is applied, which we refer to +as a benchmark classifier. As the RBC framework is agnostic to the classifier that is used, the prediction +can be the result of any type of classifier, including deep learning methods as well. The Bayes theorem +can be used to extend the generative model (RBGM) to the discriminative model (RBDM) as +ppCt|z1:tq “ ppCt|ztqppztq +ppCtq +ÿ +Ct´1PC +ppCt|Ct´1q +ř +C1 +t ppzt|C1 +tqppC1 +t|Ct´1qppCt´1|z1:t´1q +“ ppCt|ztq�� +� +ppztq +ppCtq +ÿ +Ct´1PC +ppCt|Ct´1q +ř +C1 +t +ppC1 +t|ztq + +ppztq +ppC1 +tq +ppC1 +t|Ct´1q +ppCt´1|z1:t´1q +“ ppCt|ztq +ppCtq +ÿ +Ct´1PC +ppCt|Ct´1q +ř +C1 +t +ppC1 +t|ztq +ppC1 +tq ppC1 +t|Ct´1q +ppCt´1|z1:t´1q, +(5) +where ppCtq denotes the marginal class probability. In the widely used naive Bayes classifier, the marginal +class probabilities are also used (Barber, 2011). In the absence of labeled training data and prior infor- +mation about the scene, we set their value as ppCtq “ +1 +K @Ct. In Eqs. (3) and (5), the term ppCt|Ct´1q +corresponds to the state transition probability and it can be described using the so-called state transition +probability matrix. For simplicity, in this work we assume ppCt|Ct´1q to be time invariant. Although +strong, this assumption copes with the lack of knowledge we assume regarding the studied scene. We +highlight, however, that this is so without loss of generality since prior knowledge about, e.g., seasonality +can be easily incorporated in a time dependent transition PMF. This matrix can be expressed for K +9 + +classes as +A “ +» +———– +p11 +p12 +. . . +p1K +p21 +p22 +. . . +p2K +... +... +... +... +pK1 +pK2 +. . . +pKK +fi +ffiffiffifl , +(6) +being +pij “ ppCt “ j|Ct´1 “ iq. +(7) +If we assume that pij “ ϵ for all i ‰ j, the state transition probability matrix for K classes can be +expressed as +A “ +» +———– +1 ´ pK ´ 1qϵ +ϵ +. . . +ϵ +ϵ +1 ´ pK ´ 1qϵ +. . . +ϵ +... +... +... +... +ϵ +ϵ +. . . +1 ´ pK ´ 1qϵ +fi +ffiffiffifl . +(8) +The particular case of K “ 2 makes this matrix simpler as shown in Eq. (9). +This matrix is site +dependent and may be filled by expert judgement by selecting the proper value of ϵ, which corresponds +to the probability of a pixel transitioning from one label to another. +A|K“2 “ +„ +1 ´ ϵ +ϵ +ϵ +1 ´ ϵ +ȷ +(9) +It is relevant to discuss the sensitivity of recursive Bayesian classification techniques with regard to the +pixel transition probability. If a low value of ϵ is selected, the probability of a pixel transitioning to +a different label is believed to be low. This moderates the change of the label at every pixel, making +classification more robust to atmospheric interference, discrepancies in pixels and also to extreme events. +However, lower values of ϵ also decrease the capability of the method to adapt to abrupt class changes. +Thus, the transition probabilities must be selected according to the application in order to reach a tradeoff +between robustness and accuracy. +With the approach presented in this subsection, a scene can be recursively classified using both +generative (Eq. (3)) and discriminative (Eq. (5)) models by iteratively updating a class posterior PMF. +Note that the proposed RBC framework relies in probabilistic classifiers (or generative models). Al- +though many deep learning classifiers are currently trained based on the cross-entropy loss, which leads +to a maximum likelihood estimation of the class labels (Barber, 2011), very flexible models, such as +deep neural networks, can lead to overconfident classification results (i.e., there being some j such that +ppCt “ j|ztq « 1). This can be damaging when such models are integrated in the proposed RBC frame- +work, since such overconfidence diminishes the relevance of the prior information obtained in previous +time instants through the recursion. Therefore, to remedy this issue, we propose to empirically reduce the +confidence of the predictions of deep learning models before integrating them in the proposed framework. +This is performed by using a simple relation as +p pCt|ztq “ +pNN pCt|ztq ` λ +ř +C1 +tPC ppNN pC1 +t|ztq ` λq, +(10) +where pNN pCt|ztq is the prediction of the deep learning model, being it the probability of the labels +Ct P C given the pixel value at time instant t, and λ P R` is a positive constant used to slightly push +the predicted class probabilities towards 1{K (i.e., towards a discrete uniform distribution). +3.2. Recursive Spectral Index Classification (RSIC) Algorithm +Considering the Spectral Index Classification (SIC) algorithm, we propose the use of broadband +spectral indices to generate predictive probability of occurrence of land classes, such as water or soil. The +values of these indices are of interest for a classification algorithm due to their clear interpretability and +lack of supervision. In this research, the MNDWI is used for the water mapping experiment, while the +NDVI is used for the land cover classification experiment. This is described with more detail in Section 4, +where the configuration used for each experiment is provided. +We propose the application of the recursive Bayesian framework described beforehand on top of the +SIC algorithm, which we refer to as Recursive SIC (RSIC). The values of standard broadband spectral +indices such as the MNDWI and the NDVI must be converted into probabilities in order to calculate the +10 + +posterior PMF with the RBDM (see Eq. (5)). Thus, we propose to define the class probability ppCt|ztq +as +ppCt|ztq “ +fCt pypztqq +ř +CtPC fCtpypztqq, +(11) +where ypztq corresponds to the spectral index value, which is computed as a function of the pixel zt. This +is a similar but not equivalent idea to applying a softmax function. To compute the probability value, +we use a Gaussian function as fCt “ N pµCt, σCtq, where the mean and standard deviation values are +selected based on the configuration of the experiment. A different value of mean and standard deviation +can be assigned for each class as µ “ coltµCtu and σ “ coltσCtu, being colt¨u the operator returning a +vector whose elements are µCt and σCt for Ct P C , respectively. The function fCt can be expressed as +fCt pypztqq “ +1 +σCt +? +2π exp +˜ +´1 +2 +ˆypztq ´ µCt +σCt +˙2¸ +. +(12) +The function fCt gives a measure of how close the spectral index ypztq is to the mean value of each +class Ct, which is denoted as µCt. This is used as an indication of the likelihood of zt being of class +Ct. The standard deviation σCt is used to account for the length of the spectral index interval that is +deemed to constitute class Ct. For ease of exposition, let us consider for the remainder of this section +that Ct P C “ t1, . . . , Ku, and also that the class indices are ordered in the same way as the threshold +intervals, i.e., class i corresponds to the i-th spectral index interval. +The length of intervals defining each class in the spectral index value ypztq can be highly non- +homogeneous and depends on the spectral index class thresholds τi, being i P t0, . . . , Ku. These thresholds +define a hard classification result based on the spectral index value, with pixel zt being assigned to the +i-th class if and only if ypztq P pτi´1, τis. +Their length can be calculated as Lj “ τj ´ τj´1, where +j P t1, . . . , Ku. The values of µ and σ are calculated as µj “ Lj{2 ` τj´1 and σj “ Lj{2, so that the +probability of a pixel belonging to a given class decreases smoothly as ypztq moves away from the center +of the interval and approaches one of the thresholds. The threshold values are determined empirically +and are experiment-dependent. Please refer to Section 4.1 for details on how the threshold values were +selected. +4. Results +In this section, we demonstrate the performance of the proposed Bayesian recursive classification +methodology using both spectral indices and machine learning classifiers. This is done through the two +experiments listed in Table 3. Complete results can be reproduced following the instructions in https: +//github.com/neu-spiral/RBC-SatImg, and can also be found in the supplementary material or in the +extended version of this paper (Calatrava et al., 2022a). +The first experiment aims to classify pixels as water or non-water, while the second one aims to +classify pixels as water, land and vegetation. In both experiments, we consider three static classification +methods and compare them to their recursive implementations by applying the proposed methodology. +For the water mapping experiment, we also consider two pre-trained deep learning methods. These are the +DeepWaterMap (Isikdogan et al., 2017) and the WatNet (Luo et al., 2021) algorithms. The multitemporal +classification algorithms proposed in Pelletier et al. (2019) and Rubwurm and Korner (2017) were also +evaluated. However, due to possible differences in the training data, obtained results were not superior +to the ones provided by the benchmark models and consequently are not included in this publication. +Details regarding the algorithms used in this research are provided in Table 4. Results obtained with the +probabilistic instance of the SIC algorithm introduced in Eq. (11) are also provided in this section. +Table 3: Description of the areas of study in the two experiments conducted in this research (see Figs. 2 and 3). +Experiment +Study Region (Training) +Study Area (Evaluation) +Water Mapping +1 +A and B +Land Cover Classification +2 +C +11 + +Table 4: Full name, abbreviation and note on the novelty of the algorithms used in this research. +Full Name +Abbreviation +Note +Gaussian Mixture Model +GMM +Benchmark +Logistic Regression +LR +Benchmark +Spectral Index Classification +SIC +Novel +Recursive Gaussian Mixture Model +RGMM +Novel +Recursive Logistic Regression +RLR +Novel +Recursive Spectral Index Classification +RSIC +Novel +DeepWaterMap +DWM +Benchmark (Isikdogan et al., 2017) +WatNet +WN +Benchmark (Luo et al., 2021) +Recursive DeepWaterMap +RDWM +Novel +Recursive WatNet +RWN +Novel +4.1. Experimental Setup +4.1.1. Training Dataset and Model Selection +For the water mapping experiment, 46 images from the Oroville Dam site are extracted from the +GEE with dates between 2020-09-01 and 2021-09-26 after filtering out images (dates) with cloud pixel +percentages above 10%. Each image belongs to a different date and they are mostly spaced 5 days apart +(the temporal resolution of Sentinel-2 satellites). However, some images are spaced more than 5 days +apart due to filtering of images with cloud cover or other discrepancies. We separate 4 images from the +beginning of the dataset (with dates 2020-09-01, 2020-10-06, 2020-10-11 and 2020-10-16) to train the LR +and GMM models. The remaining 42 images (with dates between 2020-10-21 and 2021-09-26) are used +for evaluation. Pixels analyzed in the training stage are the ones covered by study region 1, while pixels +analyzed in the evaluation stage are the ones covered by study areas A and B (see Fig. 2). +For the land cover classification experiment, 46 images from the Charles river basin are downloaded +from the GEE with dates between 2020-09-04 and 2021-09-26 after filtering out images (dates) with +cloud pixel percentages greater than 10%. After a visual inspection, 15 images with snow-covered land +and other large discrepancies are also removed from the dataset. Following a similar reasoning used for +splitting the training images for the Oroville Dam data, we select the first 3 images (with dates 2020-09- +04, 2020-10-01 and 2020-10-09) for training and the remaining 28 (with dates between 2020-10-14 and +2021-09-26) are used for evaluation. Pixels analyzed in the training stage are the ones covered by study +region 2, while pixels analyzed in the evaluation stage are the ones covered by study area C (see Fig. 3). +Images used for training are not used for evaluation in any of the experiments. For details on the pixel +size of training and evaluation data, please refer to Table 1. +To generate data with surrogate ground truth class labels to train the LR and GMM models, spectral +index classification algorithms are applied to the images in the training set (the MNDWI for the water +mapping experiment, and the NDVI for the land cover experiment). More precisely, pixels are classified +based on their spectral index value as a function of the class thresholds in τ W and τ LC, which are selected +accordingly so that the generated classification maps are visually close to the reference ones in Figs. 2 +and 3. To obtain the generative model ppzt|Ctq used in the RGBM (in Eq. (3)), one GMM is trained +for each class label (i.e., ppzt|Ctq is a GMM for each choice of Ct). To adequately represent the training +pixels without overfitting, we select the smallest number of components for each GMM such that the +histogram of the training distribution and the one generated by the respective GMM are visually close. +4.1.2. Water Mapping Experiment +For the water mapping experiment, Ct P C “ twater, not wateru, or, C “ t0, 1u. The spectral index +used is the MNDWI. This index uses the Green and SWIR (band 11) bands for the enhancement of open +water features, while diminishing built-up area features that are often correlated with open water in other +indices (Xu, 2006). The MNDWI is calculated as +yMNDWI pztq “ zt,Green ´ zt,SWIR +zt,Green ` zt,SWIR +, +(13) +where zt,Green and zt,SWIR denote the Green and SWIR bands of zt, respectively. The threshold for non- +water pixels is set to 0.13 after carefully comparing obtained mapping results with thresholds varying +between 0.1 and 0.15 against the reference classification map in Fig. 2. To convert the spectral index +values into a predictive probability of water ppCt|ztq, Eq. (11) is used with mean and standard deviation +12 + +µW “ r´0.435, 0.565s and σW “ r0.565, 0.435s, which are calculated with the threshold values τ W “ +r´1, 0.13, 1s as indicated in Section 3.2. The temporal class transition probability ϵ is set to 0.05 when +computing matrix A according to Eq. (9). The normalization constant λ from Eq. (10) is set to 0.3. This +value has been chosen considering a trade-off between the robustness and adaptability of the recursive +framework. +The three benchmark models and their recursive versions are also compared to two pre-trained state- +of-the-art deep learning-based methods. Both methods were trained by their authors using Sentinel 2 +images and can therefore be evaluated using the data described in Section 2. The WatNet algorithm +is based on deep semantic segmentation models, whereas the DeepWaterMap algorithm uses multiscale +CNNs. +4.1.3. Land Cover Classification +For the land cover classification experiment Ct P C “ twater, land, vegetationu, or, equivalently, +C “ t0, 1, 2u. The spectral index used is the NDVI. This index uses the red and NIR bands to observe +the presence of vegetation, soil and water (Akbar et al., 2019; Alex et al., 2017). The NDVI can be +computed as +yNDVI pztq “ zt,NIR ´ zt,Red +zt,NIR ` zt,Red +, +(14) +where zt,Red and zt,NIR denote the Red and NIR bands of zt, respectively. The spectral index thresholds +between water, land and vegetation classes are selected as τ1 “ ´0.05 and τ2 “ 0.35, respectively, +according to the recommendations found in (Akbar et al., 2019; Alex et al., 2017). +To convert the +spectral index values into a predictive probability of each class ppCt|ztq, +Eq. (11) is used with mean +and standard deviation µLC “ r´0.525, 0.149, 0.675s and σLC “ r0.475, 0.19, 0.325s, calculated with the +threshold values τ LC “ r´1, ´0.05, 0.35, 1s as indicated in Section 3.2. The temporal class transition +probability ϵ is set to 0.05 when computing matrix A according to Eq. (8), for K “ 3. +4.2. Water Mapping Experiment Results +Water mapping results for study areas A and B are presented in Figs. 5 and 6, respectively. Due +to space limitations, only a smaller representative interval of the image sequences is displayed, having +selected one every four available dates. +RGB composite images of the studied areas are shown as a +reference because they highlight changes in the scene. +Results obtained for the dam downstream region (study area A), which are depicted in Fig. 5, show a +considerable difference between some of the benchmark and recursive algorithms. For instance, the SIC, +LR and DWM methods classify a large portion of the stream as land for dates 2021-05-19, 2021-06-13, +2021-09-06 and 2021-09-26. Their recursive versions, on the other hand, adequately classify most of the +stream pixels as water. The WN algorithm classifies some portions of land pixels as water pixels in dates +between 2021-11-25 and 2021-07-08. Its recursive version, while still missclassifying a small part of the +stream, is considerably more accurate, which shows that the proposed Bayesian recursive technique can +also improve the performance of modern deep learning-based mapping algorithms. Moreover, we have +observed in multiple simulations that the DWM and WN algorithms resulted in overconfident classifica- +tion results, making the use of the strategy described in Eq. (10) necessary to obtain this performance +improvement. +13 + +2020-10-26 +SIC +GMM +LR +DWM +WN +RSIC +RGMM +RLR +RDWM +RWN +RGB +2020-11-25 +2020-12-30 +2021-02-23 +2021-04-09 +2021-05-19 +2021-06-13 +2021-07-08 +2021-08-02 +2021-09-06 +2021-09-26 +Figure 5: Water mapping results for the downstream subscene of Oroville Dam (study area A). Purple and yellow represent +water and land, respectively. Images are arranged in chronological order. +From Fig. 6, which shows results for study area B, it can be inferred that the compared methods are +able to adequately capture the decrease in water levels over time in the dam upstream. However, the SIC, +LR, DWM and WN benchmark methods missclassify a considerable amount of water pixels as land in the +date 2020-12-30. Their recursive counterparts, on the other hand, provide improved classification results +at this date. There are some dates, such as 2021-08-02, for which all methods provide adequate and +comparable classification results. The trade-off between adaptability and robustness poses a challenge to +the recursive framework for the upstream subscene of Oroville Dam. When being robust, the recursive +framework becomes less flexible and consequently, abrupt changes in the scene might be detected with +delay. This is the case of the connection between the island at the bottom of the scene and the mainland, +which is not detected by the recursive versions of the RDWM and WN algorithms in the date 2021-07-08. +This happens due to the normalization constant introduced in Eq. (10), which is used to counteract the +overconfidence of the deep learning-based classifiers, as it makes these models rely more strongly on the +data from past time instants. +14 + +2020-10-26 +SIC +GMM +LR +DWM +WN +RSIC +RGMM +RLR +RDWM +RWN +RGB +2020-11-25 +2020-12-30 +2021-02-23 +2021-04-09 +2021-05-19 +2021-06-13 +2021-07-08 +2021-08-02 +2021-09-06 +2021-09-26 +Figure 6: Water mapping results for the upstream subscene of Oroville Dam (study area B). Purple and yellow represent +water and land, respectively. Images are arranged in chronological order. +4.3. Land Cover Classification Experiment Results +Land cover classification results for study area C are presented in Fig. 7. For this experiment, we have +also selected one every four available dates. RGB composite images are also provided as a reference. +It can be seen that the seasonal variations in the distribution of land, water and vegetation are +captured well by all the algorithms. The decrease in the amount of vegetation starting from November +(through winter) with an increase in dry land (at dates 2020-12-13 and 2021-03-20) is represented by an +increase in yellow pixels until May, followed by an increase in the number of vegetation pixels through +summer and fall (from 2021-05-27 to 2021-09-14). Moreover, the recursive methods (RSIC, RGMMM and +RLR) provide significantly more robust performance when compared to their non-recursive counterparts, +being less sensitive to, e.g., atmospheric interference and illumination factors affected by solar incidence +angle. +Study area C extends over an area covering the Boston harbor, the Charles river lower, mid +and some upper basins, and the Mystic river lower basin. Since most of these correspond to urban and +suburban areas, we can find many reflective surfaces from, e.g., building terraces and metal sheds, which +lead to pixels with high spectral reflectance. Such pixels are easily mistaken for water since their NDVI +values is close to zero. This can be observed most clearly in the date 2020-12-13, where the Bayesian +recursive approaches lead to few missclassifications of reflective surface pixels close to the river when +15 + +compared to their non-recursive counterparts. Moreover, on dates 2021-05-27 and 2021-07-31, due to the +appearance of cyanobacterial blooms (due to which the water becomes diluted with chlorophyll pigments) +in the Boston harbor waters, an important portion of water pixels are classified as land by the SIC and +LR algorithms, whereas their recursive versions lead to adequate classification results. On the other +hand, GMM results are not particularly affected at this date. Among the benchmark methods, the GMM +showed the best results, which were comparable to the ones obtained with its recursive counterpart. For +some dates, the RBC methodology introduces a smoothing effect, which makes it more difficult to adapt +to sudden class changes. This can be visualized on the date 2020-03-20, as the recursive methods show +a smaller amount of vegetation pixels (i.e., more land pixels are observed) compared to the benchmark +methods. +2020-11-08 +SIC +GMM +LR +RSIC +RGMM +RLR +RGB +2020-12-13 +2021-03-20 +2021-04-24 +2021-05-27 +2021-07-31 +2021-09-14 +Figure 7: Land cover classification results for the Charles river basin (study area C). Purple, green and yellow represent +water, vegetation and land, respectively. Images are arranged in chronological order. +4.4. Model Sensitivity to Class Transition Probability +The class transition probabilities, introduced and denoted as ϵ in Section 3, are site-specific and it is +thus important to carefully select their values. In this section, we evaluate the sensitivity of the model to +the class transition probability hyperparameter. A study to understand the effect of the value assigned +to ϵ on the water mapping model is performed with images from the upstream Oroville dam site, which +corresponds to study area B from Fig. 2. In Fig. 8, variations in the number pixels classified as water +are recorded under values of ϵ between 0.1 and 0.8 for three recursive models evaluated in this research: +RGMM, RLR and RSIC. Also, results can be observed for the non-recursive version of each model for +ϵ “ 0.5 as a benchmark, in an orange color dashed line. Although the temporal resolution of Sentinel-2 +is of 5 days, due to the filtering of cloud-covered images, there are gaps between dates that are longer. +Obtained results suggest that the discriminative recursive models (RLR and RSIC) are more sensitive +to changes in the transition probabilities than the RGMM model, as the number of water pixels varies +more abruptly for the RLR and RSIC models in, for instance, dates 2020-11-10, 2020-12-30 and 2021- +01-19. In the case of the RSIC model and for ϵ “ 0.8, the amount of water pixels decreases from 7000 +to 3500, approximately, between dates 2020-12-10 and 2020-12-30. However, this variation is under 1000 +pixels for the same model with low ϵ value (ϵ “ 0.1). The RLR model is showing a very similar behavior +for these two dates. This shows how, in the case of the RLR and RSIC models, ϵ has an influence on the +robustness-adaptability trade-off presented by the proposed recursive Bayesian framework, meaning that +these models are sensitive to this hyperparameter. +Sensitivity is best evaluated on dates when there is a considerable change in class distribution caused +by natural phenomena such as draining occurring in preparation for extreme rainfall. Also, the varying +temporal gaps between dates have an impact on the distribution of classes. The use of large ϵ probability +causes the RSIC and RLR to fail during such events. In case of RLR, a small value of ϵ “ 0.1 is seen to +smooth the abrupt change in the prediction of water pixels. In the case of the RSIC model, ϵ “ 0.1 also +presents results that are much smoother when compared to the ones obtained for higher ϵ values. +16 + +The RGMM is seen to be less sensitive to changes in the values of the transition probability hyper- +parameter, as the amount of water pixels is relatively similar throughout all the dates for the different +values of ϵ. Nevertheless, we can see how on 2020-08-27 this model presents an abrupt change in the +amount of water pixels for the highest value of ϵ “ 0.8. Note that for ϵ “ 0.5, results match between +the non-recursive (orange color) and recursive (magenta color) versions of the models. We expected this +result because for this value of ϵ the posterior of the recursive algorithm should match the static classi- +fier output for the water mapping classification problem. Given the results obtained in this section, we +decided to set ϵ to a small value, as specified in Section 4.1. +2020-10-21 +2020-10-26 +2020-11-05 +2020-11-10 +2020-11-20 +2020-11-25 +2020-11-30 +2020-12-05 +2020-12-10 +2020-12-30 +2021-01-09 +2021-01-14 +2021-01-19 +2021-02-23 +2021-02-28 +2021-03-05 +2021-03-30 +2021-04-09 +2021-04-19 +2021-05-04 +2021-05-14 +2021-05-19 +2021-05-29 +2021-06-03 +2021-06-08 +2021-06-13 +2021-06-18 +2021-06-23 +2021-06-28 +2021-07-08 +2021-07-13 +2021-07-18 +2021-07-23 +2021-08-02 +2021-08-22 +2021-08-27 +2021-09-01 +2021-09-06 +2021-09-11 +2021-09-16 +2021-09-21 +2021-09-26 +2000 +4000 +6000 +8000 +10000 +Water pixels +RGMM + = 0.1 + = 0.3 + = 0.4 + = 0.45 + = 0.5 + = 0.55 + = 0.6 + = 0.8 +Benchmark +2020-10-21 +2020-10-26 +2020-11-05 +2020-11-10 +2020-11-20 +2020-11-25 +2020-11-30 +2020-12-05 +2020-12-10 +2020-12-30 +2021-01-09 +2021-01-14 +2021-01-19 +2021-02-23 +2021-02-28 +2021-03-05 +2021-03-30 +2021-04-09 +2021-04-19 +2021-05-04 +2021-05-14 +2021-05-19 +2021-05-29 +2021-06-03 +2021-06-08 +2021-06-13 +2021-06-18 +2021-06-23 +2021-06-28 +2021-07-08 +2021-07-13 +2021-07-18 +2021-07-23 +2021-08-02 +2021-08-22 +2021-08-27 +2021-09-01 +2021-09-06 +2021-09-11 +2021-09-16 +2021-09-21 +2021-09-26 +3000 +5000 +7000 +9000 +Water pixels +RLR +2020-10-21 +2020-10-26 +2020-11-05 +2020-11-10 +2020-11-20 +2020-11-25 +2020-11-30 +2020-12-05 +2020-12-10 +2020-12-30 +2021-01-09 +2021-01-14 +2021-01-19 +2021-02-23 +2021-02-28 +2021-03-05 +2021-03-30 +2021-04-09 +2021-04-19 +2021-05-04 +2021-05-14 +2021-05-19 +2021-05-29 +2021-06-03 +2021-06-08 +2021-06-13 +2021-06-18 +2021-06-23 +2021-06-28 +2021-07-08 +2021-07-13 +2021-07-18 +2021-07-23 +2021-08-02 +2021-08-22 +2021-08-27 +2021-09-01 +2021-09-06 +2021-09-11 +2021-09-16 +2021-09-21 +2021-09-26 +1000 +3000 +5000 +7000 +9000 +Water pixels +RSIC +Figure 8: Sensitivity analysis results for the three recursive models evaluated in this research. The number of pixels classified +as water for the different evaluated dates is compared between realizations of the water mapping experiment under values +of class transition probability ϵ between 0.1 and 0.8. +5. Discussion +The implications of the main findings of this research are presented hereafter. +In this paper, we +propose a recursive Bayesian classification method for robust multitemporal image classification that can +be built on top of time-independent generative or discriminative models. The recursive characteristic of +the method makes it more robust to common variations found in RS imagery such as illumination and +atmospheric interference (e.g., different aerosol concentrations or viewing angles). The proposed method +is simple, easy to use, interpretable, and controlled by a unique explainable parameter ϵ representing +the probability of transitioning among the different classification labels. As a consequence, ϵ governs +the trade-off between adaptability to natural changes in scene and robustness to outliers caused by +illumination of atmospheric interferences. As the class transition probability is specific to each site and +17 + +to the static classification algorithm, a study has been performed on the model sensitivity to the parameter +ϵ. The sensitivity analysis in Section 4.4 shows that in order to obtain robust classification results, i.e., +that are insensitive to undesired abrupt changes in the image, it is best to select small values of ϵ for +scenes that are expected to change smoothly. +In this work, we demonstrated the proposed methodology leveraging different models such as GMMs +(generative), LR (discriminative), classifiers using deep neural networks, and spectral index-based classi- +fiers. As a key contribution, we provide a clear framework to convert classifiers based on spectral indices +(such as NDVI and MNDWI), which are widely used in remote sensing applications, into robust multi- +temporal classifiers. Spectral index classifiers are a valued tool in the RS community given their simplicity +and low required computational cost. However, they are highly sensitive to changes in illumination and +pixel disturbances, which are commonly present in remotely sensed image data. To achieve this, we +provided a methodology to convert index-based classifiers into a probability measure with the mapping +presented in Eq. (11). Furthermore, the proposed approach can be easily applied to more sophisticated +methodologies such as pre-trained (static) deep learning-based classifiers without requiring additional +training data. However, models like deep neural networks are very flexible and can lead to overconfident +classification results, diminishing the impact that information from previous time instants has on the +model, thus, reducing the robustness of the algorithm. To circumvent this phenomenon, we proposed to +empirically reduce such overconfidence by inserting a positive constant to slightly push the probabilities +towards a discrete uniform distribution (see Eq. (10)), greatly improving the robustness of deep neural +network-based multitemporal classifiers. +Promising results have been obtained both with a water mapping and a land cover classification +experiment. In Figs. 5 and 6, it can be seen how the recursive versions of the models are robust to changes +in the illumination level of the scene when compared to their non-recursive counterparts. This robustness +is strongly observed when images include pixel disturbances or undesired changes in illumination, which +have a negative impact on the non-recursive models but can be successfully overcome for most of the dates +by the proposed recursive framework. Also, it is important to ensure the adaptability of the recursive +methods so that they are able to identify natural phenomena and incorporate smooth changes in the +image time-series. +As limitations of this work, we highlight the lack of ground truth multitemporal classification maps, the +non-exploitation of spatial correlations between pixels, and our choice for constant transition probabilities. +While the lack of ground truth is hard to circumvent and makes a quantitative assessment of the results +challenging, we explicitly assumed the labels from different pixels to be independent and the transition +probabilities to be time-invariant in order to simplify the method and make it easy to tune. Moreover, +this also leads to a methodology that has a low associated computational cost and can be used over large +geographical areas. +Finally, we believe that the proposed framework might have a broad impact on the RS community. +On the one hand, the simplicity of the proposed algorithm and its ability to incorporate models ranging +from spectral index-based to pre-trained deep learning classifiers, and the improved results demonstrated +in the experiments shows its potential to reliably extract crucial information from a large amount of +multitemporal data. On the other hand, there is still a need for unsupervised methods that can account +for, e,g., seasonal effects, which provides directions to improve the proposed methodology even further. +6. Conclusion +In this paper, we introduced a recursive Bayesian classification framework for satellite image time- +series that provides high accuracy while requiring low computational cost and minimal supervision, +which differs from existing recursive approaches that typically require large amounts of training data. +The proposed framework allows the conversion of static classifiers or generative models into recursive +Bayesian classifiers, making them more robust to non-informative image variations in multispectral image +sequences. A water mapping and a land cover experiment have been conducted analyzing Sentinel-2 +satellite data of two areas in the US. The performance of three different static classification algorithms, +including GMM, LR and SIC, as well as two deep learning-based methods has been compared to their +recursive counterparts. The SIC is a novel algorithm proposed in this paper that uses spectral index +values to generate predictive probability of occurrence of, e.g., land and water pixels. +In particular, +the MNDWI and NDVI are the spectral indices used as benchmark methods in the water mapping +and land cover experiments, respectively. Results suggest that the proposed framework increases the +robustness of existing static classification algorithms in a multitemporal setting, also when being applied +to pre-trained state-of-the-art deep learning-based models without requiring additional training data. +18 + +Significant improvements were observed both for the water mapping and for the land cover classification +results when compared to the non-recursive implementations of the classifiers. Future work will investigate +the incorporation of spatial information into the recursive Bayesian classification strategy to improve the +robustness against non-informative spectral variations, as well as a method to automatically determine +the class transition probabilities and their prior probabilities. +Declaration of Competing Interest +The authors of this paper declare that they have no known personal relationships or competing +financial interests that could have appeared to influence this research. +Dataset and supplementary results +A Python implementation of the proposed algorithms can be found at https://github.com/neu-spiral/ +RBC-SatImg. The dataset is also available at (Calatrava et al., 2022b). Supplementary material contain- +ing additional experimental results are also available with this paper. +Acknowledgements +This work has been partially supported by the National Geographic Society under grant NGS-86713T- +21, the National Science Foundation under Award ECCS-1845833, and the National Aeronautics and +Space Administration under Award 80NSSC20K0742. +References +Acharya, T., Subedi, A., Lee, D., 2018. Evaluation of Water Indices for Surface Water Extraction in a +Landsat 8 Scene of Nepal. Sensors 18, 2580. doi:10.3390/s18082580. +Akbar, T.A., Hassan, Q.K., Ishaq, S., Batool, M., Butt, H.J., Jabbar, H., 2019. Investigative Spatial +Distribution and Modelling of Existing and Future Urban Land Changes and Its Impact on Urban- +ization and Economy. +Remote Sensing 11, 105. +URL: http://dx.doi.org/10.3390/rs11020105, +doi:10.3390/rs11020105. +Alex, E., Ramesh, K., Hari, S., 2017. Quantification and understanding the observed changes in land +cover patterns in Bangalore. 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An enhanced spatial and temporal adaptive +reflectance fusion model for complex heterogeneous regions. +Remote Sensing of Environment 114, +2610–2623. +23 + +Supplementary Results +2020-10-26 +SIC +GMM +LR +DWM +WN +RSIC +RGMM +RLR +RDWM +RWN +RGB +2020-11-05 +2020-11-10 +2020-11-20 +2020-11-25 +2020-11-30 +2020-12-05 +2020-12-10 +2020-12-30 +2021-01-09 +2021-01-14 +2021-01-19 +2021-02-23 +Figure 9: Supplemental results from study area A including all dates (from 2020-10-26 to 2021-02-23). +24 + +1.2021-02-28 +2021-03-05 +2021-03-30 +2021-04-09 +2021-04-19 +2021-05-04 +2021-05-14 +2021-05-19 +2021-05-29 +2021-06-03 +2021-06-08 +2021-06-13 +2021-06-18 +2021-06-23 +Figure 10: Supplemental results from study area A including all dates (from 2021-02-28 to 2021-06-23). +25 + +2021-06-28 +2021-07-08 +2021-07-13 +2021-07-18 +2021-07-23 +2021-08-02 +2021-08-22 +2021-08-27 +2021-09-01 +2021-09-06 +2021-09-11 +2021-09-16 +2021-09-21 +2021-09-26 +Figure 11: Supplemental results from study area A including all dates (from 2021-06-28 to 2021-09-26). +26 + +?.2020-10-26 +SIC +GMM +LR +DWM +WN +RSIC +RGMM +RLR +RDWM +RWN +RGB +2020-11-05 +2020-11-10 +2020-11-20 +2020-11-25 +2020-11-30 +2020-12-05 +2020-12-10 +2020-12-30 +2021-01-09 +2021-01-14 +2021-01-19 +2021-02-23 +Figure 12: Supplemental results from study area B including all dates (from 2020-10-26 to 2021-02-23). +27 + +2021-02-28 +2021-03-05 +2021-03-30 +2021-04-09 +2021-04-19 +2021-05-04 +2021-05-14 +2021-05-19 +2021-05-29 +2021-06-03 +2021-06-08 +2021-06-13 +2021-06-18 +2021-06-23 +Figure 13: Supplemental results from study area B including all dates (from 2021-02-28 to 2021-06-23). +28 + +2021-06-28 +2021-07-08 +2021-07-13 +2021-07-18 +2021-07-23 +2021-08-02 +2021-08-22 +2021-08-27 +2021-09-01 +2021-09-06 +2021-09-11 +2021-09-16 +2021-09-21 +2021-09-26 +Figure 14: Supplemental results from study area B including all dates (from 2021-06-28 to 2021-09-26). +29 + +2020-11-08 +SIC +GMM +LR +RSIC +RGMM +RLR +RGB +2020-11-10 +2020-11-18 +2020-12-03 +2020-12-13 +2020-12-15 +2021-01-24 +2021-03-13 +2021-03-20 +2021-03-23 +2021-03-30 +2021-04-04 +2021-04-24 +2021-05-14 +2021-05-17 +2021-05-19 +Figure 15: Supplemental results from study area C including all dates (from 2021-11-08 to 2021-05-19). +30 + +2021-05-27 +2021-06-06 +2021-06-18 +2021-07-23 +2021-07-31 +2021-08-25 +2021-08-27 +2021-09-11 +2021-09-14 +2021-09-19 +2021-09-26 +Figure 16: Supplemental results from study area C including all dates (from 2021-05-27 to 2021-09-26). +31 + diff --git a/VNAzT4oBgHgl3EQf1P5q/content/tmp_files/load_file.txt b/VNAzT4oBgHgl3EQf1P5q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..edf0558344a75883622f2524e2f99a6dfcdde827 --- /dev/null +++ b/VNAzT4oBgHgl3EQf1P5q/content/tmp_files/load_file.txt @@ -0,0 +1,1702 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf,len=1701 +page_content='Recursive classification of satellite imaging time-series: An application to water and land cover mapping Helena Calatravaa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='˚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Bhavya Duvvuria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Haoqing Lia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Ricardo Borsoib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Edward Beighleya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Deniz Erdo˘gmu¸sa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Pau Closasa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Tales Imbiribaa aNortheastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 02215,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' USA bCRAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' University of Lorraine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Vandoeuvre-les-Nancy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' F-54000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' France Abstract A wide variety of applications of fundamental importance for security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' environmental protection and urban development need access to accurate land cover monitoring and water mapping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' for which the analysis of optical remote sensing imagery is key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Classification of time-series images, particularly with recursive methods, is of increasing interest in the current literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Nevertheless, existing recursive ap- proaches typically require large amounts of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This paper introduces a recursive classification framework that provides high accuracy while requiring low computational cost and minimal supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The proposed approach transforms a static classifier into a recursive one using a probabilistic framework that is robust to non-informative image variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' A water mapping and a land cover experiment are conducted analyzing Sentinel-2 satellite data covering two areas in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The performance of three static classification algorithms and their recursive versions is compared, including a Gaussian Mixture Model (GMM), Logistic Regression (LR) and Spectral Index Classifiers (SICs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' SICs consist in a new approach that we introduce to convert the Modified Normalized Difference Water Index (MNDWI) and the Normalized Difference Vegetation Index (NDVI) into probabilistic classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Two state-of-the-art deep learning-based classifiers are also used as benchmark models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Results show that the proposed method significantly increases the robustness of existing static classifiers in multitemporal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Our method also improves the performance of deep learning-based classifiers without the need of additional training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Keywords: Recursive Bayesian classification, spectral indices, water mapping, land cover mapping, land cover change detection and time-series analysis, unsupervised classification PACS: 0000, 1111 2000 MSC: 0000, 1111 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Introduction Given the vast amount of high resolution remotely sensed data available today, there exists a consid- erable body of literature focused on remote sensing (RS) applications involving land cover mapping and change detection (Anderson, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Karan and Samadder, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Rwanga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Some application examples are studies on land conservation, sustainable development, landscape planning and management of resources such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Changes in water dynamics can be studied by surface water mapping, with the aim of monitoring floods (Farhadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Koukoula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Proud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2011) and assessing the quality of water (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Son and Wang, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Tong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Water mapping can also be used for coastline extraction and change assessment (Ekercin, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Hannv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Pardo-Pascual et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2013), while land cover mapping is of fun- damental importance when identifying the distribution of different types of crops (Griffiths et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Tewabe and Fentahun, 2020) or the dynamical evolution of land use in urban environments (Gadrani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Several sources of remotely sensed data are currently available, presenting different characteristics when it comes to spatial, spectral, radiometric and temporal resolution (Satir and Berberoglu, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Spatial resolution typically varies from centimeters, in the case of very high resolution sensors, such ˚Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Email address: calatrava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='h@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='edu 1Indicates shared first authorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Preprint submitted to Remote Sensing of Environment November 2022 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='01796v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='IV] 4 Jan 2023 as the ones used by the GeoEye and QuickBird-2 satellites, to a few meters, in the case of sensors used by the Landsat 8 and Sentinel-2 satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Such satellites can acquire images of the same scene with a weekly temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' On the other hand, satellites equipped with the Moderate Resolution Imaging Spectroradiometer (MODIS) and the Visible Infrared Imaging Radiometer Suite (VIIRS) sensors achieve a higher temporal resolution, with daily image acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, their spatial resolution is significantly low, being in the order of hundreds of meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Considering this and in order to combine images with different spatial and temporal resolutions, multimodal image fusion techniques have been developed to generate high spatio-temporal image sequences, contributing to generate a wealth of RS data (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2010, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Spectral indices are one of the main land cover mapping tools given their simplicity and required low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' They compute scalar-valued features as a function of specific spectral bands, whose value can be used to distinguish between different land cover classes contained in a pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Besides, spectral index values can be easily interpreted or explained, as they minimize the effect of illumination in satellite imagery while enhancing different spectral features present in the scene under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For instance, the Normalized Difference Vegetation Index (NDVI) enhances the presence of trees, bushes, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This is due to the reflectance given by the spectral response of vegetation increasing for the wavelengths defined by the NDVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Water indices are used for water extraction at pixel level, given the difference in spectral reflectance of land and water in the near and middle infrared wavelengths (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The most widely used water indices are the Normalized Difference Water Index (NDWI), the Modified NDWI (MNDWI) and the Automated Water Extraction Index (AWEI) (Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' It has been shown that challenging weather conditions create problems in the extraction of water bodies with spectral index methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Nevertheless, this can be solved with modified methods like the one proposed in Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2016), which effectively mitigates mountain shadows and allows the extraction of particularly challenging small water bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Also, in Khalid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2021), they suggest that the Land surface temperature Based Water Extraction Index (LBWEI) provides high accuracy under all tested weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Although spectral indices can achieve good results using only a subset of the spectral bands provided by the sensor, it has been shown that using all Sentinel-2A bands can improve classification accuracy (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Aside from spectral index methods, there is a wide choice of land cover classification approaches based on machine learning methods available in the literature, whose main advantage is an increased flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Some of the explored machine learning methods used in RS include maximum likelihood classifiers (Frazier and Page, 2000), decision trees, support vector machines (Hannv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2013), logistic regression (LR) (Mueller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2015), random forests (Pelletier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2016, 2017), naive Bayes and clustering methods like the widely used K-means algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The taxonomy of RS image classification methods proposed in Satir and Berberoglu (2012) groups them into supervised/unsupervised, parametric/non-parametric and hard/soft classifiers, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Results in Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2019) and Skakun (2010) suggest that deep learning methods like artificial neural networks provide high accuracy results in land cover classification, even when compared to other machine learning classifiers such as support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The analysis of multitemporal or time-series data is of increasing interest for RS applications (Johnson and Iizuka, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Kuenzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Exploiting multitemporal data makes it possible to improve the performance of tasks such as classification (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' G´omez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2016) or spectral mixture analysis (Borsoi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Halabisky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2016) due to its temporal correlation, while at the same time supplying the end-user with a more complete product that shows the spatial as well as the temporal distribution of land classes or their proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The simplest approach to perform multitemporal land cover mapping is to apply a static classifier to each image in the sequence, being spectral indices such as the NDVI a popular choice (Jeevalakshmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, this does not exploit the temporal information available in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Significant effort has been dedicated to developing techniques specifically suited to process multitemporal image sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For instance, the authors in Hoberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2015) and Kenduiywo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2017) proposed classification methods based on conditional random field models, which represent the interactions between class labels in both time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Transfer and active learning were combined in Demir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2013) to adapt a pre-trained classifier to new images acquired at other time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Time-series classification accounting for missing pixels using Gaussian process regression was proposed in Constantin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2022), while other works considered 1D temporal covolutional neural networks (CNNs) (Pelletier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2019), and 3D spatio-temporal CNNs (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The aforementioned techniques require a full image time-series as input in order to produce classifica- tion maps, and are thus often referred to as batch or offline time-series classification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, in practice, instruments such as Sentinel-2 or Landsat 8 are continuously acquiring images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Generat- 2 ing classification maps online using batch algorithms require re-processing the whole image time-series every time a new image is acquired, which can be computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Thus, recursive meth- ods are of particular interest, as they can iteratively update multitemporal classification maps as new images are acquired by leveraging previously computed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' These methods can operate online and be more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The earliest recursive RS classification techniques were based on Bayesian filtering ideas, by recursively updating the probabilities of each class given the measurements after each datum is acquired (Strahler, 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Swain, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' These techniques are based on a statistical model that represents the pixel spectra given its class, called a generative model, which is non-trivial to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' More recent Bayesian approaches have proposed classification strategies that are recursive both in time, and across multiple spatial scales (multiresolution) (Hedhli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2016), using computationally expensive algorithms such as the expectation maximization method to learn parameters and a generative model for the pix- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Other recent works have leveraged deep learning strategies, in particular different instantiations of recurrent neural networks, such as long short-term memory (LSTM) networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' These have been applied to predict flood susceptibility (Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021), for crop identification (Rubwurm and Korner, 2017), and for land cover classification (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, these models need large amounts of data and long training times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Despite their widespread use in land cover mapping, the previously mentioned classification algo- rithms suffer from several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' First, they are highly sensitive to illumination and atmospheric interferences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', different aerosol concentrations or viewing angles), which can significantly impact the spectra of pixels from a given material class (Borsoi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Theiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The lack of robustness to such non-informative spectral variations is a significant limitation of spectral indices such as the MNDWI (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Moreover, due to the high sensor-to-target distances involved in RS applications, many image pixels do not belong to a single class, but are instead composed of a mixture of different material classes (Quintano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Although this can be addressed by spectral mixture analysis techniques (Keshava and Mustard, 2002) or by assigning a pixel to more than one class (Mertens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2006), it poses a significant challenge to traditional classification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Furthermore, while some deep learning-based algorithms such as artificial neural networks have sufficient flexibility to learn a classification function in the presence of these interferences, such methods require large amounts of labeled training data and have a high computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Although spectral index classifiers have more limited performance, they are widely used in RS applications because they do not rely on training data (Khalid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Also, supervised time-series classification approaches require labeled training images of the same region at multiple time instants, adding another layer of difficulty for their training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' t time index time index Data Classifier or Likelihood Model Bayesian Recursion Recursive Bayesian Classification (RBC) Framework Image time-series Time-series classification maps Recursion Generative version Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (3) Discriminative version Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (5) Discriminative or Generative Model Figure 1: Overview of the proposed recursive Bayesian classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This method is able to convert a static generative model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', which models the observation of a pixel given its class label with the likelihood function p pzt|Ctq) or a discriminative model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', which models the observation of a class label given a corresponding pixel with p pCt|ztq) into a recursive Bayesian classifier that exploits the temporal relationship of time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Knowledge about the state transition matrix A and the class prior probabilities p pCtq is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The hat operator in pCt denotes the decision from the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Finally, C is an experiment-dependent set containing K different labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This paper addresses the need for a multitemporal, multispectral land cover classification method that provides high accuracy while requiring low computational cost and minimal supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The proposed approach, termed recursive Bayesian classification (RBC), is able to convert a static generative model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', which models the observation of a pixel given its class label) or discriminative classifier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', which models the observation of a class label given a corresponding pixel) into a recursive classifier using a probabilistic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The method is based on a Markov assumption on the transition of the class labels, which states that the probabilities of a change in a class label can be determined based only on the information of labels at the most recent time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This simplifies the model, and allows to capture the 3 dynamical aspect of the classifier with a single set of class transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Opposed to previous approaches based on generative models (Hedhli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Strahler, 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Swain, 1978), the proposed method can integrate existing state-of-the-art static classification algorithms in a recursive framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Moreover, unlike deep learning methods, it does not need huge training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' An overview of the method introduced in this paper can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In our model, class transition probabilities can be tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This provides a transparent way to balance the capability of the classifier to adapt to abrupt class changes with the increase of its robustness to non-informative spectral variations originating from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', atmospheric and illumination variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Fur- thermore, a recursive version of spectral index classifiers such as the MNDWI and the NDVI is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For this matter, spectral index values are converted into probabilistic classification results, which contain class uncertainty information and can be integrated into the recursive discriminative Bayesian classifica- tion framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This results in an unsupervised, recursive classification method that is more robust in multitemporal settings when compared to the spectral indices, addressing one of their main limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The performance of the proposed approach is demonstrated through two different experiments ana- lyzing 10 meter spatial resolution Sentinel-2 satellite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Due to the lack of ground truth classification maps for the selected study regions, classification methods are trained and evaluated based only on the observed Sentinel-2 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The first experiment consisted in water mapping of a reservoir and its down- stream river, using the MNDWI as a benchmark spectral index method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The second experiment consisted in land cover classification using the NDVI as a spectral index benchmark method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Two terrain areas located in Oroville Dam, an embankment dam in California, have been evaluated in the first experi- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' These two areas pose several challenges to water mapping algorithms, such as ripples caused by changes in the water flow, variations in illumination and abrupt changes in the water level of the reser- voir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The terrain area evaluated in the second experiment is the Charles river basin, in Boston, which is also challenging given the appearance of algal blooms in the summer, variations in illumination and in atmospheric conditions, and the presence of highly reflective surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In the case of both experiments, we compared the performance of several static classification algorithms (namely, a Gaussian mixture model (GMM), LR, and spectral index classifiers) and their recursive versions obtained with the RBC approach proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The GMM belongs to the group of generative models, while the last two belong to the group of discriminative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For the water mapping experiment, we also included as benchmark models two pre-trained state-of-the-art deep learning-based discriminative classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' These are the DeepWaterMap (Isikdogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2017) and WatNet (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Experimental results show that the proposed RBC approach can significantly increase the robustness of existing classification algorithms in a multitemporal setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Moreover, for the water mapping exper- iment, the proposed framework substantially improved the performance of pre-trained state-of-the-art classification methods using deep artificial neural networks in the multitemporal setting, without requir- ing additional training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Provided results are fully reproducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' A Python implementation of the proposed algorithms can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='com/neu-spiral/RBC-SatImg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The dataset is also available at (Calatrava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The data that has been used and the area under study are described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In Section 3, the proposed recursive Bayesian classification technique based on both generative and discriminative models and also the proposed spectral index classification model are derived mathematically and described in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The experimental setup, results and a comparison with state-of-the-art approaches are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' A discussion on the implications and impact of our results is provided in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Finally, Section 6 concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Area of Study and Satellite Data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Area of Study Our research focuses on three areas of study located in the United States (US), which are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study areas A and B are located in Oroville Dam, an embankment dam on the east side of the city of Oroville, in the state of California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Being 235 meters high, it is the tallest dam in the United States and it is mostly used for water supply and flood control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study areas A and B are analyzed in the water mapping experiment using the MNDWI as one of the benchmark methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study area A is located in the dam downstream and it contains a small water stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study area B is located in the dam upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The Sentinel-2 RGB composite and ESA WorldCover Map images of the Oroville Dam region can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The sizes of study areas A and B are 200 ˆ 500 and 150 ˆ 110 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4 Table 1: Description of the three evaluation areas and the two training regions used in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Data size is given in pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study Area (Evaluation) Location Data Size (px) A (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2) Oroville Dam (Downstream) in California, US 200 ˆ 500 B (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2) Oroville Dam (Upstream) in California, US 150 ˆ 110 C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 3) Charles river in Boston, Massachusetts, US 700 ˆ 1241 Study Region (Training) Location Data Size (px) 1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2) Oroville Dam in California, US 2229 ˆ 3341 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 3) Charles river in Boston, Massachusetts, US 927 ˆ 2041 1 Figure 2: Sentinel-2 RGB composite (upper figures) and ESA WorldCover Map (Zanaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021) (lower figure) images of study areas A and B, located in the downstream and upstream of the Oroville Dam and with approximate coordinates 39˝36102N 121˝27102W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Models are trained with pixel data covering study region 1 and evaluated with pixel data covering study areas A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study area C extends over the Charles and Mystic rivers and the Boston harbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' To demonstrate the performance of the proposed approach for land cover mapping, images of the Charles river basin are analyzed using the NDVI as one of the benchmark methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This site contains a big permanent water body, urban vegetation and built-up area, which are interesting for land cover detection and classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The Sentinel-2 RGB composite and ESA WorldCover Map images of the Charles river area 5 esa2 km 1 C Study Area (Evaluation) 2 Study Region (Training) Figure 3: Sentinel-2 RGB composite (upper figure) and ESA WorldCover Map (Zanaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021) (lower figure) images of study area C, located in the Charles river basin and with approximate coordinates 71˝5102W 42˝22102N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Models are trained with pixel data covering study region 2 and evaluated with pixel data covering study area C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The size of study area C is 700 ˆ 1241 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study areas A, B and C pose significant challenges to multitemporal classification algorithms due to seasonal changes in the land cover and variations in illumination and atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study area A, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4 (first row), suffers from the effect of illumination factors given by varying solar incidence angles and the time of day at which images are captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Moreover, ripples and other artifacts caused by the high flows in the river stream make the classification of water pixels challenging, particularly to spectral indices such as the MDNWI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study area B is located in a reservoir where the water level changes seasonally based on reservoir storage data obtained from the NWIS USGS website (https://waterdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='usgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='gov/ nwis/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In October 2020, the recorded water storage was 200,485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='8 hc-m, decreasing to 160,783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2 hc-m in December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This was followed by changes that resulted in recorded water storage of 183,223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='8 and 97,542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='6 hc-m in May and September of 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4 shows Sentinel-2 RGB composite images of study area B that clearly illustrate the significant changes caused by the varying water storage of this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Challenges in study area C include the seasonal cyanobacterial bloom in the Charles river and in the Boston harbor waters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The presence of reflective surfaces from buildings in the area are also difficult to classify, as they are easily mistaken for water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Algal blooms mostly occur during summer (Rome et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021), as shown by the image captured on 2021-07-31 from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 6 esaesaesaestesaScattered clouds Algal blooms Pixel discrepancies 2020-10-16 2021-01-19 2021-05-04 2021-09-01 Challenges in Area B: Abrupt water level changes in the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Challenges in Area A: Ripples are caused by the water flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' See the challenging variations in illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Challenges in Area C: Appearance of algal blooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' See the challenging variations in illumination and atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2021-03-23 2021-07-31 2020-10-21 2021-06-23 Figure 4: Sentinel-2 RGB composite images obtained with the Google Earth Engine (GEE) (Gorelick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2017) showing the challenges posed by the selected evaluation areas A, B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The evaluation areas are selected to be smaller than the training regions in order to facilitate the analysis of results since the temporal behavior of the images in these regions can be more easily interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Also, the overall computational cost decreases when lowering the number of pixels to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The GMM and LR models need to be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Training is performed using a weakly supervised approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' First, pseudo-labels for a small set of images are obtained as classification maps from spectral index classifiers, which are carefully checked so as to accurately represent the study areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Then, these pseudo- labels are used to train the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In the case of the Oroville Dam scenario, images covering the training region shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2 are used, with size 2229 ˆ 3341 in pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Alternatively, for the Charles river scenario, images with a size of 927ˆ2041 pixels are used, covering the training region shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Satellite Data In this research, Sentinel-2 images at the Coastal Aerosol, Blue, Green, Red, Near-Infrared (NIR, band 8 for Sentinel-2), Narrow NIR (band 8A for Sentinel-2), Shortwave Infrared (SWIR) 1 and SWIR 2 bands are used to evaluate the proposed recursive Bayesian classification technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The Sentinel-2 mission is designed to give a high revisit frequency of 5 days alternating between two twin satellites, where each twin satellite systematically acquires optical imagery from the target scene every 10 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Besides, the high spatial resolution of the data (10, 20 or 60 m) aids in observing the changing patterns in the land more accurately than compared to lower resolution satellites such as the ones using the MODIS or the VIIRS sensors, which provide resolutions of hundreds of meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Taking this into account, the Sentinel-2 data is of our interest due to its high temporal and spatial resolution, in addition to its availability via the Google Earth Engine (GEE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The resolution and central wavelength of the considered bands can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Regarding the RGB bands, band 4 (Red) is useful for identifying soil, water and many urban features, band 3 (Green) gives excellent contrast between clear and turbid waters, and band 2 (Blue) is useful for identifying vegetation and also human-made features (Maciej Huk, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' When it comes to the SWIR bands, they are useful for measuring vegetation, water and soil moisture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Table 2: Resolution (in meters) and central wavelength (in nanometers) of the Sentinel-2 spectral bands used in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Band Description Resolution (m) Central wavelength (nm) 1 Coastal Aerosol 60 443 2 Blue 10 490 3 Green 10 560 4 Red 10 665 8 NIR 10 842 8A Narrow NIR Edge 20 865 11 SWIR 1 20 1610 12 SWIR 2 20 2190 7 The Sentinel-2 level-2A (surface reflectance) data was downloaded with the GEE from the COPERNI- CUS/S2 SR collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' GEE atmospherically corrects the images using the standard SEN2COR software package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Only images with at most 10% cloud cover as indicated by the GEE are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In the post-processing stage, the resolution of bands 8A, 11, 12 (20 meters) and band 1 (60 meters) is increased to 10 meters by nearest-neighbor interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The pixel values are scaled by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='0001 such that the surface reflectance values are between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' We consider images acquired between dates 2020-09-01 and 2021-09-26 for the Oroville Dam scenario, and images between dates 2020-09-04 and 2021-09-29 for the Charles river region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Images of the Charles river area with significant snow cover are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The preprocessed data used in our paper was made available at (Calatrava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Recursive Bayesian Classification In this paper, we propose a recursive Bayesian classification technique for water mapping and land cover classification using multispectral and multitemporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The proposed technique is considered to be unsupervised because it does not need labeled training data for multiple time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' It may be applied on top of other classifiers regardless of their supervised, semi-supervised or unsupervised nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Therefore, it can be stated that the recursion technique is agnostic to the classifier that is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' When applied on top of an unsupervised classifier, the resulting technique can be considered to be completely unsupervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Let us denote by Zt P RBˆN an image with B bands and N pixels observed at time instant t “ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The images at the different time instants are supposed to be coregistered, that is, they constitute observations of the same geographical scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For each pixel zt,n P RB, being n “ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , Nu, we associate a label Ct,n P C , where C is an experiment-dependent set containg the possible K labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For a set of images Zt over time, the most likely label Ct,n for each pixel zt,n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', the n-th column of Zt) can be estimated based on all the previously observed data tZt, Zt´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , Z1u by maximizing the posterior probability ppCt,n|Zt, Zt´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , Z1q as pCt,n “ arg max Ct,nPC ppCt,n|Zt, Zt´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , Z1q , (1) where the hat operator denotes the decision from the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This expression is powerful, as it considers both temporal and spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, learning the posterior Probability Mass Function (PMF) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (1) can be hard, specially with high dimensional images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' A spatial independence assumption can be applied to reduce the computational cost when calculating the conditional PMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' We propose to treat the label of every pixel as independent of the data from other pixels, meaning that Ct,n only depends on zt,n, zt´1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , z1,n, or, equivalently, on z1:t,n fi tzt,n, zt´1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , z1,nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This is without loss of generality, as the proposed approach can be directly extended to consider spatial information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', from multiple pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Thus, the posterior in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (1) becomes ppCt,n|z1:t,nq, disregarding spatial information, and leading to pCt “ arg max CtPC ppCt|z1:tq, (2) where the pixel index n is omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The classifier proposed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2) still considers a temporal dependence on previous data, meaning that the labels and images at previous time instants influence the results of the current time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The posterior PMF in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2) can be computed recursively using Bayes theorem under conditional independence assumptions and assuming knowledge about the prior ppC0q, the sate transition ppCt|Ct´1q 8 and the likelihood distribution ppzt|Ctq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Thus, the posterior PMF can be computed as ppCt|z1:tq “ ÿ Ct´1PC ppCt, Ct´1|z1:tq paq “ ÿ Ct´1PC ppCt|Ct´1, ztqppCt´1|z1:t´1q pbq “ ÿ Ct´1PC ppzt|CtqppCt|Ct´1q ppzt|Ct´1q ppCt´1|z1:t´1q “ ÿ Ct´1PC ppzt|CtqppCt|Ct´1q ř C1 t ppzt|C1 tqppC1 t|Ct´1qppCt´1|z1:t´1q “ ppzt|Ctq ÿ Ct´1PC ppCt|Ct´1q ř C1 t ppzt|C1 tqppC1 t|Ct´1qppCt´1|z1:t´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (3) where in equality paq we assumed a first-order Markov model, that is, given Ct´1 and zt, the class labels Ct are independent of z1:t´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' in equality pbq we applied the Bayes theorem followed by the same conditional independence assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' We refer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (3) as the Recursive Bayesian Generative Model (RBGM) due to its dependence on the likelihood function ppzt|Ctq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The term ppCt´1|z1:t´1q denotes the posterior PMF of the previous time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This term shows in the equation because states are assumed to be independent of future measurements, meaning that Ct´1 depends on z1:t´1 but not on zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' When t “ 1, ppCt´1|z1:t´1q becomes equivalent to the class prior probabilities since ppC0|z0q “ ppC0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (3) uses the Bayes theorem, where the posterior probability is given by the product of the likelihood and the prior divided by the marginal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The denominator in this expression considers the probability of all the possible transitions that a pixel can go through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The transition PMF ppCt|Ct´1q is described later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The posterior probability can also be computed as a function of the probability of the labels given the pixel values, which allows existing classification algorithms to be used in the RBC framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In this case, the posterior probability is computed by following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (5) and this is referred to as Recursive Bayesian Discriminative Model (RBDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Applying the Bayes theorem to the likelihood ppzt|Ctq we obtain ppzt|Ctq “ ppCt|ztqppztq ppCtq , (4) where ppCt|ztq is the prediction of the classifier to which the RBC framework is applied, which we refer to as a benchmark classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' As the RBC framework is agnostic to the classifier that is used, the prediction can be the result of any type of classifier, including deep learning methods as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The Bayes theorem can be used to extend the generative model (RBGM) to the discriminative model (RBDM) as ppCt|z1:tq “ ppCt|ztqppztq ppCtq ÿ Ct´1PC ppCt|Ct´1q ř C1 t ppzt|C1 tqppC1 t|Ct´1qppCt´1|z1:t´1q “ ppCt|ztq�� � ppztq ppCtq ÿ Ct´1PC ppCt|Ct´1q ř C1 t ppC1 t|ztq\x18\x18 \x18 ppztq ppC1 tq ppC1 t|Ct´1q ppCt´1|z1:t´1q “ ppCt|ztq ppCtq ÿ Ct´1PC ppCt|Ct´1q ř C1 t ppC1 t|ztq ppC1 tq ppC1 t|Ct´1q ppCt´1|z1:t´1q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (5) where ppCtq denotes the marginal class probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In the widely used naive Bayes classifier, the marginal class probabilities are also used (Barber, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In the absence of labeled training data and prior infor- mation about the scene, we set their value as ppCtq “ 1 K @Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (3) and (5), the term ppCt|Ct´1q corresponds to the state transition probability and it can be described using the so-called state transition probability matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For simplicity, in this work we assume ppCt|Ct´1q to be time invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Although strong, this assumption copes with the lack of knowledge we assume regarding the studied scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' We highlight, however, that this is so without loss of generality since prior knowledge about, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', seasonality can be easily incorporated in a time dependent transition PMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This matrix can be expressed for K 9 classes as A “ » ———– p11 p12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' p1K p21 p22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' p2K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' pK1 pK2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' pKK fi ffiffiffifl , (6) being pij “ ppCt “ j|Ct´1 “ iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (7) If we assume that pij “ ϵ for all i ‰ j, the state transition probability matrix for K classes can be expressed as A “ » ———– 1 ´ pK ´ 1qϵ ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' ϵ ϵ 1 ´ pK ´ 1qϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' ϵ ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 1 ´ pK ´ 1qϵ fi ffiffiffifl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (8) The particular case of K “ 2 makes this matrix simpler as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This matrix is site dependent and may be filled by expert judgement by selecting the proper value of ϵ, which corresponds to the probability of a pixel transitioning from one label to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' A|K“2 “ „ 1 ´ ϵ ϵ ϵ 1 ´ ϵ ȷ (9) It is relevant to discuss the sensitivity of recursive Bayesian classification techniques with regard to the pixel transition probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' If a low value of ϵ is selected, the probability of a pixel transitioning to a different label is believed to be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This moderates the change of the label at every pixel, making classification more robust to atmospheric interference, discrepancies in pixels and also to extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, lower values of ϵ also decrease the capability of the method to adapt to abrupt class changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Thus, the transition probabilities must be selected according to the application in order to reach a tradeoff between robustness and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' With the approach presented in this subsection, a scene can be recursively classified using both generative (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (3)) and discriminative (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (5)) models by iteratively updating a class posterior PMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Note that the proposed RBC framework relies in probabilistic classifiers (or generative models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Al- though many deep learning classifiers are currently trained based on the cross-entropy loss, which leads to a maximum likelihood estimation of the class labels (Barber, 2011), very flexible models, such as deep neural networks, can lead to overconfident classification results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', there being some j such that ppCt “ j|ztq « 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This can be damaging when such models are integrated in the proposed RBC frame- work, since such overconfidence diminishes the relevance of the prior information obtained in previous time instants through the recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Therefore, to remedy this issue, we propose to empirically reduce the confidence of the predictions of deep learning models before integrating them in the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This is performed by using a simple relation as p pCt|ztq “ pNN pCt|ztq ` λ ř C1 tPC ppNN pC1 t|ztq ` λq, (10) where pNN pCt|ztq is the prediction of the deep learning model, being it the probability of the labels Ct P C given the pixel value at time instant t, and λ P R` is a positive constant used to slightly push the predicted class probabilities towards 1{K (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', towards a discrete uniform distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Recursive Spectral Index Classification (RSIC) Algorithm Considering the Spectral Index Classification (SIC) algorithm, we propose the use of broadband spectral indices to generate predictive probability of occurrence of land classes, such as water or soil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The values of these indices are of interest for a classification algorithm due to their clear interpretability and lack of supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In this research, the MNDWI is used for the water mapping experiment, while the NDVI is used for the land cover classification experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This is described with more detail in Section 4, where the configuration used for each experiment is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' We propose the application of the recursive Bayesian framework described beforehand on top of the SIC algorithm, which we refer to as Recursive SIC (RSIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The values of standard broadband spectral indices such as the MNDWI and the NDVI must be converted into probabilities in order to calculate the 10 posterior PMF with the RBDM (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Thus, we propose to define the class probability ppCt|ztq as ppCt|ztq “ fCt pypztqq ř CtPC fCtpypztqq, (11) where ypztq corresponds to the spectral index value, which is computed as a function of the pixel zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This is a similar but not equivalent idea to applying a softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' To compute the probability value, we use a Gaussian function as fCt “ N pµCt, σCtq, where the mean and standard deviation values are selected based on the configuration of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' A different value of mean and standard deviation can be assigned for each class as µ “ coltµCtu and σ “ coltσCtu, being colt¨u the operator returning a vector whose elements are µCt and σCt for Ct P C , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The function fCt can be expressed as fCt pypztqq “ 1 σCt ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2π exp ˜ ´1 2 ˆypztq ´ µCt σCt ˙2¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (12) The function fCt gives a measure of how close the spectral index ypztq is to the mean value of each class Ct, which is denoted as µCt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This is used as an indication of the likelihood of zt being of class Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The standard deviation σCt is used to account for the length of the spectral index interval that is deemed to constitute class Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For ease of exposition, let us consider for the remainder of this section that Ct P C “ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , Ku, and also that the class indices are ordered in the same way as the threshold intervals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', class i corresponds to the i-th spectral index interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The length of intervals defining each class in the spectral index value ypztq can be highly non- homogeneous and depends on the spectral index class thresholds τi, being i P t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , Ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' These thresholds define a hard classification result based on the spectral index value, with pixel zt being assigned to the i-th class if and only if ypztq P pτi´1, τis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Their length can be calculated as Lj “ τj ´ τj´1, where j P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' , Ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The values of µ and σ are calculated as µj “ Lj{2 ` τj´1 and σj “ Lj{2, so that the probability of a pixel belonging to a given class decreases smoothly as ypztq moves away from the center of the interval and approaches one of the thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The threshold values are determined empirically and are experiment-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Please refer to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1 for details on how the threshold values were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Results In this section, we demonstrate the performance of the proposed Bayesian recursive classification methodology using both spectral indices and machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This is done through the two experiments listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Complete results can be reproduced following the instructions in https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='com/neu-spiral/RBC-SatImg, and can also be found in the supplementary material or in the extended version of this paper (Calatrava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The first experiment aims to classify pixels as water or non-water, while the second one aims to classify pixels as water, land and vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In both experiments, we consider three static classification methods and compare them to their recursive implementations by applying the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For the water mapping experiment, we also consider two pre-trained deep learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' These are the DeepWaterMap (Isikdogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2017) and the WatNet (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The multitemporal classification algorithms proposed in Pelletier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (2019) and Rubwurm and Korner (2017) were also evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, due to possible differences in the training data, obtained results were not superior to the ones provided by the benchmark models and consequently are not included in this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Details regarding the algorithms used in this research are provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Results obtained with the probabilistic instance of the SIC algorithm introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (11) are also provided in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Table 3: Description of the areas of study in the two experiments conducted in this research (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Experiment Study Region (Training) Study Area (Evaluation) Water Mapping 1 A and B Land Cover Classification 2 C 11 Table 4: Full name, abbreviation and note on the novelty of the algorithms used in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Full Name Abbreviation Note Gaussian Mixture Model GMM Benchmark Logistic Regression LR Benchmark Spectral Index Classification SIC Novel Recursive Gaussian Mixture Model RGMM Novel Recursive Logistic Regression RLR Novel Recursive Spectral Index Classification RSIC Novel DeepWaterMap DWM Benchmark (Isikdogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2017) WatNet WN Benchmark (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2021) Recursive DeepWaterMap RDWM Novel Recursive WatNet RWN Novel 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Experimental Setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Training Dataset and Model Selection For the water mapping experiment, 46 images from the Oroville Dam site are extracted from the GEE with dates between 2020-09-01 and 2021-09-26 after filtering out images (dates) with cloud pixel percentages above 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Each image belongs to a different date and they are mostly spaced 5 days apart (the temporal resolution of Sentinel-2 satellites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, some images are spaced more than 5 days apart due to filtering of images with cloud cover or other discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' We separate 4 images from the beginning of the dataset (with dates 2020-09-01, 2020-10-06, 2020-10-11 and 2020-10-16) to train the LR and GMM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The remaining 42 images (with dates between 2020-10-21 and 2021-09-26) are used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Pixels analyzed in the training stage are the ones covered by study region 1, while pixels analyzed in the evaluation stage are the ones covered by study areas A and B (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For the land cover classification experiment, 46 images from the Charles river basin are downloaded from the GEE with dates between 2020-09-04 and 2021-09-26 after filtering out images (dates) with cloud pixel percentages greater than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' After a visual inspection, 15 images with snow-covered land and other large discrepancies are also removed from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Following a similar reasoning used for splitting the training images for the Oroville Dam data, we select the first 3 images (with dates 2020-09- 04, 2020-10-01 and 2020-10-09) for training and the remaining 28 (with dates between 2020-10-14 and 2021-09-26) are used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Pixels analyzed in the training stage are the ones covered by study region 2, while pixels analyzed in the evaluation stage are the ones covered by study area C (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Images used for training are not used for evaluation in any of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For details on the pixel size of training and evaluation data, please refer to Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' To generate data with surrogate ground truth class labels to train the LR and GMM models, spectral index classification algorithms are applied to the images in the training set (the MNDWI for the water mapping experiment, and the NDVI for the land cover experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' More precisely, pixels are classified based on their spectral index value as a function of the class thresholds in τ W and τ LC, which are selected accordingly so that the generated classification maps are visually close to the reference ones in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' To obtain the generative model ppzt|Ctq used in the RGBM (in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (3)), one GMM is trained for each class label (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', ppzt|Ctq is a GMM for each choice of Ct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' To adequately represent the training pixels without overfitting, we select the smallest number of components for each GMM such that the histogram of the training distribution and the one generated by the respective GMM are visually close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Water Mapping Experiment For the water mapping experiment, Ct P C “ twater, not wateru, or, C “ t0, 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The spectral index used is the MNDWI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This index uses the Green and SWIR (band 11) bands for the enhancement of open water features, while diminishing built-up area features that are often correlated with open water in other indices (Xu, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The MNDWI is calculated as yMNDWI pztq “ zt,Green ´ zt,SWIR zt,Green ` zt,SWIR , (13) where zt,Green and zt,SWIR denote the Green and SWIR bands of zt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The threshold for non- water pixels is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='13 after carefully comparing obtained mapping results with thresholds varying between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='15 against the reference classification map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' To convert the spectral index values into a predictive probability of water ppCt|ztq, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (11) is used with mean and standard deviation 12 µW “ r´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='435, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='565s and σW “ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='565, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='435s, which are calculated with the threshold values τ W “ r´1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='13, 1s as indicated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The temporal class transition probability ϵ is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='05 when computing matrix A according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The normalization constant λ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (10) is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This value has been chosen considering a trade-off between the robustness and adaptability of the recursive framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The three benchmark models and their recursive versions are also compared to two pre-trained state- of-the-art deep learning-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Both methods were trained by their authors using Sentinel 2 images and can therefore be evaluated using the data described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The WatNet algorithm is based on deep semantic segmentation models, whereas the DeepWaterMap algorithm uses multiscale CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Land Cover Classification For the land cover classification experiment Ct P C “ twater, land, vegetationu, or, equivalently, C “ t0, 1, 2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The spectral index used is the NDVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This index uses the red and NIR bands to observe the presence of vegetation, soil and water (Akbar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Alex et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The NDVI can be computed as yNDVI pztq “ zt,NIR ´ zt,Red zt,NIR ` zt,Red , (14) where zt,Red and zt,NIR denote the Red and NIR bands of zt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The spectral index thresholds between water, land and vegetation classes are selected as τ1 “ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='05 and τ2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='35, respectively, according to the recommendations found in (Akbar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Alex et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' To convert the spectral index values into a predictive probability of each class ppCt|ztq, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (11) is used with mean and standard deviation µLC “ r´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='525, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='149, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='675s and σLC “ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='475, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='19, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='325s, calculated with the threshold values τ LC “ r´1, ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='35, 1s as indicated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The temporal class transition probability ϵ is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='05 when computing matrix A according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (8), for K “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Water Mapping Experiment Results Water mapping results for study areas A and B are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Due to space limitations, only a smaller representative interval of the image sequences is displayed, having selected one every four available dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' RGB composite images of the studied areas are shown as a reference because they highlight changes in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Results obtained for the dam downstream region (study area A), which are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 5, show a considerable difference between some of the benchmark and recursive algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For instance, the SIC, LR and DWM methods classify a large portion of the stream as land for dates 2021-05-19, 2021-06-13, 2021-09-06 and 2021-09-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Their recursive versions, on the other hand, adequately classify most of the stream pixels as water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The WN algorithm classifies some portions of land pixels as water pixels in dates between 2021-11-25 and 2021-07-08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Its recursive version, while still missclassifying a small part of the stream, is considerably more accurate, which shows that the proposed Bayesian recursive technique can also improve the performance of modern deep learning-based mapping algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Moreover, we have observed in multiple simulations that the DWM and WN algorithms resulted in overconfident classifica- tion results, making the use of the strategy described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (10) necessary to obtain this performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 13 2020-10-26 SIC GMM LR DWM WN RSIC RGMM RLR RDWM RWN RGB 2020-11-25 2020-12-30 2021-02-23 2021-04-09 2021-05-19 2021-06-13 2021-07-08 2021-08-02 2021-09-06 2021-09-26 Figure 5: Water mapping results for the downstream subscene of Oroville Dam (study area A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Purple and yellow represent water and land, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Images are arranged in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 6, which shows results for study area B, it can be inferred that the compared methods are able to adequately capture the decrease in water levels over time in the dam upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, the SIC, LR, DWM and WN benchmark methods missclassify a considerable amount of water pixels as land in the date 2020-12-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Their recursive counterparts, on the other hand, provide improved classification results at this date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' There are some dates, such as 2021-08-02, for which all methods provide adequate and comparable classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The trade-off between adaptability and robustness poses a challenge to the recursive framework for the upstream subscene of Oroville Dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' When being robust, the recursive framework becomes less flexible and consequently, abrupt changes in the scene might be detected with delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This is the case of the connection between the island at the bottom of the scene and the mainland, which is not detected by the recursive versions of the RDWM and WN algorithms in the date 2021-07-08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This happens due to the normalization constant introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (10), which is used to counteract the overconfidence of the deep learning-based classifiers, as it makes these models rely more strongly on the data from past time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 14 2020-10-26 SIC GMM LR DWM WN RSIC RGMM RLR RDWM RWN RGB 2020-11-25 2020-12-30 2021-02-23 2021-04-09 2021-05-19 2021-06-13 2021-07-08 2021-08-02 2021-09-06 2021-09-26 Figure 6: Water mapping results for the upstream subscene of Oroville Dam (study area B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Purple and yellow represent water and land, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Images are arranged in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Land Cover Classification Experiment Results Land cover classification results for study area C are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For this experiment, we have also selected one every four available dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' RGB composite images are also provided as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' It can be seen that the seasonal variations in the distribution of land, water and vegetation are captured well by all the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The decrease in the amount of vegetation starting from November (through winter) with an increase in dry land (at dates 2020-12-13 and 2021-03-20) is represented by an increase in yellow pixels until May, followed by an increase in the number of vegetation pixels through summer and fall (from 2021-05-27 to 2021-09-14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Moreover, the recursive methods (RSIC, RGMMM and RLR) provide significantly more robust performance when compared to their non-recursive counterparts, being less sensitive to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', atmospheric interference and illumination factors affected by solar incidence angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Study area C extends over an area covering the Boston harbor, the Charles river lower, mid and some upper basins, and the Mystic river lower basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Since most of these correspond to urban and suburban areas, we can find many reflective surfaces from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', building terraces and metal sheds, which lead to pixels with high spectral reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Such pixels are easily mistaken for water since their NDVI values is close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This can be observed most clearly in the date 2020-12-13, where the Bayesian recursive approaches lead to few missclassifications of reflective surface pixels close to the river when 15 compared to their non-recursive counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Moreover, on dates 2021-05-27 and 2021-07-31, due to the appearance of cyanobacterial blooms (due to which the water becomes diluted with chlorophyll pigments) in the Boston harbor waters, an important portion of water pixels are classified as land by the SIC and LR algorithms, whereas their recursive versions lead to adequate classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' On the other hand, GMM results are not particularly affected at this date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Among the benchmark methods, the GMM showed the best results, which were comparable to the ones obtained with its recursive counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' For some dates, the RBC methodology introduces a smoothing effect, which makes it more difficult to adapt to sudden class changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This can be visualized on the date 2020-03-20, as the recursive methods show a smaller amount of vegetation pixels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', more land pixels are observed) compared to the benchmark methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2020-11-08 SIC GMM LR RSIC RGMM RLR RGB 2020-12-13 2021-03-20 2021-04-24 2021-05-27 2021-07-31 2021-09-14 Figure 7: Land cover classification results for the Charles river basin (study area C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Purple, green and yellow represent water, vegetation and land, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Images are arranged in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Model Sensitivity to Class Transition Probability The class transition probabilities, introduced and denoted as ϵ in Section 3, are site-specific and it is thus important to carefully select their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In this section, we evaluate the sensitivity of the model to the class transition probability hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' A study to understand the effect of the value assigned to ϵ on the water mapping model is performed with images from the upstream Oroville dam site, which corresponds to study area B from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 8, variations in the number pixels classified as water are recorded under values of ϵ between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='8 for three recursive models evaluated in this research: RGMM, RLR and RSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Also, results can be observed for the non-recursive version of each model for ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='5 as a benchmark, in an orange color dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Although the temporal resolution of Sentinel-2 is of 5 days, due to the filtering of cloud-covered images, there are gaps between dates that are longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Obtained results suggest that the discriminative recursive models (RLR and RSIC) are more sensitive to changes in the transition probabilities than the RGMM model, as the number of water pixels varies more abruptly for the RLR and RSIC models in, for instance, dates 2020-11-10, 2020-12-30 and 2021- 01-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In the case of the RSIC model and for ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='8, the amount of water pixels decreases from 7000 to 3500, approximately, between dates 2020-12-10 and 2020-12-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, this variation is under 1000 pixels for the same model with low ϵ value (ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The RLR model is showing a very similar behavior for these two dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This shows how, in the case of the RLR and RSIC models, ϵ has an influence on the robustness-adaptability trade-off presented by the proposed recursive Bayesian framework, meaning that these models are sensitive to this hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Sensitivity is best evaluated on dates when there is a considerable change in class distribution caused by natural phenomena such as draining occurring in preparation for extreme rainfall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Also, the varying temporal gaps between dates have an impact on the distribution of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The use of large ϵ probability causes the RSIC and RLR to fail during such events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In case of RLR, a small value of ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1 is seen to smooth the abrupt change in the prediction of water pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In the case of the RSIC model, ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1 also presents results that are much smoother when compared to the ones obtained for higher ϵ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 16 The RGMM is seen to be less sensitive to changes in the values of the transition probability hyper- parameter, as the amount of water pixels is relatively similar throughout all the dates for the different values of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Nevertheless, we can see how on 2020-08-27 this model presents an abrupt change in the amount of water pixels for the highest value of ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Note that for ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='5, results match between the non-recursive (orange color) and recursive (magenta color) versions of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' We expected this result because for this value of ϵ the posterior of the recursive algorithm should match the static classi- fier output for the water mapping classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Given the results obtained in this section, we decided to set ϵ to a small value, as specified in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-10-21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-10-26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-11-05 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-11-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-11-20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-11-25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-11-30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-12-05 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2021-09-26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='Water pixels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='RGMM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='45 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='55 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='Benchmark ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-10-21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-10-26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-11-05 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='7000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='9000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='Water pixels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='RLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-10-21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2020-10-26 ' metadata={'source': 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+page_content='2021-08-27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2021-09-01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2021-09-06 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2021-09-11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2021-09-16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2021-09-21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2021-09-26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='7000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='9000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='Water pixels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='RSIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='Figure 8: Sensitivity analysis results for the three recursive models evaluated in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The number of pixels classified as water for the different evaluated dates is compared between realizations of the water mapping experiment under values of class transition probability ϵ between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Discussion The implications of the main findings of this research are presented hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In this paper, we propose a recursive Bayesian classification method for robust multitemporal image classification that can be built on top of time-independent generative or discriminative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The recursive characteristic of the method makes it more robust to common variations found in RS imagery such as illumination and atmospheric interference (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', different aerosol concentrations or viewing angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The proposed method is simple, easy to use, interpretable, and controlled by a unique explainable parameter ϵ representing the probability of transitioning among the different classification labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' As a consequence, ϵ governs the trade-off between adaptability to natural changes in scene and robustness to outliers caused by illumination of atmospheric interferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' As the class transition probability is specific to each site and 17 to the static classification algorithm, a study has been performed on the model sensitivity to the parameter ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The sensitivity analysis in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='4 shows that in order to obtain robust classification results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', that are insensitive to undesired abrupt changes in the image, it is best to select small values of ϵ for scenes that are expected to change smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In this work, we demonstrated the proposed methodology leveraging different models such as GMMs (generative), LR (discriminative), classifiers using deep neural networks, and spectral index-based classi- fiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' As a key contribution, we provide a clear framework to convert classifiers based on spectral indices (such as NDVI and MNDWI), which are widely used in remote sensing applications, into robust multi- temporal classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Spectral index classifiers are a valued tool in the RS community given their simplicity and low required computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, they are highly sensitive to changes in illumination and pixel disturbances, which are commonly present in remotely sensed image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' To achieve this, we provided a methodology to convert index-based classifiers into a probability measure with the mapping presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Furthermore, the proposed approach can be easily applied to more sophisticated methodologies such as pre-trained (static) deep learning-based classifiers without requiring additional training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' However, models like deep neural networks are very flexible and can lead to overconfident classification results, diminishing the impact that information from previous time instants has on the model, thus, reducing the robustness of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' To circumvent this phenomenon, we proposed to empirically reduce such overconfidence by inserting a positive constant to slightly push the probabilities towards a discrete uniform distribution (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' (10)), greatly improving the robustness of deep neural network-based multitemporal classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Promising results have been obtained both with a water mapping and a land cover classification experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 5 and 6, it can be seen how the recursive versions of the models are robust to changes in the illumination level of the scene when compared to their non-recursive counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' This robustness is strongly observed when images include pixel disturbances or undesired changes in illumination, which have a negative impact on the non-recursive models but can be successfully overcome for most of the dates by the proposed recursive framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Also, it is important to ensure the adaptability of the recursive methods so that they are able to identify natural phenomena and incorporate smooth changes in the image time-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' As limitations of this work, we highlight the lack of ground truth multitemporal classification maps, the non-exploitation of spatial correlations between pixels, and our choice for constant transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' While the lack of ground truth is hard to circumvent and makes a quantitative assessment of the results challenging, we explicitly assumed the labels from different pixels to be independent and the transition probabilities to be time-invariant in order to simplify the method and make it easy to tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Moreover, this also leads to a methodology that has a low associated computational cost and can be used over large geographical areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Finally, we believe that the proposed framework might have a broad impact on the RS community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' On the one hand, the simplicity of the proposed algorithm and its ability to incorporate models ranging from spectral index-based to pre-trained deep learning classifiers, and the improved results demonstrated in the experiments shows its potential to reliably extract crucial information from a large amount of multitemporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' On the other hand, there is still a need for unsupervised methods that can account for, e,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', seasonal effects, which provides directions to improve the proposed methodology even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Conclusion In this paper, we introduced a recursive Bayesian classification framework for satellite image time- series that provides high accuracy while requiring low computational cost and minimal supervision, which differs from existing recursive approaches that typically require large amounts of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The proposed framework allows the conversion of static classifiers or generative models into recursive Bayesian classifiers, making them more robust to non-informative image variations in multispectral image sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' A water mapping and a land cover experiment have been conducted analyzing Sentinel-2 satellite data of two areas in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The performance of three different static classification algorithms, including GMM, LR and SIC, as well as two deep learning-based methods has been compared to their recursive counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The SIC is a novel algorithm proposed in this paper that uses spectral index values to generate predictive probability of occurrence of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', land and water pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' In particular, the MNDWI and NDVI are the spectral indices used as benchmark methods in the water mapping and land cover experiments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Results suggest that the proposed framework increases the robustness of existing static classification algorithms in a multitemporal setting, also when being applied to pre-trained state-of-the-art deep learning-based models without requiring additional training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 18 Significant improvements were observed both for the water mapping and for the land cover classification results when compared to the non-recursive implementations of the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Future work will investigate the incorporation of spatial information into the recursive Bayesian classification strategy to improve the robustness against non-informative spectral variations, as well as a method to automatically determine the class transition probabilities and their prior probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Declaration of Competing Interest The authors of this paper declare that they have no known personal relationships or competing financial interests that could have appeared to influence this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Dataset and supplementary results A Python implementation of the proposed algorithms can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='com/neu-spiral/ RBC-SatImg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' The dataset is also available at (Calatrava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Supplementary material contain- ing additional experimental results are also available with this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Acknowledgements This work has been partially supported by the National Geographic Society under grant NGS-86713T- 21, the National Science Foundation under Award ECCS-1845833, and the National Aeronautics and Space Administration under Award 80NSSC20K0742.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1007/s11633-018-1143-x, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='1007/s11633-018-1143-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', Cai, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', Tian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=', Williams, T.' metadata={'source': 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adaptive reflectance fusion model for complex heterogeneous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' Remote Sensing of Environment 114, 2610–2623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 23 Supplementary Results 2020-10-26 SIC GMM LR DWM WN RSIC RGMM RLR RDWM RWN RGB 2020-11-05 2020-11-10 2020-11-20 2020-11-25 2020-11-30 2020-12-05 2020-12-10 2020-12-30 2021-01-09 2021-01-14 2021-01-19 2021-02-23 Figure 9: Supplemental results from study area A including all dates (from 2020-10-26 to 2021-02-23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='2021-02-28 2021-03-05 2021-03-30 2021-04-09 2021-04-19 2021-05-04 2021-05-14 2021-05-19 2021-05-29 2021-06-03 2021-06-08 2021-06-13 2021-06-18 2021-06-23 Figure 10: Supplemental results from study area A including all dates (from 2021-02-28 to 2021-06-23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 25 2021-06-28 2021-07-08 2021-07-13 2021-07-18 2021-07-23 2021-08-02 2021-08-22 2021-08-27 2021-09-01 2021-09-06 2021-09-11 2021-09-16 2021-09-21 2021-09-26 Figure 11: Supplemental results from study area A including all dates (from 2021-06-28 to 2021-09-26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 26 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content='.2020-10-26 SIC GMM LR DWM WN RSIC RGMM RLR RDWM RWN RGB 2020-11-05 2020-11-10 2020-11-20 2020-11-25 2020-11-30 2020-12-05 2020-12-10 2020-12-30 2021-01-09 2021-01-14 2021-01-19 2021-02-23 Figure 12: Supplemental results from study area B including all dates (from 2020-10-26 to 2021-02-23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 27 2021-02-28 2021-03-05 2021-03-30 2021-04-09 2021-04-19 2021-05-04 2021-05-14 2021-05-19 2021-05-29 2021-06-03 2021-06-08 2021-06-13 2021-06-18 2021-06-23 Figure 13: Supplemental results from study area B including all dates (from 2021-02-28 to 2021-06-23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 28 2021-06-28 2021-07-08 2021-07-13 2021-07-18 2021-07-23 2021-08-02 2021-08-22 2021-08-27 2021-09-01 2021-09-06 2021-09-11 2021-09-16 2021-09-21 2021-09-26 Figure 14: Supplemental results from study area B including all dates (from 2021-06-28 to 2021-09-26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 29 2020-11-08 SIC GMM LR RSIC RGMM RLR RGB 2020-11-10 2020-11-18 2020-12-03 2020-12-13 2020-12-15 2021-01-24 2021-03-13 2021-03-20 2021-03-23 2021-03-30 2021-04-04 2021-04-24 2021-05-14 2021-05-17 2021-05-19 Figure 15: Supplemental results from study area C including all dates (from 2021-11-08 to 2021-05-19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 30 2021-05-27 2021-06-06 2021-06-18 2021-07-23 2021-07-31 2021-08-25 2021-08-27 2021-09-11 2021-09-14 2021-09-19 2021-09-26 Figure 16: Supplemental results from study area C including all dates (from 2021-05-27 to 2021-09-26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQf1P5q/content/2301.01796v1.pdf'} +page_content=' 31' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..3ceb0dced4047a31058f0ac039c920df8523c267 --- /dev/null +++ b/Y9E2T4oBgHgl3EQfZAfl/content/tmp_files/2301.03861v1.pdf.txt @@ -0,0 +1,528 @@ +Axisymmetric gravity-capillary standing waves on the surface of a fluid +Jules Fillette,1, 2 St´ephan Fauve,1 and Eric Falcon2, ∗ +1Laboratoire de Physique de l’´Ecole Normale Sup´erieure, CNRS, PSL Research University, +Sorbonne Universit´e, Universit´e Paris Cit´e, F-75005 Paris, France +2Universit´e Paris Cit´e, CNRS, MSC, UMR 7057, F-75013 Paris, France +We report on the experimental study of axisymmetric gravity-capillary standing waves generated +by a vertically vibrating ring partially immersed into a fluid. Different regimes of standing waves are +highlighted at the basin center depending on the forcing parameters: linear, nonlinear and ejection +regimes. +For weak forcing, the standing waves display a resonant response, close to a natural +frequency of the circular basin, predicted by the linear theory. For stronger forcing, we observed +that the experimental spatial profile of standing waves breaks the up-down symmetry, and is well +described by a third-order nonlinear theory. When the forcing is further increased, the maximum +height of the axisymmetric wave crest at the basin center is found to increase linearly with its +wavelength, due to the saturation of its steepness, a result well captured by a proposed model. +I. +INTRODUCTION +One of the most common wave observations in everyday life is the propagation of concentric waves after a stone +has disturbed the interface between water and air [1]. Looking at this pattern, one could wonder what happens when +the waves converge instead of diverging. This phenomenon, characterized by the concentration of a finite amount of +energy in an infinitely small area, is called wave focusing. Wave focusing has been studied in optics since the 19th +century, in the neighborhood of a caustic [2], or a Huygens cusp of light [3]. In particular, diffraction theory states +that (in a homogeneous medium with no source) the diffraction limit, i.e., the shortest spatial wave-field fluctuations, +is precisely one-half wavelength, λ/2 [4] and focusing is known to shift the phase of the wave [5]. In acoustics, wave +focusing has been used to develop tools for trapping or tweezers [6, 7], whereas time-reversal techniques overcome the +diffraction limit and reduce the size of the focal spot as narrow as λ/14 [8]. Although hydrodynamic systems have +several advantages compared to optics or acoustics (macroscopic, slow dynamics, and direct space-and-time resolved +wave-field measurement), hydrodynamic focusing has not been studied in detail except for spatial focusing with a +parabolic shaped wave maker [9], wave control by time reversal and holography methods [10] or three-dimensional +wave breakings [11]. Nevertheless, directional focusing has also been suggested as a candidate for the formation of +rogue waves [12] and amplification of tsunamis [13] in the ocean. +Axisymmetric surface waves have been routinely studied in the past. Indeed, the behavior of standing waves in a +circular basin is of primary interest, in particular to the study of sloshing in cylindrical tanks or harbor oscillations +[14, 15]. Experiments in large-scale basins were also reported in which converging axisymmetric gravity waves are +generated by several wavemakers, driven in unison, surrounding the tank [16–18]. These studies mainly focus on +the transient phenomenon of jetting occurring at the center of the tank. Such hyperbolic-shape jet eruptions on +a fluid surface have also been investigated, within more feasible setups, either by drop or projectile impact [19– +22], bubble bursting at a free surface [23] or parametric forcing (Faraday instability) of cylindrical containers [24, +25]. These observations are usually compared to gravity-wave profiles from linear [26] or nonlinear [27–29] theories. +Recently, numerical simulations investigated the decay of axisymmetric gravity-capillary waves initially generated +by a zeroth-order Bessel-function deformation [30, 31]. All these experimental studies mainly concern the transient +regimes, and axisymmetric gravity-capillary standing waves have been much less experimentally investigated. +Here, we propose an original model experiment to study axisymmetric gravity-capillary standing waves generated +by a vertically vibrating ring on a fluid surface. Under weak sinusoidal forcing, the spatial pattern of the waves is +found to agree with linear predictions [26]. For high enough forcing, the up-and-down symmetry of the spatial profile is +broken as predicted by a nonlinear theory [27, 29], and a divergence of the wave amplitude occurs at the basin center, +sometimes with the ejection of a drop. We show in particular that the maximum height reached by the axisymmetric +gravity-wave crest, at the basin center, increases linearly with its wavelength, due to the saturation of its steepness to +5/(2π). To the best of our knowledge, this saturation has not been previously reported. It should not be confused with +the stability limit of a periodic sharp-crested wave derived for unidimensional [32–34] or axisymmetric [27] standing +waves of finite amplitude, and tested experimentally [18, 24, 35, 36]. +∗ E-mail: eric.falcon@u-paris.fr (corresponding author) +arXiv:2301.03861v1 [physics.flu-dyn] 10 Jan 2023 + +2 +FIG. 1. Experimental setup of axisymmetric gravity-capillary wave focusing. The surface elevation η(r, t) is measured using a +capacitive probe mounted on a translation stage, and a side camera. +II. +EXPERIMENTAL SETUP +The experiment consists of a cubic transparent tank (L = 19 cm wide) filled with distilled water (density ρ = +1000 kg m−3) up to a depth h = 7 cm (see Fig. 1). We add surfactants to fix its surface tension γ to a constant value +of 37 mN m−1 by using Trimethyl(tetradecyl)ammonium bromide at a concentration higher than the critical micelle +concentration [37, 38]. Axisymmetric convergent waves are generated by the vertical oscillations of a solid ring made +of plexiglass (internal radius R = 8.25 cm, vertical thickness 2 cm) half immersed into the fluid at rest. The ring is +mechanically connected to an electromagnetic shaker (Dynamic Solution VTS-100) driven by a sinusoidal voltage from +a power supply (Kepco 36V/6A) leading to a vertical ring motion z(t) = a sin(2πft)/2 where f and a are the forcing +frequency and amplitude, respectively (f ∈ [5, 9.3] Hz, i.e., λ ∈ [2.3, 6.5] cm and a ∈ [0, 1] cm). A point of the free +surface is referred to by its polar coordinates (O, r, θ). Nevertheless, θ is not considered hereafter as the phenomenon +is mainly axisymmetric. Indeed, special attention is paid to adjusting the horizontality of the ring, and limiting the +transverse vibrations, that break this symmetry. After a few forcing periods, the transient regime vanishes. The +surface elevation η(r, t) of the stationary wave field is then measured at a single location over time t, thanks to a +home-made capacitive probe (10 µm in vertical resolution) [39]. We iterate this temporal measurement for every r +along the ring diameter (with a 1 mm step) using a translation stage with a stepper motor driven by a computer. +Moving the probe along a diameter, therefore, gives access to the wave profile resolved in time and the wave envelope +resolved both in space and time. The corresponding vertical resolution is 100 µm. The nonlinear parameter, namely +the wave steepness, is ε ≡ ηmax/λ where λ is the wavelength and ηmax is the maximum elevation at the container +center r = 0. ε is varied by almost two decades in the range ε ∈ [0.01, 1]. +FIG. 2. Typical regimes of axisymmetric standing waves for increasing forcing amplitudes (side views). (a) Linear regime +(ε ≃ 0.07, a = 0.04 cm). (b) Nonlinear regime (ε ≃ 0.2, a = 0.18 cm). (c) Ejection regime (ε ≃ 0.7, a = 0.36 cm). The white +bar corresponds to 1 cm. Sinusoidal forcing f = 6.75 Hz. Part of the transparent solid ring is visible at the back. + +3 +III. +PATTERNS +Different typical axisymmetric patterns of the free surface are observed depending on the control parameters a and +f. We show in Fig. 2 the qualitative influence of increasing the forcing amplitude a (from left to right), for a fixed +forcing frequency f (see also movies in Supplemental Material [40]). At low a (ε ≃ 0.07), standing axisymmetric +oscillations are gentle in particular near the center (see Fig. 2a). We call below this regime the linear regime. For +high enough a (ε > 0.1), nonlinearities arise and the up-and-down central deformation is more prominent and much +higher than the periphery ones (see Fig. 2b). This regime is called afterwards the nonlinear regime. When the forcing +amplitude a is further increased (ε ≳ 0.7), we observe an ejection regime characterized by the formation of a thin and +intense jet at the center, with the ejection of at least one droplet. +5 +6 +7 +8 +9 +10 +0 +0.2 +0.4 +0.6 +0.8 +10-2 +10-1 +100 +FIG. 3. Wave steepness, ε = ηmax/λ, of the central deformation as a function of the forcing amplitude a and frequency f. +Logscale colorbar. Vertical lines: circular basin eigenfrequencies fn from J′ +0(knR) = 0 (see text) where fn and kn are related +by Eq. (1). Solid line corresponds to the same value of ε ≃ 0.1 as a function of f. (■): maximum amplitude of the central +deformation before ejection as a function of f. +IV. +PHASE DIAGRAM +We now explore in more detail the phase diagram of the three regimes found in §III as a function of the control +parameters. We report in Fig. 3 the measured values of the wave steepness ε for each accessible pair (f, a) of the +forcing parameters. Following a virtually vertical line (from bottom to top) in Fig. 3, ε is found to increase with a +from very weak values (≃ 10−2 – in blue) to values close to unity (in red). The ejection regime occurs when crossing +the black-dotted line. The influence of the forcing frequency is highlighted by following the black solid line, which +links data with the same value of ε (namely ε ≃ 0.1). The curve minima point out frequencies for which the central +deformation reaches this specific steepness although the forcing amplitude is weak, and thus correspond to resonances +of the system. Moreover, these resonance frequencies appear to be in good agreement with the main theoretical +eigenmodes of the ring marked by vertical dash-dotted lines. Indeed, the axisymmetric eigenmodes of a circular basin +are obtained by considering an inviscid, irrotational, and incompressible fluid whose velocity potential ϕ satisfies the +Laplace equation ∆ϕ = 0. The solution implies the Bessel function of the first kind of order α, Jα(kr), where k is the +wave vector k = 2π/λ [41]. Moreover, imposing that the fluid cannot penetrate the solid boundary at r = R over t +leads, in the linear approximation, to J′ +0(x)|knR = 0, where the prime stands for the spatial derivative of J0(x) [26]. +This quantifies the modes of the system to discrete wave vectors kn. Using the linear dispersion relation of inviscid +deep-water gravity-capillary waves [26] (as kh > 8 for λ < 5 cm), +ω = +� +gk + γ +ρk3 , +(1) + +4 +the corresponding axisymmetric eigenfrequencies fn read 5.70, 6.64, 7.55, 8.47, and 9.39 Hz for n = 3, 4, 5, 6, and +7, respectively (g = 9.81 m s−2 is the acceleration of gravity). Note that the linearized kinematic condition at the +interface z = η(r, t) leads to ∂ϕ/∂z = ∂η/∂t, where ϕ is the velocity potential, and implies that ϕ and η have the +same dependence on r. Moreover, experiments show that the fundamental angular frequency ω of waves coincides +with the forcing pulsation, 2πf, leading thus to the same notation. Note that the vertical thickness of the ring is +finite and the no-penetration condition is not fully verified below the ring (see §V). However, we have verified that +the initial immersed depth of the ring into the water does not impact qualitatively the wave shape (see Appendix A), +but could explain slight departures between the resonances and eigenfrequencies in Fig. 3. We have also verified, +using a ring of smaller diameter, that the resonances correspond to the eigenfrequencies, although the latter change +(see Appendix B). The ejection threshold is hardly visible in Fig. 3 outside the range f ∈ [5, 9] Hz mainly because, +for smaller f, the wavelength becomes comparable to the system size [e.g., λ|5 Hz = 6.5 cm from Eq. (1)], so that the +standing wave pattern is modified by finite size effects of the container. For f > 9 Hz, the ejection regime cannot be +reached due to the mechanical limitations of the wavemaker. +-R +-60 +-40 +-20 +0 +20 +40 +60 +R +-1 +-0.5 +0 +0.5 +1 +1.5 +-R +-60 +-40 +-20 +0 +20 +40 +60 +R +-1 +-0.5 +0 +0.5 +1 +1.5 +0 +0.05 +0.1 +0 +1 +2 +3 +FIG. 4. Dimensionless wave envelope along a basin diameter. All curves have been rescaled by the asymmetry coefficient +N = (ηmax − ηmin)/2. Red dotted line: experimental wave envelope showing both ηmax(r) > 0 and ηmin(r) < 0 (f = 6.9 Hz, +a = 0.38 cm, ε = 0.36). (a) Blue solid line: Linear prediction from [26]. (b) Black solid line: Third-order nonlinear prediction +computed numerically from [27]. Inset: N vs. the nonlinear parameter ε for a fixed f = 6.9 Hz. Red dots correspond to data, +black dotted line to the linear prediction, and black solid line to the nonlinear theory [27]. +V. +STATIONARY SPATIAL PROFILE +We denote ηmax(r) the maximum of the wave elevation, η(r, t), over time t at position r, and the maximum central +elevation ηmax ≡ ηmax(0). In the same way, we define the quantities ηmin(r) and ηmin ≡ ηmin(0) for the minimum of +η(r, t). We plot in Fig. 4 the experimental wave envelope [i.e., ηmax(r) and ηmin(r)] rescaled by N = (ηmax − ηmin)/2 +as a function of r along a diameter (see red-dotted line in Fig. 4a-b). We superimposed in Fig. 4a the prediction from +the linear theory (blue solid line) when imposing two boundary conditions: (i) ∂η/∂r |r=0 = 0 to ensure continuity +at the basin center r = 0, and (ii) η(±R, t) = b cos(2πft)/2 as the fluid must follow the ring oscillations at r = ±R, +and the fluid elevation b may differ from the ring amplitude a. This leads to the envelope equation η(r) = ηmaxJ0(kr) +where J0 is the Bessel function of the first kind and k is computed from Eq. (1) for fixed f, whereas ηmax is not +needed thanks to the rescaling by N. We first notice that the linear theoretical profile in Fig. 4a does not show a zero +slope at r = ±R indicating that the no penetrability condition used in Sect. IV is indeed debatable. Moreover, several +differences are visible between the experiment and the linear model. First, the amplitude of the central deformation +is measured to be asymmetric which is not captured by the linear theory. Second, the shift of the zeros suggests +that the dispersion relation does not hold as is for nonlinear waves. Third, the linear prediction shows local minima + +5 +with strictly zero vibration whereas the experiment shows nonzero minima of the envelope. These differences are +significantly reduced when using a third-order nonlinear theory of axisymmetric gravity standing waves [27]. Indeed, +Fig. 4b shows that the experimental and theoretical local minima occur at the same positions evidencing the relevance +of using this nonlinear theory. The latter also confirms that the wave elevation at these nodes does not have to go to +zero (although getting closer than in the experiment), in particular near the basin center. Indeed, the nonlinear theory +predicts a slight horizontal oscillation of the locations of the zeros over a period so that the surface never goes flat and +the water level in any location r is nonzero at least for a fraction of time. Moreover, close to the focus, i.e., r → 0, +the up-and-down asymmetry (which is a classical signature of the nonlinearity [32]) is well fitted by the nonlinear +theory [27] (see black solid line). More precisely, the inset of Fig. 4b shows the asymmetry, N = (ηmax − ηmin)/2, +as a function of ε = ηmax/λ. From the linear theory, one should have ηmax = −ηmin leading to N = ηmax = ελ as +displayed by the dotted line in the inset of Fig. 4b. The nonlinear theory computed numerically from Ref. [27] is also +shown (solid line) and is found to be in good agreement with the experiments. Note that the departure between linear +and nonlinear theories reaches 10% for ε = 0.064 which is close to the arbitrary criterion ε = 0.1 used in Sect. III to +distinguish the linear and nonlinear regimes. Finally, Fig. 4 shows that the experimental profile near r = ±R does +not satisfy the condition ∂η/∂r |r=±R = 0 as imposed to the system when computing its eigenmodes in §IV. This +effect is experimentally confirmed for other forcing frequencies and could contribute to small departures between the +resonances and eigenfrequencies observed on the phase diagram in Fig. 3. +VI. +CENTRAL DEFORMATION AMPLITUDE +The maximum elevation at the center (r = 0) is now investigated. We measure it either by the capacitive probe for +weak and moderate forcing amplitude, a, or by using a side camera (Basler 2048 × 1536 px2, 120 fps) for higher a. +Error bars are the statistical average of data from a few similar jets, as the jet eruption often deviates from the +vertical. Figure 5 then shows the maximum height, ηmax, reached by the fluid at the center rescaled by λ as a function +of the dimensionless forcing acceleration, aω2/g, when varying the forcing amplitude a. The three different curves +correspond to three different forcing frequencies f ≡ ω/(2π). In the linear and weakly nonlinear regimes (ε < 0.3), +ηmax/λ grows linearly with aω2/g regardless of f as confirmed by the inset of Fig. 5. In the ejection regime (ε ≳ 0.5), +the rescaled maximal height of the jet (not taking into account possible drop ejection) increases strongly with aω2/g, +then is found to saturate to a value denoted by εsat = ηsat/λ, roughly independent of the acceleration, but depending +on f. A model described below will explain this saturation. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +10-2 +10-1 +100 +10-2 +10-1 +100 +FIG. 5. +Rescaled central maximum height, ηmax/λ, as a function of the forcing acceleration, aω2/g (i.e., various forcing +amplitudes), for three different forcing frequencies f = (♦) 5.70, (□) 6.64, and (◦) 7.55 Hz. +Horizontal dot-dashed lines +correspond to εmax = 0.3 and 5/(2π). Inset: Same in log-log scales. Solid line has a unit slope. Best fit has a 1.13 slope. + +6 +VII. +SATURATION MODEL +To explain the vertical saturation of the jet steepness, we approximate the jet surface at any time t, by a cone +of height η(t) and radius λ/4 as shown in the top-right hand Fig. 6 (λ/2 is the natural diameter of the central +deformation). First, we estimate the dominant forces of the problem. Given ρ, γ, g, µ = 10−3 Pa s the dynamic +viscosity of water, L the typical size of the cone, and v its typical vertical velocity, we compute the Weber (We), +Reynolds (Re) and Bond (Bo) numbers. +Taking L = ηmax ≈ 2 cm and v = 2πfηmax ≈ 0.8 m s−1, one obtains +We = ρv2L/γ ≈ 350, Re = ρvL/µ ≈ 1.6 × 104, and Bo = ρgL2/γ ≈ 100. The Weber number is defined as the +ratio between the inertial and surface tension forces and shows that inertia dominates surface tension effects. The +Reynolds number (comparing inertia forces to viscous ones) shows that viscosity can be neglected here. The Bond +number shows that gravitational forces are two orders of magnitude larger than surface tension ones. Since We, Re, +and Bo ≫ 1, viscous and surface tension forces can be neglected. The following model thus takes into account only +inertial and gravitational forces through kinetic and potential energies. We now consider the energy balance between +a final state where the jet of height ηmax has a finite conical volume V , a positive potential energy and is motionless, +and an initial state where the surface is flat and the same volume of the fluid is enclosed within a downwards cone of +same dimensions located under the free surface (see bottom-right hand Fig. 6). Leaving out the t notation for clarity, +the cone edge equation thus reads +z(r) = ±ηmax ∓ 4ηmax +λ +r , +i.e., +r(z) = (ηmax ∓ z) +λ +4ηmax +, +(2) +where the upper (resp. lower) signs describe the top (resp. bottom) cone and z is the vertical coordinate. The +volume of such a cone reads V = πλ2ηmax +� +48. The fluid velocity in the bottom cone is unknown, but we keep only +its vertical component u(z) and approximate it by a linear dependence on z between u(0) = 0 and u(−ηmax) = U +(upward velocity at its lowest depth), as proposed and proven sufficient in Ref. [21]. This yields +u(z) = − +z +ηmax +U , +for z ∈ [−ηmax, 0] , +(3) +Then, we can express in a general way the kinetic energy Ek = +� 0 +−ηmax ρπr(z)2u(z)2dz/2 and the potential energy +Ep = +� 0 +−ηmax ρπr(z)2gzdz of the bottom cone. Substituting Eq. (2) and (3) in these expressions, we end up with +Ek = ρV U 2� +20 and Ep = − ρgηmaxV /4. For the final state (top cone), Ek = 0 as U = 0 and Ep = + ρgηmaxV /4. +Neglecting as justified above, viscous dissipation and surface tension, the conservation of energy between the final +state and the initial state yields ηmax = 10g +� +U 2 . To find the dependence of ηmax on the wavelength λ, we assume +that the dispersion relation for linear waves of Eq. (1) makes a first approximation even for these nonlinear and +nonsinusoidal deformations. Taking U of order αωηmax with α a fitting parameter, this yields +ηmax = +5 +α2π +λ +[1 + (λc/λ)2] , +(4) +where λc/(2π) ≡ +� +γ/ρg = 1.9 mm is the capillary length separating the capillary (λ ≪ λc) and gravity (λ ≫ λc) +wave regimes, the gravity-capillary regime occurring in between. Note that, in the capillary-wave regime, the surface +tension cannot be neglected anymore and a different model must be applied when Bo ≲ 1, i.e., λ ≲ 2 cm. We plot +ηmax(λ) from Eq. (4) in Fig. 6 together with the experimental maximum heights reached at the basin center for our +gravity-capillary range (λ ∈ [2.5, 4.5] cm). As in §VI, error bars come from the statistical average of data from a few +similar jets. Figure 6 then shows that our model well captures the experimental saturation heights using α = +√ +2. α > 1 +means that the linear approximation U = ωηmax underestimates the cone velocity at saturation and that nonlinear +effects tend to increase it. However, the saturation of the jet height indicates that beyond a certain forcing amplitude +the energy injected in the system does not contribute to the central jet velocity, which should be directly converted +into potential energy, but is mainly dissipated by the meniscus at the ring boundaries and/or transferred within the +fluid bulk in the form of flows. Equation (4) with α = +√ +2 then tends asymptotically in the pure gravity regime to +ηg +sat = 5λ/(2π), corresponding thus to a saturation of the wave steepness towards εg +sat ≡ ηg +sat/λ = 5/(2π). The inset +indeed shows that this predicted saturation of the wave steepness occurs experimentally for large enough wavelengths. +Beyond the good agreement, the comparison is experimentally limited by viscous dissipation and mechanical limitation +of the shaker for smaller wavelengths, and by the system size for larger wavelengths. +Finally, note that for existing large-scale axisymmetric tanks (from 1.6 m to 25 m diameters) surrounded by several +wavemakers, driven in unison [16–18], Eq. (4) would give an upper bound of the height of the central wave crest, which +would be either larger than the size of the tank building, or not reachable due to wavemaker limitations. Furthermore, + +7 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +7 +2 +4 +6 +0 +0.5 +1 +FIG. 6. Saturation wave height at the basin center, ηsat, as a function of the rescaled wavelength, λ/λc . (◦) experimental data +for various forcing frequencies f ∈ [5.5, 8.5] Hz. (Blue solid line) prediction from the model of Eq. (4) with α = +√ +2. (Blue +dashed line) asymptotic trend of Eq. (4) with α = +√ +2, i.e., ηg +sat = 5/(2π)λ, valid in the pure gravity regime (λ ≫ λc). Inset: +Corresponding wave steepness at the basin center, εsat ≡ ηsat/λ, vs. the dimensionless wavelength, λ/λc . For large enough +λ, the saturation steepness εsat tends to 5/(2π). Same curves as in the main figure. Right: Schemes and quantities used in +the model: (top) crude jet-shape approximation as a cone in its final state, (bottom) initial state with a flat surface and an +upwards underwater-conical flow. +Eq. (4) should not be confused with another limit commonly discussed in the literature concerning the stability of a +periodic sharp-crested wave. Indeed, for unidimensional progressive gravity waves of finite amplitude, this limiting +angle β of the crest was first derived by Stokes to be 120° assuming a steady crest profile [32], and to be 90° for +the standing wave case [33], the latter value being confirmed experimentally [35]. For axisymmetric standing gravity +waves of finite amplitude, a limiting angle of 109.47° was derived analytically [27] and tested experimentally [18, 36], +corresponding thus to a rescaled maximum height ηmax/λ = 1/(4 tan β/2) = 0.25. As expected, this value is much +smaller than 0.8 as it corresponds to a stability limit (and not to a maximum height). +VIII. +CONCLUSION +We reported on the experimental study of the focusing of axisymmetric gravity-capillary waves generated by a +vertically vibrating ring partially immersed in a fluid. Different regimes of standing waves are observed at the basin +center depending on the forcing parameters: linear, nonlinear, and ejection regimes. For weak forcing, and close to +a natural frequency of the circular basin predicted by the linear theory [26], the standing waves display a resonant +response. For stronger forcing, we observed that the spatial profile of standing waves breaks the up-down symmetry, +and exhibits nonzero local minima, which are both well taken into account by a nonlinear theory of axisymmetric +standing waves up to third order in amplitude [27]. Finally, for an even stronger forcing, we observed a jet together +with possible drop ejections. The maximum elevation reached experimentally by the wave at the center of the basin +is found to saturate at ηsat, even for stronger forcing amplitudes. For gravity waves, ηsat increases linearly with the +wavelength, due to the saturation of its steepness to 5/(2π). This maximum wave height is well captured using a +crude model, based on an energy balance with strong hypotheses concerning the forces and the shape of the jet. This +is a first step towards a more elaborated one. In the future, we will address the origin of the jet. Does it arise out +of a deep depression of the free surface leading to the collapse of this cavity coupled to a singularity, or/and the +collapse of a bubble entrapped underneath [20, 25]? The dynamical properties of the focusing will be also investigated +by tracking the propagation of axisymmetric gravity-capillary propagating waves converging towards the center to +explore open questions, such as which mechanisms drive their central interaction, and how the power injected by the +ring is dissipated at the central singularity? + +8 +ACKNOWLEDGMENTS +We thank G. Michel and M. Roch´e for fruitful discussions. We thank A. Di Palma, and Y. Le Goas for technical help. +This work is supported by the French National Research Agency (ANR DYSTURB project No. ANR-17-CE30-0004, +ANR SOGOOD project No. ANR-21-CE30-0061-04), and by the Simons Foundation MPS No. 651463-Wave Turbu- +lence (USA). +Appendix A: Role of the immersed depth of the ring +We quantify the influence of the ring immersion into the fluid on the wave properties. We measure the maximum +water elevation, ηmax, at the center, for different relative positions d of the ring to the water surface at rest, all other +things being equal (especially the forcing parameters f and a). The experimental data are displayed in Fig. 7 and +show that ηmax depends on the fluid volume moved by the ring. Indeed, ηmax is maximum when the ring is fully +immersed into the fluid and flushing to the surface (d = 20 mm). The water volume moved by the ring oscillation, +when the ring is closer to the free surface, is thus combined with the meniscus movement (same for d ≈ 0 mm). On +the other hand, when the ring is partially immersed (d ≈ 10 mm), ηmax reaches a plateau where only the meniscus +effect plays a role in the wave forcing. ηmax decreases as expected when the ring plunges deeper and deeper (d larger +than the ring thickness 20 mm) and the two forcing effects disappear. Beyond this dependence of the wave amplitude +on d, we have furthermore verified that the initial immersed depth of the ring does not impact other wave properties. +0 +10 +20 +30 +60 +80 +100 +120 +FIG. 7. (Top) Sectional schemes defining d as the ring distance to the water surface at rest (horizontal blue line). The ring +thickness (gray area) is 20 mm. (Bottom) Maximum water elevation, ηmax, measured at the center when increasing d for fixed +sinusoidal forcing parameters (f = 6.9 Hz and a = 0.13 cm), and plotted as a percentage of the initial value. +Appendix B: Experiments with a different ring size +We perform the same experiments as in the main text, but with a ring of different radius. We now use a ring radius +of R = 5.1 cm instead of 8.25 cm as in the main text. Figure 8 shows the corresponding phase diagram as a function of +the control parameters. The resonance frequencies with this smaller ring are f = 4.37, 6.07, 7.59, and 9.11 Hz, which +differ from those with the larger ring in Fig. 3 (i.e., 5.7, 6.66, 7.59, 8.53, 9.48 Hz), but correspond to the circular basin +eigenmodes. It thus confirms that the system resonance frequencies are indeed given by the ring eigenfrequencies. +[1] B. Le M´ehaut´e, Gravity–capillary rings generated by water drops, J. Fluid Mech. 197, 415 (1988) +[2] G. B. 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Dias, Numerical modeling of extreme rogue waves generated by directional energy focusing, +Wave motion 44, 395 (2007). +[13] M. V. Berry, Focused tsunami waves, Proc. R. Soc. A 463, 3055 (2007). +[14] R. A. Ibrahim, Liquid Sloshing Dynamics (Cambridge University Press, New York, 2005). +[15] G. Michel, F. P´etr´elis, and S. Fauve, Observation of nonlinear sloshing induced by wetting dynamics, Phys. Rev. Fluids +2, 022801(R) (2017). +[16] M. Minoura, S. Naito, T. Muto, and O. Etsuro, Generation of arbitrary wave field in arbitrarily configured wave basin +composed of element-absorbing wavemakers, Int. J. Offshore Polar Eng. 21 (2011). +[17] K. Maeda, N. Hosotani, K. Tamura and H. Ando, Wave making properties of circular basin, Proceedings of the 2004 Int. +Symp. on Underwater Tech. (IEEE Cat. No.04EX869), Taipei, Taiwan (IEEE, Piscataway, NJ, 2004), pp. 349-354. +[18] M. L. McAllister, S. Draycott, T. Davey, Y. Yang, T. A. A. Adcock, S. Liao, and T. S. van den Bremer, Wave breaking +and jet formation on axisymmetric surface gravity waves, J. Fluid Mech. 935, A5-1 (2022). +[19] A. M. Worthington, On impact with a liquid surface, Proc. R. Soc. London 34, 217 (1883); D. Bartolo, C. Josserand, and +D. Bonn, Singular Jets and Bubbles in Drop Impact, Phys. Rev. Lett. 96, 124501 (2006); J. M. Gordillo and S. Gekle, +Generation and breakup of Worthington jets after cavity collapse. Part 2. Tip breakup of stretched jets, J. Fluid Mech. +663, 331 (2010); C. J. M. van Rijn, J. Westerweel, B. van Brummen, A. Antkowiak, and D. Bonn, Self-similar jet evolution +after drop impact on a liquid surface, Phys. Rev. Fluids 6, 034801 (2021). +[20] G.-J. Michon, C. Josserand, and T. S´eon, Jet dynamics post drop impact on a deep pool, Phys. Rev. Fluids 2, 023601 +(2017). +[21] Z. Che and O. K. Matar, Impact of droplets on immiscible liquid films, Soft Matter 14, 1540 (2018). +[22] T. T. Truscott, B. P. Epps, and J. Belden, Water entry of projectiles, Annu. Rev. Fluid Mech. 46, 355 (2014). +[23] E. Ghabache, A. Antkowiak, C. Josserand, and T. S´eon, On the Physics of fizziness: How bubble bursting controls droplets +ejection, Phys. of Fluids 26, 121701 (2014). +[24] M. S. Longuet-Higgins, Bubbles, breaking waves and hyperbolic jets at a free surface, J. Fluid Mech. 127 103 (1983). + +linear +nonlinear +0.610 +[25] B. W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, Singularity dynamics in curvature collapse and jet eruption on a +fluid surface, Nature, 403, 401 (2000); J. E. Hogrefe, N. L. Peffley, C. L. Goodridge, W. T. Shi, H. G. E. Hentschel, and +D. P. Lathrop, Power-law singularities in gravity-capillary waves, Physica D (Amsterdam) 123, 183 (1998); D. K. Raja, +S. P. Das, and E. J. Hopfinger, On standing gravity wave-depression cavity collapse and jetting, J. Fluid Mech. 866, 112 +(2019). +[26] H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, 1932). +[27] L. R. Mack, Periodic, Finite-Amplitude, Axisymmetric Gravity Waves, J. Geophys. Research 67, 829 (1962). +[28] J. F. Dalzell, A note on finite depth second-order wave-wave interactions, Appl. Ocean Res. 21, 105 (1999). +[29] W.-T. Tsai and D. K. P. Yue, Numerical calculation of nonlinear axisymmetric standing waves in a circular basin, Phys. +Fluids 30, 3441 (1987). +[30] P. K. Farsoiya, Y. S. Mayya, and R. Dasgupta, Axisymmetric viscous interfacial oscillations – theory and simulations, J. +Fluid Mech. 826, 797 (2017). +[31] S. Basak, P. K. Farsoiya, and R. Dasgupta, Jetting in finite-amplitude, free, capillary-gravity waves, J. Fluid Mech. 909, +A3 (2021). +[32] G. Stokes, On the theory of the oscillatory waves, Trans. Cambridge Philos. Soc. 8, 441 (1847); J. H. Michell, The highest +waves in water, Lond. Edinb. Dublin Philos. Mag. J. Sci. 36, 430 (1893) +[33] W. G. Penney, A. T. Price, J. C. Martin, W. J. Moyce, and C. K. Thornhill, Part II. Finite Periodic Stationary Gravity +Waves in a Perfect Liquid, Philos. Trans. R. Soc. A, 244, 254 (1952). +[34] M. A. Grant, Standing Stokes waves of maximum height, J. Fluid Mech. 60, 593 (1973). +[35] G. Taylor, An experimental study of standing waves, Proc. R. Soc. A 218, 44 (1953). +[36] D. Fultz and T. S. Murty, Experiments on the frequency of finite-amplitude axisymmetric gravity waves in a circular +cylinder, J. Geophys. Res. 68, 1457 (1963), and reference therein. +[37] B. W. Barry, J. C. Morrison, and G. F. J. Russel, Prediction of the Critical Micelle Concentration of Mixtures of +Alkyltrimethylammonium Salts, J. of Coll. and Interf. Sci. 33, 554 (1970). +[38] U. More, Z. Vaid, P. Bhamoria, A. Kumar, and N. I. Malek, Effect of [Cnmim][Br] Based Ionic Liquids on the Aggregation +Behavior of Tetradecylmethylammonium Bromide in Aqueous Medium, J. Solution Chem. 44, 850 (2015). +[39] E. Falcon, C. Laroche, and S. Fauve, Observation of Gravity-Capillary Wave Turbulence, Phys. Rev. Lett. 98, 094503 +(2007). +[40] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevFluids.7.124801 for movies. +[41] Note that the Bessel function minima are not exactly periodic, in particular for the first two ones (departures of 0.8% and +0.2%, respectively), but are approximately periodic for the following ones (departures less than 0.01%). + diff --git a/Y9E2T4oBgHgl3EQfZAfl/content/tmp_files/load_file.txt b/Y9E2T4oBgHgl3EQfZAfl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75b146b2692353a4a6cd1b8fbcbcc5728056510c --- /dev/null +++ b/Y9E2T4oBgHgl3EQfZAfl/content/tmp_files/load_file.txt @@ -0,0 +1,737 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf,len=736 +page_content='Axisymmetric gravity-capillary standing waves on the surface of a fluid Jules Fillette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 2 St´ephan Fauve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1 and Eric Falcon2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' ∗ 1Laboratoire de Physique de l’´Ecole Normale Sup´erieure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' PSL Research University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Sorbonne Universit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Universit´e Paris Cit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' F-75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' France 2Universit´e Paris Cit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' MSC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' UMR 7057,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' F-75013 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' France We report on the experimental study of axisymmetric gravity-capillary standing waves generated by a vertically vibrating ring partially immersed into a fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Different regimes of standing waves are highlighted at the basin center depending on the forcing parameters: linear, nonlinear and ejection regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For weak forcing, the standing waves display a resonant response, close to a natural frequency of the circular basin, predicted by the linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For stronger forcing, we observed that the experimental spatial profile of standing waves breaks the up-down symmetry, and is well described by a third-order nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' When the forcing is further increased, the maximum height of the axisymmetric wave crest at the basin center is found to increase linearly with its wavelength, due to the saturation of its steepness, a result well captured by a proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' INTRODUCTION One of the most common wave observations in everyday life is the propagation of concentric waves after a stone has disturbed the interface between water and air [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Looking at this pattern, one could wonder what happens when the waves converge instead of diverging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' This phenomenon, characterized by the concentration of a finite amount of energy in an infinitely small area, is called wave focusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Wave focusing has been studied in optics since the 19th century, in the neighborhood of a caustic [2], or a Huygens cusp of light [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' In particular, diffraction theory states that (in a homogeneous medium with no source) the diffraction limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', the shortest spatial wave-field fluctuations, is precisely one-half wavelength, λ/2 [4] and focusing is known to shift the phase of the wave [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' In acoustics, wave focusing has been used to develop tools for trapping or tweezers [6, 7], whereas time-reversal techniques overcome the diffraction limit and reduce the size of the focal spot as narrow as λ/14 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Although hydrodynamic systems have several advantages compared to optics or acoustics (macroscopic, slow dynamics, and direct space-and-time resolved wave-field measurement), hydrodynamic focusing has not been studied in detail except for spatial focusing with a parabolic shaped wave maker [9], wave control by time reversal and holography methods [10] or three-dimensional wave breakings [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Nevertheless, directional focusing has also been suggested as a candidate for the formation of rogue waves [12] and amplification of tsunamis [13] in the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Axisymmetric surface waves have been routinely studied in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Indeed, the behavior of standing waves in a circular basin is of primary interest, in particular to the study of sloshing in cylindrical tanks or harbor oscillations [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Experiments in large-scale basins were also reported in which converging axisymmetric gravity waves are generated by several wavemakers, driven in unison, surrounding the tank [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' These studies mainly focus on the transient phenomenon of jetting occurring at the center of the tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Such hyperbolic-shape jet eruptions on a fluid surface have also been investigated, within more feasible setups, either by drop or projectile impact [19– 22], bubble bursting at a free surface [23] or parametric forcing (Faraday instability) of cylindrical containers [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' These observations are usually compared to gravity-wave profiles from linear [26] or nonlinear [27–29] theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Recently, numerical simulations investigated the decay of axisymmetric gravity-capillary waves initially generated by a zeroth-order Bessel-function deformation [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' All these experimental studies mainly concern the transient regimes, and axisymmetric gravity-capillary standing waves have been much less experimentally investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Here, we propose an original model experiment to study axisymmetric gravity-capillary standing waves generated by a vertically vibrating ring on a fluid surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Under weak sinusoidal forcing, the spatial pattern of the waves is found to agree with linear predictions [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For high enough forcing, the up-and-down symmetry of the spatial profile is broken as predicted by a nonlinear theory [27, 29], and a divergence of the wave amplitude occurs at the basin center, sometimes with the ejection of a drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We show in particular that the maximum height reached by the axisymmetric gravity-wave crest, at the basin center, increases linearly with its wavelength, due to the saturation of its steepness to 5/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' To the best of our knowledge, this saturation has not been previously reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' It should not be confused with the stability limit of a periodic sharp-crested wave derived for unidimensional [32–34] or axisymmetric [27] standing waves of finite amplitude, and tested experimentally [18, 24, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' ∗ E-mail: eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='falcon@u-paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='fr (corresponding author) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='03861v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='flu-dyn] 10 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Experimental setup of axisymmetric gravity-capillary wave focusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The surface elevation η(r, t) is measured using a capacitive probe mounted on a translation stage, and a side camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' EXPERIMENTAL SETUP The experiment consists of a cubic transparent tank (L = 19 cm wide) filled with distilled water (density ρ = 1000 kg m−3) up to a depth h = 7 cm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We add surfactants to fix its surface tension γ to a constant value of 37 mN m−1 by using Trimethyl(tetradecyl)ammonium bromide at a concentration higher than the critical micelle concentration [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Axisymmetric convergent waves are generated by the vertical oscillations of a solid ring made of plexiglass (internal radius R = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='25 cm, vertical thickness 2 cm) half immersed into the fluid at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The ring is mechanically connected to an electromagnetic shaker (Dynamic Solution VTS-100) driven by a sinusoidal voltage from a power supply (Kepco 36V/6A) leading to a vertical ring motion z(t) = a sin(2πft)/2 where f and a are the forcing frequency and amplitude, respectively (f ∈ [5, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='3] Hz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', λ ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5] cm and a ∈ [0, 1] cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' A point of the free surface is referred to by its polar coordinates (O, r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Nevertheless, θ is not considered hereafter as the phenomenon is mainly axisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Indeed, special attention is paid to adjusting the horizontality of the ring, and limiting the transverse vibrations, that break this symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' After a few forcing periods, the transient regime vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The surface elevation η(r, t) of the stationary wave field is then measured at a single location over time t, thanks to a home-made capacitive probe (10 µm in vertical resolution) [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We iterate this temporal measurement for every r along the ring diameter (with a 1 mm step) using a translation stage with a stepper motor driven by a computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Moving the probe along a diameter, therefore, gives access to the wave profile resolved in time and the wave envelope resolved both in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The corresponding vertical resolution is 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The nonlinear parameter, namely the wave steepness, is ε ≡ ηmax/λ where λ is the wavelength and ηmax is the maximum elevation at the container center r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' ε is varied by almost two decades in the range ε ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='01, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Typical regimes of axisymmetric standing waves for increasing forcing amplitudes (side views).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (a) Linear regime (ε ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='07, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='04 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (b) Nonlinear regime (ε ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='2, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='18 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (c) Ejection regime (ε ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='7, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='36 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The white bar corresponds to 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Sinusoidal forcing f = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='75 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Part of the transparent solid ring is visible at the back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' PATTERNS Different typical axisymmetric patterns of the free surface are observed depending on the control parameters a and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 2 the qualitative influence of increasing the forcing amplitude a (from left to right), for a fixed forcing frequency f (see also movies in Supplemental Material [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' At low a (ε ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='07), standing axisymmetric oscillations are gentle in particular near the center (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We call below this regime the linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For high enough a (ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1), nonlinearities arise and the up-and-down central deformation is more prominent and much higher than the periphery ones (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' This regime is called afterwards the nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' When the forcing amplitude a is further increased (ε ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='7), we observe an ejection regime characterized by the formation of a thin and intense jet at the center, with the ejection of at least one droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 5 6 7 8 9 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='8 10-2 10-1 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Wave steepness, ε = ηmax/λ, of the central deformation as a function of the forcing amplitude a and frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Logscale colorbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Vertical lines: circular basin eigenfrequencies fn from J′ 0(knR) = 0 (see text) where fn and kn are related by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Solid line corresponds to the same value of ε ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1 as a function of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (■): maximum amplitude of the central deformation before ejection as a function of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' PHASE DIAGRAM We now explore in more detail the phase diagram of the three regimes found in §III as a function of the control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We report in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 3 the measured values of the wave steepness ε for each accessible pair (f, a) of the forcing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Following a virtually vertical line (from bottom to top) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 3, ε is found to increase with a from very weak values (≃ 10−2 – in blue) to values close to unity (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The ejection regime occurs when crossing the black-dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The influence of the forcing frequency is highlighted by following the black solid line, which links data with the same value of ε (namely ε ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The curve minima point out frequencies for which the central deformation reaches this specific steepness although the forcing amplitude is weak, and thus correspond to resonances of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Moreover, these resonance frequencies appear to be in good agreement with the main theoretical eigenmodes of the ring marked by vertical dash-dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Indeed, the axisymmetric eigenmodes of a circular basin are obtained by considering an inviscid, irrotational, and incompressible fluid whose velocity potential ϕ satisfies the Laplace equation ∆ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The solution implies the Bessel function of the first kind of order α, Jα(kr), where k is the wave vector k = 2π/λ [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Moreover, imposing that the fluid cannot penetrate the solid boundary at r = R over t leads, in the linear approximation, to J′ 0(x)|knR = 0, where the prime stands for the spatial derivative of J0(x) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' This quantifies the modes of the system to discrete wave vectors kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Using the linear dispersion relation of inviscid deep-water gravity-capillary waves [26] (as kh > 8 for λ < 5 cm), ω = � gk + γ ρk3 , (1) 4 the corresponding axisymmetric eigenfrequencies fn read 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='70, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='64, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='55, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='47, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='39 Hz for n = 3, 4, 5, 6, and 7, respectively (g = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='81 m s−2 is the acceleration of gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Note that the linearized kinematic condition at the interface z = η(r, t) leads to ∂ϕ/∂z = ∂η/∂t, where ϕ is the velocity potential, and implies that ϕ and η have the same dependence on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Moreover, experiments show that the fundamental angular frequency ω of waves coincides with the forcing pulsation, 2πf, leading thus to the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Note that the vertical thickness of the ring is finite and the no-penetration condition is not fully verified below the ring (see §V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' However, we have verified that the initial immersed depth of the ring into the water does not impact qualitatively the wave shape (see Appendix A), but could explain slight departures between the resonances and eigenfrequencies in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We have also verified, using a ring of smaller diameter, that the resonances correspond to the eigenfrequencies, although the latter change (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The ejection threshold is hardly visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 3 outside the range f ∈ [5, 9] Hz mainly because, for smaller f, the wavelength becomes comparable to the system size [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', λ|5 Hz = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5 cm from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (1)], so that the standing wave pattern is modified by finite size effects of the container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For f > 9 Hz, the ejection regime cannot be reached due to the mechanical limitations of the wavemaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' R 60 40 20 0 20 40 60 R 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5 R 60 40 20 0 20 40 60 R 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1 0 1 2 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Dimensionless wave envelope along a basin diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' All curves have been rescaled by the asymmetry coefficient N = (ηmax − ηmin)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Red dotted line: experimental wave envelope showing both ηmax(r) > 0 and ηmin(r) < 0 (f = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='9 Hz, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='38 cm, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (a) Blue solid line: Linear prediction from [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (b) Black solid line: Third-order nonlinear prediction computed numerically from [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Inset: N vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' the nonlinear parameter ε for a fixed f = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='9 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Red dots correspond to data, black dotted line to the linear prediction, and black solid line to the nonlinear theory [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' STATIONARY SPATIAL PROFILE We denote ηmax(r) the maximum of the wave elevation, η(r, t), over time t at position r, and the maximum central elevation ηmax ≡ ηmax(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' In the same way, we define the quantities ηmin(r) and ηmin ≡ ηmin(0) for the minimum of η(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 4 the experimental wave envelope [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', ηmax(r) and ηmin(r)] rescaled by N = (ηmax − ηmin)/2 as a function of r along a diameter (see red-dotted line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 4a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We superimposed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 4a the prediction from the linear theory (blue solid line) when imposing two boundary conditions: (i) ∂η/∂r |r=0 = 0 to ensure continuity at the basin center r = 0, and (ii) η(±R, t) = b cos(2πft)/2 as the fluid must follow the ring oscillations at r = ±R, and the fluid elevation b may differ from the ring amplitude a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' This leads to the envelope equation η(r) = ηmaxJ0(kr) where J0 is the Bessel function of the first kind and k is computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (1) for fixed f, whereas ηmax is not needed thanks to the rescaling by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We first notice that the linear theoretical profile in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 4a does not show a zero slope at r = ±R indicating that the no penetrability condition used in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' IV is indeed debatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Moreover, several differences are visible between the experiment and the linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' First, the amplitude of the central deformation is measured to be asymmetric which is not captured by the linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Second, the shift of the zeros suggests that the dispersion relation does not hold as is for nonlinear waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Third, the linear prediction shows local minima 5 with strictly zero vibration whereas the experiment shows nonzero minima of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' These differences are significantly reduced when using a third-order nonlinear theory of axisymmetric gravity standing waves [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Indeed, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 4b shows that the experimental and theoretical local minima occur at the same positions evidencing the relevance of using this nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The latter also confirms that the wave elevation at these nodes does not have to go to zero (although getting closer than in the experiment), in particular near the basin center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Indeed, the nonlinear theory predicts a slight horizontal oscillation of the locations of the zeros over a period so that the surface never goes flat and the water level in any location r is nonzero at least for a fraction of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Moreover, close to the focus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', r → 0, the up-and-down asymmetry (which is a classical signature of the nonlinearity [32]) is well fitted by the nonlinear theory [27] (see black solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' More precisely, the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 4b shows the asymmetry, N = (ηmax − ηmin)/2, as a function of ε = ηmax/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' From the linear theory, one should have ηmax = −ηmin leading to N = ηmax = ελ as displayed by the dotted line in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The nonlinear theory computed numerically from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' [27] is also shown (solid line) and is found to be in good agreement with the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Note that the departure between linear and nonlinear theories reaches 10% for ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='064 which is close to the arbitrary criterion ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1 used in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' III to distinguish the linear and nonlinear regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 4 shows that the experimental profile near r = ±R does not satisfy the condition ∂η/∂r |r=±R = 0 as imposed to the system when computing its eigenmodes in §IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' This effect is experimentally confirmed for other forcing frequencies and could contribute to small departures between the resonances and eigenfrequencies observed on the phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' CENTRAL DEFORMATION AMPLITUDE The maximum elevation at the center (r = 0) is now investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We measure it either by the capacitive probe for weak and moderate forcing amplitude, a, or by using a side camera (Basler 2048 × 1536 px2, 120 fps) for higher a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Error bars are the statistical average of data from a few similar jets, as the jet eruption often deviates from the vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Figure 5 then shows the maximum height, ηmax, reached by the fluid at the center rescaled by λ as a function of the dimensionless forcing acceleration, aω2/g, when varying the forcing amplitude a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The three different curves correspond to three different forcing frequencies f ≡ ω/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' In the linear and weakly nonlinear regimes (ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='3), ηmax/λ grows linearly with aω2/g regardless of f as confirmed by the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' In the ejection regime (ε ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5), the rescaled maximal height of the jet (not taking into account possible drop ejection) increases strongly with aω2/g, then is found to saturate to a value denoted by εsat = ηsat/λ, roughly independent of the acceleration, but depending on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' A model described below will explain this saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='8 1 10-2 10-1 100 10-2 10-1 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Rescaled central maximum height, ηmax/λ, as a function of the forcing acceleration, aω2/g (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', various forcing amplitudes), for three different forcing frequencies f = (♦) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='70, (□) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='64, and (◦) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='55 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Horizontal dot-dashed lines correspond to εmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='3 and 5/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Inset: Same in log-log scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Solid line has a unit slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Best fit has a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='13 slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 6 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' SATURATION MODEL To explain the vertical saturation of the jet steepness, we approximate the jet surface at any time t, by a cone of height η(t) and radius λ/4 as shown in the top-right hand Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 6 (λ/2 is the natural diameter of the central deformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' First, we estimate the dominant forces of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Given ρ, γ, g, µ = 10−3 Pa s the dynamic viscosity of water, L the typical size of the cone, and v its typical vertical velocity, we compute the Weber (We), Reynolds (Re) and Bond (Bo) numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Taking L = ηmax ≈ 2 cm and v = 2πfηmax ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='8 m s−1, one obtains We = ρv2L/γ ≈ 350, Re = ρvL/µ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='6 × 104, and Bo = ρgL2/γ ≈ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The Weber number is defined as the ratio between the inertial and surface tension forces and shows that inertia dominates surface tension effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The Reynolds number (comparing inertia forces to viscous ones) shows that viscosity can be neglected here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The Bond number shows that gravitational forces are two orders of magnitude larger than surface tension ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Since We, Re, and Bo ≫ 1, viscous and surface tension forces can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The following model thus takes into account only inertial and gravitational forces through kinetic and potential energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We now consider the energy balance between a final state where the jet of height ηmax has a finite conical volume V , a positive potential energy and is motionless, and an initial state where the surface is flat and the same volume of the fluid is enclosed within a downwards cone of same dimensions located under the free surface (see bottom-right hand Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Leaving out the t notation for clarity, the cone edge equation thus reads z(r) = ±ηmax ∓ 4ηmax λ r , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', r(z) = (ηmax ∓ z) λ 4ηmax , (2) where the upper (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' lower) signs describe the top (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' bottom) cone and z is the vertical coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The volume of such a cone reads V = πλ2ηmax � 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The fluid velocity in the bottom cone is unknown, but we keep only its vertical component u(z) and approximate it by a linear dependence on z between u(0) = 0 and u(−ηmax) = U (upward velocity at its lowest depth), as proposed and proven sufficient in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' This yields u(z) = − z ηmax U , for z ∈ [−ηmax, 0] , (3) Then, we can express in a general way the kinetic energy Ek = � 0 −ηmax ρπr(z)2u(z)2dz/2 and the potential energy Ep = � 0 −ηmax ρπr(z)2gzdz of the bottom cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (2) and (3) in these expressions, we end up with Ek = ρV U 2� 20 and Ep = − ρgηmaxV /4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For the final state (top cone), Ek = 0 as U = 0 and Ep = + ρgηmaxV /4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Neglecting as justified above, viscous dissipation and surface tension, the conservation of energy between the final state and the initial state yields ηmax = 10g � U 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' To find the dependence of ηmax on the wavelength λ, we assume that the dispersion relation for linear waves of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (1) makes a first approximation even for these nonlinear and nonsinusoidal deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Taking U of order αωηmax with α a fitting parameter, this yields ηmax = 5 α2π λ [1 + (λc/λ)2] , (4) where λc/(2π) ≡ � γ/ρg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='9 mm is the capillary length separating the capillary (λ ≪ λc) and gravity (λ ≫ λc) wave regimes, the gravity-capillary regime occurring in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Note that, in the capillary-wave regime, the surface tension cannot be neglected anymore and a different model must be applied when Bo ≲ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', λ ≲ 2 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We plot ηmax(λ) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (4) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 6 together with the experimental maximum heights reached at the basin center for our gravity-capillary range (λ ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5] cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' As in §VI, error bars come from the statistical average of data from a few similar jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Figure 6 then shows that our model well captures the experimental saturation heights using α = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' α > 1 means that the linear approximation U = ωηmax underestimates the cone velocity at saturation and that nonlinear effects tend to increase it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' However, the saturation of the jet height indicates that beyond a certain forcing amplitude the energy injected in the system does not contribute to the central jet velocity, which should be directly converted into potential energy, but is mainly dissipated by the meniscus at the ring boundaries and/or transferred within the fluid bulk in the form of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Equation (4) with α = √ 2 then tends asymptotically in the pure gravity regime to ηg sat = 5λ/(2π), corresponding thus to a saturation of the wave steepness towards εg sat ≡ ηg sat/λ = 5/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The inset indeed shows that this predicted saturation of the wave steepness occurs experimentally for large enough wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Beyond the good agreement, the comparison is experimentally limited by viscous dissipation and mechanical limitation of the shaker for smaller wavelengths, and by the system size for larger wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Finally, note that for existing large-scale axisymmetric tanks (from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='6 m to 25 m diameters) surrounded by several wavemakers, driven in unison [16–18], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (4) would give an upper bound of the height of the central wave crest, which would be either larger than the size of the tank building, or not reachable due to wavemaker limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Furthermore, 7 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Saturation wave height at the basin center, ηsat, as a function of the rescaled wavelength, λ/λc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (◦) experimental data for various forcing frequencies f ∈ [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='5] Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (Blue solid line) prediction from the model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (4) with α = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (Blue dashed line) asymptotic trend of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (4) with α = √ 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', ηg sat = 5/(2π)λ, valid in the pure gravity regime (λ ≫ λc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Inset: Corresponding wave steepness at the basin center, εsat ≡ ηsat/λ, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' the dimensionless wavelength, λ/λc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For large enough λ, the saturation steepness εsat tends to 5/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Same curves as in the main figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Right: Schemes and quantities used in the model: (top) crude jet-shape approximation as a cone in its final state, (bottom) initial state with a flat surface and an upwards underwater-conical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (4) should not be confused with another limit commonly discussed in the literature concerning the stability of a periodic sharp-crested wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Indeed, for unidimensional progressive gravity waves of finite amplitude, this limiting angle β of the crest was first derived by Stokes to be 120° assuming a steady crest profile [32], and to be 90° for the standing wave case [33], the latter value being confirmed experimentally [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For axisymmetric standing gravity waves of finite amplitude, a limiting angle of 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='47° was derived analytically [27] and tested experimentally [18, 36], corresponding thus to a rescaled maximum height ηmax/λ = 1/(4 tan β/2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' As expected, this value is much smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='8 as it corresponds to a stability limit (and not to a maximum height).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' CONCLUSION We reported on the experimental study of the focusing of axisymmetric gravity-capillary waves generated by a vertically vibrating ring partially immersed in a fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Different regimes of standing waves are observed at the basin center depending on the forcing parameters: linear, nonlinear, and ejection regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For weak forcing, and close to a natural frequency of the circular basin predicted by the linear theory [26], the standing waves display a resonant response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For stronger forcing, we observed that the spatial profile of standing waves breaks the up-down symmetry, and exhibits nonzero local minima, which are both well taken into account by a nonlinear theory of axisymmetric standing waves up to third order in amplitude [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Finally, for an even stronger forcing, we observed a jet together with possible drop ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The maximum elevation reached experimentally by the wave at the center of the basin is found to saturate at ηsat, even for stronger forcing amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' For gravity waves, ηsat increases linearly with the wavelength, due to the saturation of its steepness to 5/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' This maximum wave height is well captured using a crude model, based on an energy balance with strong hypotheses concerning the forces and the shape of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' This is a first step towards a more elaborated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' In the future, we will address the origin of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Does it arise out of a deep depression of the free surface leading to the collapse of this cavity coupled to a singularity, or/and the collapse of a bubble entrapped underneath [20, 25]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The dynamical properties of the focusing will be also investigated by tracking the propagation of axisymmetric gravity-capillary propagating waves converging towards the center to explore open questions, such as which mechanisms drive their central interaction, and how the power injected by the ring is dissipated at the central singularity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 8 ACKNOWLEDGMENTS We thank G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Michel and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Roch´e for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Di Palma, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Le Goas for technical help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' This work is supported by the French National Research Agency (ANR DYSTURB project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' ANR-17-CE30-0004, ANR SOGOOD project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' ANR-21-CE30-0061-04), and by the Simons Foundation MPS No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 651463-Wave Turbu- lence (USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Appendix A: Role of the immersed depth of the ring We quantify the influence of the ring immersion into the fluid on the wave properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We measure the maximum water elevation, ηmax, at the center, for different relative positions d of the ring to the water surface at rest, all other things being equal (especially the forcing parameters f and a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The experimental data are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 7 and show that ηmax depends on the fluid volume moved by the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Indeed, ηmax is maximum when the ring is fully immersed into the fluid and flushing to the surface (d = 20 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The water volume moved by the ring oscillation, when the ring is closer to the free surface, is thus combined with the meniscus movement (same for d ≈ 0 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' On the other hand, when the ring is partially immersed (d ≈ 10 mm), ηmax reaches a plateau where only the meniscus effect plays a role in the wave forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' ηmax decreases as expected when the ring plunges deeper and deeper (d larger than the ring thickness 20 mm) and the two forcing effects disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Beyond this dependence of the wave amplitude on d, we have furthermore verified that the initial immersed depth of the ring does not impact other wave properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 0 10 20 30 60 80 100 120 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (Top) Sectional schemes defining d as the ring distance to the water surface at rest (horizontal blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The ring thickness (gray area) is 20 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (Bottom) Maximum water elevation, ηmax, measured at the center when increasing d for fixed sinusoidal forcing parameters (f = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='9 Hz and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='13 cm), and plotted as a percentage of the initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Appendix B: Experiments with a different ring size We perform the same experiments as in the main text, but with a ring of different radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' We now use a ring radius of R = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1 cm instead of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='25 cm as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Figure 8 shows the corresponding phase diagram as a function of the control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' The resonance frequencies with this smaller ring are f = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='37, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='07, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='59, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='11 Hz, which differ from those with the larger ring in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=', 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='7, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='66, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='59, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='53, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='48 Hz), but correspond to the circular basin eigenmodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' It thus confirms that the system resonance frequencies are indeed given by the ring eigenfrequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Le M´ehaut´e, Gravity–capillary rings generated by water drops, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 197, 415 (1988) [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Airy, On the intensity of light in the neighbourhoud of a caustic, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Cambridge Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 6, 379 (1838).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Wave steepness, ε = ηmax/λ, of the central deformation as a function of the forcing amplitude a and frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Logscale colorbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Vertical lines: circular basin eigenfrequencies fn from J′ 0(knR) = 0 (here R = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1 cm, compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 3) and fn and kn are related by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Solid line corresponds to the same value of ε ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1 as a function of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' (■): maximum amplitude of the central deformation before ejection vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' [3] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Pearcey, The structure of an electromagnetic field in the neighborhood of a cusp of a caustic, Phylos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 37, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Falcon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Laroche, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Fauve, Observation of Gravity-Capillary Wave Turbulence, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' 98, 094503 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' [40] See Supplemental Material at http://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='org/supplemental/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='1103/PhysRevFluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='124801 for movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content=' [41] Note that the Bessel function minima are not exactly periodic, in particular for the first two ones (departures of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='8% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='2%, respectively), but are approximately periodic for the following ones (departures less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} +page_content='01%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E2T4oBgHgl3EQfZAfl/content/2301.03861v1.pdf'} diff --git a/Z9E1T4oBgHgl3EQfKQM3/vector_store/index.faiss b/Z9E1T4oBgHgl3EQfKQM3/vector_store/index.faiss new file mode 100644 index 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b/bNFPT4oBgHgl3EQfAjSZ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9c98323ecad9a369832f8facfe3802e2c5a32ed77177e1e8812fe2326168f903 +size 2555949 diff --git a/bdAyT4oBgHgl3EQfivj4/content/tmp_files/2301.00404v1.pdf.txt b/bdAyT4oBgHgl3EQfivj4/content/tmp_files/2301.00404v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b48490cfad6a8992b3bf89d6bbd0fa7bdd802ce5 --- /dev/null +++ b/bdAyT4oBgHgl3EQfivj4/content/tmp_files/2301.00404v1.pdf.txt @@ -0,0 +1,617 @@ + +1 +Template Mediated Formation of Colloidal Two- +Dimensional Tin Telluride Nanosheets and the Role +of the Ligands +Fagui He,1 Eugen Klein,1 Stephan Bartling,2 Siavash Saeidpour,3,4 Björn Corzilius,3,4,2 +Rostyslav Lesyuk,1,5 Christian Klinke1,4,6* +1 Institute of Physics, University of Rostock, Albert-Einstein-Straße 23, 18059 Rostock, +Germany +2 Leibniz Institute for Catalysis (LIKAT), Albert-Einstein-Straße 29a, 18059 Rostock, Germany +3 Institute of Chemistry, University of Rostock, Albert-Einstein-Straße 27, 18059 Rostock, +Germany +4 Department “Life, Light & Matter”, University of Rostock, Albert-Einstein-Straße 25, 18059 +Rostock, Germany +5 Pidstryhach Institute for applied problems of mechanics and mathematics of NAS of Ukraine, +Naukowa str. 3b, 79060 Lviv, Ukraine & Department of Photonics, Lviv Polytechnic National +University, Bandery str. 12, 79000 Lviv, Ukraine +6 Department of Chemistry, Swansea University – Singleton Park, Swansea SA2 8PP, United +Kingdom +KEYWORDS colloidal synthesis, tin telluride, ligands, two-dimensional nanostructure, template- +assisted growth + + +2 +ABSTRACT We report the colloidal synthesis of 2D SnTe nanosheets through precursor hot- +injection in a nonpolar solvent. During the reaction, an important intermediate – Sn-template – is +formed which defines the confined growth of SnTe. This "flake-like" structure gives the first +evidence for the possible 2D morphology formation prior to the anion precursor injection (TOP- +Te). Additionally, we explore the role of each ligand in the reaction process. Thus, we explain the +formation and morphology evolution of 2D SnTe nanostructures from a mechanism perspective as +well as the role of each ligand on the molecular scale. The interplay of ligands provides the +necessary conditions for the realization of stable low-dimensional SnTe nanomaterials with +tunable size and shape. + + + + + +3 +Introduction +Low-dimensional thermoelectric materials such as tin telluride (SnTe) can adopt advantages +based on the quantum confinement effect, suggesting great potential for heat-electricity +conversion.1 As a IV-VI narrow bandgap semiconductor (0.18 eV, bulk), SnTe exhibits an +intrinsically high charge carrier concentration, which results in a relatively low Seebeck +coefficient,2, 3 but optimization of the material through doping and alloying offers great promise +for thermoelectric applications of this material.2 SnTe has also gained significant interest due to +its exciting properties as a topological crystalline insulator, IR detection and radiation receivers +material, as well as photovoltaic absorber.4-8 So far, attempts to obtain solution-based SnTe +nanocrystals (NCs) mainly yielded zero-dimensional (0D) or one-dimensional (1D) +nanostructures.9-11 Recently we reported a synthesis protocol for two-dimensional (2D) colloidal +SnTe nanostructures.12 Now, we disclose the important role of ligands in the colloidal synthesis +process information of the 2D morphology and the observed faceting of nanocrystals. The obtained +nanomaterials are investigated by means of scanning electron microscopy (SEM), transmission +electron microscope (TEM), high-resolution TEM, powder X-ray diffraction (XRD), X-ray +photoelectron spectroscopy (XPS), energy-dispersive X-ray (EDX) analysis and Fourier transform +infrared (FT-IR) measurements.13 Since SnTe has a cubic crystal structure with m3̅m symmetry +and octahedral coordination geometry, the formation of anisotropic shapes has to be conducted by +molecular templates. These templates are well-organized domains consisting of the cation and a +specific combination of ligands/counterions. When the tellurium precursor is injected into the +reaction mixture, the tin/halogen arrangement of the template is replaced by the Sn-Te bond, and +a bilayer oleic acid shell is formed at the same time. These findings show the importance of co- +ligands and emphasize the formation of templates in the syntheses of colloidal nanomaterials. + + + +4 +Methods + +Chemicals and Materials: Tin (II) acetate (Sn(CH3CO2)2, anhydrous, ≥99.99%, stored in a +nitrogen filled glovebox), tellurium shots (Te, amorphous, 1-2 mm; 99.999 %; stored in a nitrogen +filled glovebox), oleic acid (90%), trio-n-octylphosphine (TOP; 97%; stored in a nitrogen filled +glovebox), diphenyl ether (≥99%), and 1-chlorotetradecane (1-CTD, 98%) were purchased from +Sigma-Aldrich. Ethanol and toluene were purchased from Honeywell. All the chemicals were used +as-received without additional purification. Trio-n-ctylphosphine-Te (TOP-Te) was prepared and +stored in a glovebox and all the syntheses were carried out applying standard air-free Schlenk-line +techniques. +Synthesis of 2D SnTe nanosheets: In a three necked flask equipped with a condenser, a septum and +a thermocouple in a glass mantle, 59.5 mg of Sn (CH3CO2)2 (0.25 mmol), 2 mL of oleic acid (6.25 +mmol), and 0.5 mL of tri-n-octylphosphine (1.1 mmol) were mixed in 10 mL of diphenyl ether, +stirred at 130 °C for 5 min and degassed under vacuum at 80 °C for 1.5 h. Afterwards, the reaction +solution was heated to reaction temperature 210°C under nitrogen after adding 0.15 mL of 1- +chlorotetradecane (0.5 mmol). 10 min later, 0.8 mL of a 0.65 M tri-n-octylphosphine-tellurium +(Te–P(octyl)3, TOP-Te) precursor solution was injected. The solution turned from clear yellow +to greyish yellow. The reaction was quenched after 1 min by removal of the heating mantle. The +resultant nanostructures were purified by precipitation with toluene, centrifugation at 4000 rpm +for 3 min (3 times), removal of the supernatant and re-suspension in toluene for further +characterization or storage. +Synthesis of Sn templates: In a typical synthesis of Sn template, 59.5 mg of Sn (CH3CO2)2 (0.25 +mmol), 2 mL of oleic acid (6.25 mmol), 0.5 mL of tri-n-octylphosphine (1.1 mmol) and 10 mL of +diphenyl ether were mixed and dissolved in a 50 mL three-neck flask, stirred at 130 °C for 5 min + + +5 +and degassed under vacuum at 80 °C for 1.5 h. Afterwards, the reaction solution was heated to +reaction temperature 210°C under nitrogen after adding 0.15 mL of 1-chlorotetradecane (0.5 +mmol). 10 min later, the heating mantle was removed and when the temperature reached to 75°C, +15 mL of ethanol was injected into the reaction. The resultant white powder was then purified by +centrifugation with ethanol at 6000 rpm for 3 min (3 times). The product could then be re- +suspended in ethanol for further characterization or storage. +Scanning electron microscope (SEM) and Energy Dispersive X-ray Spectroscopy (EDX): Standard +SEM images and EDX elemental mapping were performed on a Zeiss EVO MA 10 microscope at +an acceleration voltage of 10 kV. Samples were prepared by dropping 10 μL of the dilute NS +solution onto a silicon wafer. +Transmission Electron Microscopy (TEM): Standard TEM images were performed on a JEOL +Jem-1011 microscope at an acceleration voltage of 100 kV. Samples were prepared by dropping +10 μL of the dilute NS solution onto carbon-covered copper grids. +Powder X-ray Diffraction (XRD): The crystal structure of the SnTe nanosheets was determined by +XRD measurements, which were carried out on a Philips X’Perts PRO MPD diffractometer with +monochromatic X-Ray radiation from a copper anode with a wavelength of 1.54 A (CuKα). The +samples were prepared by dropping 10 μL of the dilute NS solution onto a silicon wafer. +X-ray Photoelectron Spectroscopy (XPS): measurements were performed on an ESCALAB +220iXL (Thermo Fisher Scientific) with monochromated Al Kα radiation (E = 1486.6 eV). +Samples are prepared on a stainless-steel holder with conductive double-sided adhesive carbon +tape. The electron binding energies were obtained with charge compensation using a flood electron +source and referenced to the C 1s core level of carbon at 284.8 eV (C-C and C-H bonds). For +quantitative analysis the peaks were deconvoluted with Gaussian-Lorentzian curves using the + + +6 +software Unifit 2021. A background model composed of a Shirley background + polynomial +background was used and the parameters are varied during the fit together with the peak parameters +to find the optimal background. The peak areas were normalized by the transmission function of +the spectrometer and the element specific sensitivity factor of Scofield1. +Sensitivity-enhanced magic-angle spinning nuclear magnetic resonance (MAS NMR) by dynamic +nuclear polarization (DNP): 1H-31P CPMAS NMR measurements were performed by suspending +the dried nanomaterials in a 1,1,2,2-tetrachloroethane solution containing 15 mM TEKPol +polarizing agent. 37 Experiments were carried out using a Bruker Biospin ASCENT 400DNP wide +bore magnet operating at 9.4 T (400 MHz 1H frequency) with a Bruker AVANCE III HD +spectrometer and a Bruker 3.2 mm LTMAS DNP probe tuned to 1H and 31P in dual channel mode. +Spectra were recorded at an MAS frequency of 15 kHz at a sample temperature of 100 K utilizing +cross-polarization from hyperpolarized 1H under microwave irradiation from a Bruker/CPI 263 +GHz gyrotron source operating at 130 mA beam current. The 1H-DNP enhancements for the frozen +dispersions of SnTe-template and SnTe-nanosheets were 23 and 152, respectively. +Fourier transform infrared (FTIR): FTIR measurements were carried out by drying the +nanomaterials and putting the powders on a diamond-ATR unit (PerkinElmer Lambda 1050+). +The FTIR measurements are performed with a range from 500 to 4000 cm-1. + +Results and Discussion + +The coordinating ligands can modify the surface energy of exposed crystallographic facets +(which assume an increasingly prominent role with increasing surface-to-volume ratio) or binding +selectively to particular facets, thereby favoring specific morphologies or yielding altogether +different polymorphs.14 In our previous work, the type and content of ligands used in the colloidal + + +7 +synthesis have been optimized; however, a fundamental investigation and understanding of the +mechanism would be very beneficial for the precise control of the formation and faceting +processes. In this report, first we used a large amount of ethanol to prevent the reaction before the +hot injection of the tellurium precursor. As a result, a white powder, here called the Sn-template, +was obtained. Figure 1a shows a SEM image of the Sn-templates. We observe a flake-like +structure giving the first evidence for the possible 2D morphology formation prior to the tri-n- +octylphosphine (TOP)–Te injection, which is similar to the Zn-soft template formation.15 The +selected area (red frame) in Figure 1a contains a large amount of Sn and carbon from the Sn-oleate +complex, as well as a smaller amount of Cl which stems from the 1-chlorotetradecane (1-CTD) +ligands (Figure S1). Figures 1b and c depict a TEM image and a XRD pattern of SnTe nanosheets +(NSs) after synthesis and purification steps. The sample consist mainly of large nanosheets with +lateral dimensions between 1 and 5 µm and a thickness of 57 nm calculated from XRD data using +the Scherrer formula and 48 nm measured by atomic force microscopy (Figure S2). The difference +in thickness stems from the three-dimensional cubic side products which contribute to the XRD +signals and have edge lengths bigger than 100 nm. The XRD pattern shows only 2 signals due to +a strong texture effect attributed to the (200) and (400) reflections. These signals can be attributed +to the cubic crystal structure with the Fm3̅m space group. + +20 +30 +40 +50 +60 +70 +20 +30 +40 +50 +60 +70 +(200) +(400) +SnTe nanosheets +(c) +Intensity (a.u.) +2 theta (°) +(200) +(220) +(222) +(400) +(420) +Cubic, SnTe +00-046-1210 + + +(a) +(b) +2um +8 +Figure 1. (a) SEM image of the Sn templates with flake-like structure; (b) TEM image of the +product containing mainly large nanosheets and minor fraction cubic particles; (c) XRD pattern of +the SnTe nanosheets with a strong texture effect. +In previous studies, we observed that halogenated compounds have an important influence on +the shape of semiconductor nanomaterials.12, 13, 16, 17 The active species promoting the shape +transformation and confined growth were halide ions produced in situ which influence both the +nucleation and ripening of the nanostructures.13 We also evidenced that the amount of 1- +chlorotetradecane (1-CTD) is one of the key factors for obtaining relative uniform and well- +defined 2D SnTe nanostripes while only agglomerates were produced if no haloalkanes were used +during the synthesis of SnTe nanostructures.12 The importance of 1-bromoheptane (Br-Hep) in the +formation of CdSe nanosheets was earlier revealed, suggesting that two active species, Br-Hep and +ionic Br +-, were present on the surface of the CdSe nanosheets and partially replace the carboxylate +ligands.13 As the new surface ligands, the formation of Cd-Br bonds can regulate the growth rate +by modulation of the relative surface energies of those facets and consequently influence the shape +of CdSe nanosheets.13 However, in the present work we find that 1-CTD influences the formation +and final shape of SnTe nanosheets in a different way. +X-ray photoelectron spectra (XPS) were recorded to analyze the composition and the chemical +states of all elements in Sn-templates and SnTe nanosheets. The survey spectra of both samples +are shown in Figure S3a and b, respectively. As main elements C, O, and F as well as Sn, Te (for +SnTe nanosheets), and Cl (for Sn-templates) with minor contributions can be identified. The XPS +quantification data are shown in Table S1. High-resolution XPS scans of Sn 3d and Cl 2p of the +Sn-templates, as well as Sn 3d and Te 3d of the SnTe nanosheets are presented in Figure 2. The +Sn 3d5/2 peak of Sn-template can be found at a binding energy of 487.2 eV (Figure 2a), which can + + +9 +be assigned to Sn2+. Figure 2b shows a rather small Cl 2p peak of Sn-templates at a binding energy +of 199.5 eV which might be attributed to SnCl2 as minor Sn species. We speculate that Cl does not +exist as free Cl¯ ions. This is in good accordance with the previous observations that Cl¯ ions, +which are introduced to the synthesis by chloroalkanes, are attached to the surface of the +nanostructures in the form of a halogen–metal complex.18 After the Te precursor (TOP-Te) was +injected into the reaction, the product was further analyzed. Figure 2c shows the XPS spectrum of +Sn 3d for SnTe nanosheets (see Table S2 for details of the fit). The peaks at 487.0 and 495.5 eV +correspond to the binding energies of Sn 3d5/2 and Sn 3d3/2 of SnTe (Sn2+). However as shown for +the template, the binding energy of Sn2+ attributed to tin oleate is located nearly at the same spectral +position. The peaks located at 488.8 and 497.3 eV indicate the oxidation state Sn4+ observed in the +SnO2 crystals.19 The weak peaks at 485.4 and 494.0 eV are assigned to elemental Sn. Figure 2d +shows the XPS spectrum of Te 3d (see Table S3 for details of the fit). The peaks at 574.4 and +584.9 eV are assigned to Te 3d5/2 and Te 3d3/2 in SnTe, respectively. The peaks at 572.4 and 582.9 +eV could be assigned to the Te 3d5/2 and Te 3d3/2 from elemental Te and the weak peaks at 576.1 +and 588.4 eV could be assigned to the oxidized species of Te4+ in TeO2.20-22 The results express +the clear propensity for oxidation of the SnTe nanostructures.23 However, due to the formation of +an oxide shell on SnTe nanocrystals, the synthesized material could be protected from being +oxidized further and promote the stability of the nanocrystals.11, 12 The data reveal also fluorine as +very intense component of the surface, identified as contamination originated in the grease used +for the glassware during the synthesis procedure. In fact, a majority of the F is bond to C as can be +confirmed by the C 1s spectra shown in Figure S4 with strong peaks at 291 and 293 eV +corresponding to C-F2 and C-F3, respectively. + + +10 + +Figure 2. XPS spectra: Sn 3d (a) and Cl 2p (b) of Sn template; Sn 3d (c) and Te 3d (d) of SnTe +nanosheets. +In the context of understanding the bonding mode of the ligands with the nanomaterial surfaces, +FTIR spectroscopy has been employed. As seen in Figure 3a and b, the bands at 2853 and 2922 +cm-1 were attributed to the asymmetric CH2 stretch and the symmetric CH2 stretch modes of oleic +acid, respectively.24 It is worth noting that the band at 1710 cm-1, corresponding to stretching +vibration of C=O in pure oleic acid (see Supporting Information, Figure S5), was absent in the +spectrum of Sn-templates. Instead, two new bands at 1457 and 1583 cm-1 appeared in the FTIR +spectrum of the Sn-templates (Figure 3a). They were attributed to the asymmetric (-COO¯) and +symmetric (-COO¯) stretch vibration bands, respectively.24, 25 This indicates that there is no free +oleic acid in the Sn template sample and a complexation between the carboxyl and Sn was formed, +which is in good accordance with the XPS results. As a result, oleic acid molecules are chemically +adsorbed on the surface of Sn template through the chemical interaction between their -COO− +groups and Sn atoms, meanwhile, the hydrophobic tails of oleic acid molecules face outwards + +(a +Sn3d +Sn-templates +(b) +CI2p +Sn-templates +Sn2+ 3ds/2 +Intensity (a.u.) +Intensity (a.u.) +Sn2* 3d3/2 +505 +500 +495 +490 +485 +480 +215 +210 +205 +200 +195 +190 +Bindingenergy (eV) +Binding energy (eV) +(c) +Sn3d +(d) +SnTenanosheets +Te3d +SnTenanosheets +3d5/2 +3d3/2 +3d5/2 +Sn2+ +Intensity (a.u.) +3d31/2 +Intensity (a.u.) +Te2. +Sn4+ +Te (0) +Sn (0) +505 +500 +495 +490 +485 +480 +475 +595 +590 +585 +580 +575 +570 +565 +560 +Bindingenergy (eV) +Bindingenergy(eV) +11 +(Figure 3c) and form a nonpolar shell which supports the single layer coated Sn-templates that +can be dispersed in nonpolar carrier liquids. After TOP-Te was injected into the reaction, the band +at 1710 cm-1 + appeared in the FTIR spectrum of SnTe nanosheets while the band at 1457 cm-1, the +asymmetric (-COO¯) stretch vibration band, became very weak (Figure 3b). Thus, the FTIR signal +changes also verify that the injection of TOP-Te to the synthesized oleic acid-coated Sn-template +changes the kind of bonding between -COO¯ groups of the oleic acid and the Sn atoms. The band +at 1710 cm-1 is the characteristic band of a secondary layer in bilayer oleic acid-coated +nanomaterials which was already demonstrated by Wen et al.24 The primary layer in the bilayer +coated structure is chemically adsorbed on the surface of nanoparticles, and the secondary layer is +physically adsorbed on the primary layer through the interpenetration of the tails of the primary +and secondary surfactants at their interface (Figure S6).24, 26 Because the oleic acid of the +secondary layer was only physically adsorbed on the primary layer, the 1710 cm-1 + band due to the +stretching vibration of C=O in oleic acid should appear in the FTIR spectrum of bilayer oleic acid- +coated SnTe nanosheets. +The tri-n-octylphosphine (TOP)–Te complex was used as a tellurium precursor in the synthesis. +It has a higher cleavage rate compared to TOP-S and TOP-Se, which leads to the formation of +more nuclei and faster exhaustion of the monomers. This fast cleavage rate and a higher number +of nuclei in the reaction showed an adverse effect on the anisotropic growth and resulted in 3D +bulk structures instead of a 2D morphology.27 Interestingly, the introduction of an appropriate +amount of TOP is the guarantee for obtaining a two-dimensional structure along with the presence +of the haloalkane. For comparison, reference syntheses with different amounts of TOP were also +performed. No product could be obtained under these conditions without TOP while only +agglomerates were produced if less or more amount of TOP compared to the standard procedure + + +12 +was injected into the synthesis (Figure S7). TOP plays an important role in slowing down particle +growth, likely by blocking surface binding sites.28, 29 When TOP was used as a capping ligand in +the synthesis of nanomaterials, the coordination with the TOP ligands could effectively stop the +growth of the nanoparticles (even in the presence of unreacted precursors) which would influence +the final shape of nanomaterials.30 In addition, TOP could act as a reducing agent as well, leading +to producing SnO through an autocatalytic process at the SnTe NSs surface.31 +Another significance of TOP that can ensure the acquisition of 2D structures is that TOP affects +not only the reaction pathway for the halide transfer but also the dissolution of tin halides on SnTe +nanocrystals which is similar to the structural development of CdSe tetrapods.32 As shown in +Figure S8, three resonances at ∼-12, ∼30 and ∼50 ppm in the 31P NMR spectrum of both the Sn +template and SnTe nanosheets indicate the occurrence of ligand exchange and elimination when +halide and TOP are present in the solution: alkylphosphonium halides (R4P+(halide)− where R = +alkyl chain), formed from the reaction between alkylhalides (thermal decomposition product) and +neat TOP, and halide-bound Sn is dissolved by forming (R3P)2Sn(halide)2.32 When the binding +between Sn and Cl was replaced by Sn–Te bonds, a certain quantity of Cl− ions is released into the +solution, resulting in more ligand exchange between TOP and halide. Therefore, stronger +resonance at around -12 ppm (R3P) and 30 ppm (R4P+) are observed. On the other hand, in reactions +where oleic acid was used as a surfactant molecule, the formation of II-VI nanocrystals was +accompanied by the formation of trialkylphosphine oxides and (OA)2O.33 In particular, the +conversion of TOP-Te to its corresponding phosphine oxides is linked to the formation of +anhydrides of oleic acid, suggesting that phosphonic and carboxylic acids are responsible for the +cleavage of the phosphorus-chalcogen double bond.33 Thus, after the hot injection of TOP-Te, +stronger resonance at around 50 ppm (corresponding to the conversion of TOP-Te to TOPO-Te) + + +13 +is observed due to the cleavage of TOP-Te. As H-OA is responsible for the cleavage of the P=Te +bond, changing the concentration of these surfactants will likely change the TOP-Te cleavage +kinetics in addition to the binding of surfactants to the nanocrystal surface. This is especially +important because the rate of this cleavage will influence particle nucleation and growth and offers +an explanation of the influence of the amount of OA on the morphology of the ultimate product +(Figure S9). +Additionally, we investigated the influence of the other ligands and the synthesis temperature. +The amount of oleic acid varies the lateral dimensions. Low amounts of oleic acid yield small +nanosheets while larger amounts caused big nanosheets (Figure S9). Since the Sn templates +consist mainly of tin cations and a combination of oleate and oleic acid molecules, a higher amount +of oleic acid means larger and better-organized templates. On the other hand, the co-ligand 1-CTD +also influences the synthesis. After a cleavage of the Cl¯ ion from the alkane rest, these small ions +replace some of the oleate ligands in the template structure which alters the reactivity of the +templates. An excessive amount of haloalkane molecules yield solely three dimensional (3D) +structures while low amounts favor a large shape and size distribution (Figure S10). It is important +to note that the cleavage of this molecule does depend on the temperature and the time it spends +in the reaction mixture, which means that the same amount of 1-CTD could produce different +products when changing the reaction conditions. The temperature mainly determines the reactivity +in a direct and indirect way as mentioned before through the cleavage rate of the 1-CTD. When +the reaction temperature is below 170 °C no reaction takes place for this material. Low reaction +temperatures like 170 °C or 180 °C yield mainly 3D structures (Figure S11) while high +temperatures like 250 °C produce products with a high size and shape distribution.12 The results +indicate that temperatures below 190 °C are not enough to separate the Cl¯ ions from the alkane + + +14 +rest, which means that the synthesis is basically performed without this co-ligand and yields solely +3D structures. At high temperatures, the cleavage rate of the 1-CTD is very high. Thus, the +structure of the templates becomes more unstable resulting in templates with different shapes and +reactivities. +After hot injection of the TOP-Te precursor, the colorless solution transiently turned to yellow +(the color of TOP-Te precursor) and the morphology evolution was investigated by taking aliquots +during the reaction followed by TEM analysis. At the early stage, when the color of the solution +starts to change (from yellow to dark yellow at 32 s after hot injection of TOP-Te), the SnTe +nanoplatelets with hexagonal shape (≈2 μm length, ≈400 nm width) could be observed (Figure +S12a, Supporting Information). As the reaction proceeds, the nanoplatelets grow longer and the +widths become larger (≈3 μm length, 600-700 nm width) (Figure S12b). Based on evidence from +XPS and FTIR measurements shown above, we propose a possible formation process of SnTe +nanosheets. As depicted in Figure 3d, the two ligands of oleic acid and 1-CTD help Sn to form a +flat structure, the Sn template. After introduction of the Te precursor, the Cl¯ ions are replaced by +Te to form Sn-Te bonds. Meanwhile, the SnTe nanostructure is coated with a well-organized +primary oleic acid molecule. Then, the excess oleic acid was weakly adsorbed on the primary layer +of the oleate-coated SnTe nanosheets to form a double layer shell through the steric intermolecular +interaction between the subsequent molecule and the hydrophobic tail of oleate of the primary +layer. + + +15 + +Figure 3. FTIR spectrum of Sn template (a); FTIR spectrum of final product SnTe nanosheets (b); +scheme of the formation of Sn oleate (c); scheme of the proposed mechanism of the formation of +SnTe nanosheets (d). +Conclusion +In this work, we discussed the interaction of ligands with metal precursors at different stages of +the synthesis and investigated the formation process of SnTe nanosheets. We see from the FTIR +analysis that the formation of an OA-double layer is already evidenced at the stage of the Sn-white- +complex (template). At the same time, the Cl-interaction with Sn is evidenced by XPS, suggesting +that both OA and halogen ensure conditions for the 2D templating. Later on, Cl disappears and +only OA and TOP stay on the surface leading to growth in three dimensions with different +velocities. Thus, the collective interplay of different ligation agents is needed for obtaining the 2D +morphology in the cubic system of SnTe. Based on the experiments, we explained the formation +and morphology evolution of 2D SnTe nanostructure from a mechanism perspective as well as the +role of each ligand at the molecular scale which provides the building block for the realization of +stable low-dimensional SnTe nanomaterials with tunable size and shape. + +HO +(a) +Sn-template +C(CH2)7CH=CH(CH2)7CH3 +C(CH2)7CH-CH(CH2)CH +1583 +1457 +Transmittance (a.u.) +(1) +(b) +SnTenanosheets +1710 +3 +400035003000 +2500 +2000 +1500 +1000 +500 +Wavenumber(cm-1) +Sn +RAAURAURAL +16 +ASSOCIATED CONTENT +Supporting Information. +EDS analysis of Sn templates; AFM images and measured height images of synthesized SnTe +nanosheets; XPS spectra and XPS quantification data; fit parameters of the Sn 3d spectra and Te +3d spectra; FT-IR spectrum of pure oleic acid; TEM images; 31P NMR spectrum (PDF) +AUTHOR INFORMATION +Corresponding Author +* E-mail: christian.klinke@uni-rostock.de +Funding Sources +This work was supported by the China Scholarship Council and German Academic Exchange +Service (DAAD). Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research +Foundation) – SFB 1477 "Light-Matter Interactions at Interfaces", project number 441234705. +Notes +The authors declare no competing financial interest. +ACKNOWLEDGMENT +We would like to thank Fabian Strunk for the help with the XPS measurements. We +acknowledge the European Regional Development Fund of the European Union for funding the +PL +spectrometer +(GHS-20-0035/P000376218) +and +X-ray +diffractometer +(GHS-20- +0036/P000379642) and the Deutsche Forschungsgemeinschaft (DFG) for funding an electron +microscope Jeol NeoARM TEM (INST 264/161-1 FUGG) and an electron microscope +ThermoFisher Talos L120C (INST 264/188-1 FUGG) and for supporting the collaborative +research center LiMatI (SFB 1477). + + +17 +REFERENCES +1. +Li, Y.; Wu, M. N.; Ding, T.; Ma, K.; Liu, F. S.; Ao, W. Q.; Li, J. 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Journal of the American Chemical +Society 2007, 129 (2), 305-312. + + diff --git a/btAyT4oBgHgl3EQfi_gf/content/2301.00405v1.pdf b/btAyT4oBgHgl3EQfi_gf/content/2301.00405v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c299bd926ca6d0bed73b95c282ab65fc6ce8f094 --- /dev/null +++ b/btAyT4oBgHgl3EQfi_gf/content/2301.00405v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4da8e0b11d23241f53513cbb8671cbbdf6ec4335c244c9d7943ab4c12b7b58a4 +size 297145 diff --git a/btAyT4oBgHgl3EQfi_gf/vector_store/index.faiss b/btAyT4oBgHgl3EQfi_gf/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a4c382afe2d355e2b54a87caa7547932ef988926 --- /dev/null +++ b/btAyT4oBgHgl3EQfi_gf/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b94fbe2d8fdecc8f6bac2ab0ff094f8cad283580c7877cc04dbb46fece49f7b3 +size 4128813 diff --git a/btAyT4oBgHgl3EQfi_gf/vector_store/index.pkl b/btAyT4oBgHgl3EQfi_gf/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..cde94bb0ab390bdb620517369c417d5c557bd439 --- /dev/null +++ b/btAyT4oBgHgl3EQfi_gf/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:88f057552c86a7d8916f1820014613798847b48581d2ef516f825da0078d47b9 +size 137679 diff --git a/c9AzT4oBgHgl3EQfLvtS/content/2301.01119v1.pdf b/c9AzT4oBgHgl3EQfLvtS/content/2301.01119v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..92d79204388b2a9edfafac7cbac6a2eca693e046 --- /dev/null +++ b/c9AzT4oBgHgl3EQfLvtS/content/2301.01119v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:91c05306d8fb8507906558d2d45f27e2253fb1f49cc397a84d9a54a0de0c6c2c +size 669243 diff --git a/cNFAT4oBgHgl3EQf5h65/vector_store/index.faiss b/cNFAT4oBgHgl3EQf5h65/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..0b289e886aaec5e0a85e228c7eebebd22267bd6e --- /dev/null +++ b/cNFAT4oBgHgl3EQf5h65/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cd4a8cedfdfd4ffc0c8959b3be75d19ed7a6e8f18bb2ef6c886aedad1444a3ed +size 4259885 diff --git a/d9AzT4oBgHgl3EQfL_tB/content/tmp_files/2301.01123v1.pdf.txt b/d9AzT4oBgHgl3EQfL_tB/content/tmp_files/2301.01123v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ef4dfd287ba33368b75db9cbec69e5ebdeff793 --- /dev/null +++ b/d9AzT4oBgHgl3EQfL_tB/content/tmp_files/2301.01123v1.pdf.txt @@ -0,0 +1,2340 @@ +MGTAB: A Multi-Relational Graph-Based Twitter Account Detection +Benchmark +Shuhao Shi1,†, Kai Qiao1,†, Jian Chen1 Shuai Yang1, Jie Yang1, +Baojie Song1, Linyuan Wang1, Bin Yan1,∗ +1Henan Key Laboratory of Imaging and Intelligence Processing, +PLA strategy support force information engineering university +Abstract +The development of social media user stance detection +and bot detection methods rely heavily on large-scale and +high-quality benchmarks. However, in addition to low an- +notation quality, existing benchmarks generally have in- +complete user relationships, suppressing graph-based ac- +count detection research. To address these issues, we pro- +pose a Multi-Relational Graph-Based Twitter Account De- +tection Benchmark (MGTAB), the first standardized graph- +based benchmark for account detection. +To our knowl- +edge, MGTAB was built based on the largest original data +in the field, with over 1.55 million users and 130 million +tweets. +MGTAB contains 10,199 expert-annotated users +and 7 types of relationships, ensuring high-quality annota- +tion and diversified relations. In MGTAB, we extracted the +20 user property features with the greatest information gain +and user tweet features as the user features. In addition, +we performed a thorough evaluation of MGTAB and other +public datasets. Our experiments found that graph-based +approaches are generally more effective than feature-based +approaches and perform better when introducing multiple +relations. By analyzing experiment results, we identify ef- +fective approaches for account detection and provide poten- +tial future research directions in this field. Our benchmark +and standardized evaluation procedures are freely available +at: https://github.com/GraphDetec/MGTAB. +1. Introduction +With the continuous development of the Internet, social +networks have become an essential part of people’s daily +social life. Twitter is one of the most visited social net- +works worldwide, providing online news and information +exchange to billions of users worldwide. Due to the avail- +ability, many account detection benchmarks are constructed +based on Twitter data [9,15,17,47]. +Stance detection and bot detection are essential tasks in +account detection. Stance detection aims at detecting the +user’s stance on a topic or claim. It is a critical technique +in applications such as fake news detection [25,31], claims +validation [1, 27], and analyzing public opinion on social +media. Bot detection is crucial for detecting information +manipulation on social media. Social bots are automated +user accounts operated by computer programs [60] and are +often used to abuse social media platforms [10, 19] to ma- +nipulate public opinion [9–11,60]. +Most account detection methods use only part of the in- +formation in social media, such as posts, registration infor- +mation, etc., for classification. Rarely consider the connec- +tion between users [24], making it challenging to ensure +detection accuracy. In stance detection, silent users often +do not post directly but express their stance through be- +haviors, such as following others and favoring posts [24]. +However, most studies focus only on the posting content +of active users and ignore silent users [24]. The features +of social graphs need to be used to detect the silent users’ +stance better [1]. In bot detection, since most studies ignore +bots’ social graph features, bots can simulate genuine users +through complex strategies to evade feature-based detection +methods [10]. +Recent work [14, 18, 38] in account detection has fo- +cused on exploiting relationships between users, with +performance improvement compared to feature-based ap- +proaches. However, the existing datasets have several draw- +backs to supporting graph-based methods, as follows: +• (a) Low annotation quality. Previous account detec- +tion datasets were mostly annotated by crowdsourc- +ing, while crowdworkers’ lack of domain knowledge +resulted in significant noise in the annotation [15]. +• (b) Incomplete user relationships. +None of the +stance detection datasets explicitly provide the graph +structure among users, and only the bot detection +datasets Cresci-15 [9], TwiBot-20 [17], and TwiBot- +22 [15] contain explicit graph structures. Moreover, +Cresci-15 and TwiBot-20 contain only 2 types of user +arXiv:2301.01123v1 [cs.CV] 3 Jan 2023 + +relationships, which is insufficient for graph-based de- +tection methods. +• (c) Complex user information. Social media user in- +formation is diverse and voluminous, but most infor- +mation has little effect on account detection. Existing +datasets lack the extraction and organization of essen- +tial user information, making account detection a dif- +ficult problem. +To address the shortcomings above, we presented Multi- +Relational Graph-Based Twitter Account Detection Bench- +mark (MGTAB), a large standardized expert-annotated +dataset for stance and bot detection. +MGTAB contains +10,199 users manually annotated by experts and 400,000 +closely related unannotated users. +Further, MGTAB ex- +tracted 20 most effective user property features by calcu- +lating the information gain (IG) and user tweet features. Fi- +nally, MGTAB simplified the social graph and constructed +a user network with 7 types of relationships. The contribu- +tions of this paper are as follows: +• We presented MGTAB, a large-scale expert-annotated +benchmark for stance detection and bot detection. All +annotations are carried out by experts and improve an- +notation quality by cross-validation. The annotation +quality has been substantially improved compared to +the previous dataset. +• We released the first standardized dataset containing +the property features, user tweet features and 7 types +of user relationships. We constructed a user-level so- +cial graph that can be applied to state-of-the-art graph- +based account detection methods, making account de- +tection simpler. The release of the MGTAB dataset +will facilitate the development of new methods for +graph-based account detection. +• To build MGTAB, we collect over 1.55 million Twit- +ter users and 135 million tweets. To the best of our +knowledge, it is the biggest data in the domain. We +carried out meticulous data cleaning, retaining 400,000 +closely related unlabeled users, which supports semi- +supervised learning to merge with account detection +research. +• Our experiments show that graph-based detection +methods are more effective than feature-based meth- +ods in most cases. In addition, We found that the per- +formance of graph-based approaches improved when +multiple relationships were introduced. +The results +suggest that future research should focus on using mul- +tiple relationships. +2. Related Work +2.1. Stance Detection +The existing stance detection methods can be divided +into feature-based methods and graph-based methods. +Feature-based methods. +Previous research works +[56, +58, 62] used machine learning algorithms and deep learn- +ing methods such as Support Vector Machines (SVM), Re- +current Neural Networks (RNNs) [62], and Convolutional +Neural Networks (CNNs) to automatically learn latent fea- +tures from a large amount of raw data. Several recent works +[31, 39, 40, 45, 57] focused on the use of bidirectional en- +coder representations from transformers (BERT) [12] on +stance detection. Ghosh et al. [20] explored stance detec- +tion based on transfer learning, and Li et al. [39] explored +BERT-based data augmentation models. +Graph-based methods. Most studies on stance detection +use text-based features [40,47,62]. Some recent work has +shown the effectiveness of using user network graphs as +features [1, 35]. Graph Neural Networks (GNNs) [34, 55] +have become the model of choice in account detection due +to their excellent ability to process graph information. Li et +al. [38] first achieved stance and rumor detection through +a GNN-based architecture, which could capture user inter- +action characteristics efficiently. Although GNNs perform +well in stance mining, the lack of graph structure in exist- +ing stance detection datasets constrains the development of +graph-based detection methods. +Stance Detection Dataset. +We summarize the existing +Twitter stance detection dataset in Tab. 1. The SemEval- +2016 T6 dataset [47] is the first dataset for Twitter stance +detection, which contains topic-tweet pairs annotated by +crowdworkers. +The SemEval-2019 T7 [25] contains ru- +mors about various incidents from Reddit posts and tweets. +COVID-19-Stance [23] consists of manually annotated +tweets covering users’ stances towards four targets rele- +vant to COVID-19 health mandates. +COVIDLies [30], +COVMis-Stance [31] are also COVID-related datasets. P- +STANCE [40] is a large stance detection dataset in the polit- +ical domain collected during the 2020 U.S. elections. Con- +forti et al. [7] constructed WT-WT, a financial dataset con- +taining tweets and annotations carried out by experts. Mo- +hammad et al. [46] presented the Stance Dataset consisting +of target pairs annotated for the stance of tweeters toward +the target. +We present MGTAB, the first stance detection dataset +with user network graphs. The large-scale and high-quality +annotation of MGTAB will facilitate the development of +user stance detection. In addition, MGTAB provides op- +portunities for studying graph-based methods in stance de- +tection. + +Dataset +Samples +Annotation +Graph +Instance +Expert- +annotated +SemEval-2016 T6 +4,870 +tweet +� +� +SemEval-2019 T7 +7,730 +tweet +� +� +COVIDLies +8,937 +tweet +� +� +COVID-19-Stance +7,122 +tweet +� +� +COVMis-Stance +2,631 +tweet +� +� +WT-WT +51,284 +tweet +� +� +P-STANCE +21,574 +tweet +� +� +Stance Dataset +4,870 +tweeter +� +� +MGTAB (ours) +410,199 +tweeter +� +� +Table 1. Statistics about our benchmark versus existing stance de- +tection datasets. Compared to other datasets, MGTAB explicitly +provide the graph structure among users. +2.2. Bot Detection +The existing bot detection methods can be divided into +feature-based methods and graph-based methods. +Feature-based methods. +Feature-based methods extract +and design features from the user’s metadata and then use +traditional classifiers for bot detection. Early works [9, 53] +used simple features such as followers count, friends count, +tweets count, and creation date, etc. Some studies have used +more complex features, such as features based on social re- +lationships [11, 59]. There are also research using the fea- +tures of user tweets [29, 53]. For extracted user features, +many studies [3, 29, 33, 48, 52] use machine learning algo- +rithms for bot detection. Adaboost (AB) [28], Random For- +est (RF) [6], Decision Tree (DT) [42], and SVM [5] have +all been applied to bot detection. However, the bot may +change the registration information according to the fea- +tures designed for detection to evade feature-based detec- +tion methods [10,15]. +Graph-based methods. Graph-based methods are gener- +ally more effective than feature-based methods [15]. SA- +TAR [16] is constructed based on the social graph of Twit- +ter users in a feature engineering manner. GNNs could ex- +tract latent representation from complex relations. Inspired +by the success of GNNs, Alhosseini et al. [2] first attempt +to use Graph Convolutional Neural Networks (GCN) [34] +for spam bot detection that efficiently exploits the graph- +ical structure and relationships of Twitter accounts. Guo +et al. [26] symmetrically combine BERT and GCN, uti- +lizing text and graph-based features. +Some recent stud- +ies [4,14,18,49] investigate multiple relationships in social +graphs. BotRGCN [18] constructs a heterogeneous graph +through a user network and applies a relational graph convo- +lutional network to bot detection. RGT [14] uses relational +graph transformers to model the interaction between users +in the heterogeneous social graph. However, limited by the +lack of relationships in bot detection datasets, the previous +researches have used only two types of relationships, friend +and follower. Using multiple relationships in social graphs +for bot detection remains unexplored. +Bot detection dataset. Despite the highest quality of expert +annotation, only the Varol-icwsm is fully annotated by ex- +perts due to high costs. Most of the datasets are annotated +by crowdsourcing, while others are created using automated +techniques based on account behavior, filters on metadata, +or others more sophisticated procedures. We summarize the +existing bot detection datasets, as shown in Tab. 2. +Dataset +Users +Semantic +Expert- +Graph +annotated +Caverlee +30,316 +� +� +� +Varol-icwsm +2,228 +� +� +� +Gilani-17 +2,484 +� +� +� +Midterm-18 +50,538 +� +� +� +Cresci-stock +13,276 +� +� +� +Cresci-rtbust +693 +� +� +� +Botometer-feedback +518 +� +� +� +Kaiser +1,374 +� +� +� +Cresci-17 +14,368 +� +� +� +Cresci-15 +5,301 +� +� +� +TwiBot-20 +229,580 +� +� +� +TwiBot-22 +1,000,000 +� +� +� +MGTAB (ours) +410,199 +� +� +� +Table 2. Statistics about our benchmark versus existing bot detec- +tion datasets. Semantic represents datasets including tweets. +The Caverlee [36] consists of bot accounts lured by hon- +eypot accounts, verified human accounts, and their most re- +cent tweets. Varol-icwsm [22] dataset consists of manually +labeled Twitter accounts sampled from different Botome- +ter score deciles [54]. In Gilani-17 [21], Twitter accounts +were grouped into four categories based on the number +of followers. Apart from that, Midterm-18 [61], Cresci- +17 [10], Botometer-feedback [60], Cresci-stock [8], Cresci- +rtbust [44], Kaiser [50] are also bot detection datasets with +various annotation methods and information completeness. +Although there are many bot detection datasets, few have +graph structure. Only three publicly available bot detec- +tion datasets provide social graphs: Cresci-15 [9], TwiBot- +20 [17], and TwiBot-22 [15]. Cresci-15 and TwiBot-20 +contain only two types of relationships, friend and follower, +making it difficult to support the research of multi-relational +graph-based detection. In TwiBot-22, 1,000 manually la- +beled accounts are used to train models to get the labels of +the remaining accounts, resulting in label deviations. Our +proposed MGTAB is fully expert-annotated and has 7 types +of relationships. Compared with most previous datasets, it +has a larger scale, higher quality annotations, and richer re- +lationships. + +3. Dataset Preprocess +3.1. Data Collection and Cleaning +We adopt breadth-first search (BFS) to obtain the user +network of MGTAB, which is based on the selection of 100 +seed accounts that are closely involved in the discussion of +an online event in 2021. We collected 10,000 most recent +tweets for each user, sufficient for the account detection. +The collected data contains a total of 1,554,000 users and +135,450,000 tweets. +We first removed the noisy data and outlier nodes to con- +struct a compact graph. Specifically, users without follow- +ers or friends were removed. We then discarded users that +were not closely relevant to the target online event and even- +tually preserved 410,199 accounts and more than 40 million +tweets. +3.2. Expert Annotation +We invited 12 experts in bot detection and stance detec- +tion with more than ten years of working experience to an- +notate the user stance manually and determine if it is a bot. +To further improve the annotation quality, each Twitter user +was independently labeled by nine annotators, and annota- +tions for all users were obtained using majority voting. The +stances were labeled into three classes: neutral, against, and +support, and the categories were labeled into two types: hu- +man and bot. The annotation of the entire dataset took about +four months. The distribution of the annotation labels is +shown in Tab. 3. Following TwiBot-20, we use the remain- +ing 400,000 unlabeled users as the support set for research +on semi-supervised learning methods. +Table 3. Distribution of Labels in annotations. +Stance +Bot +Label +Count +% +Label +Count +% +neutral +3,776 +37.02 +human +7,451 +73.06 +against +3,637 +35.66 +bot +2,748 +26.94 +support +2,786 +27.32 +3.3. Quality Assessment +The remaining three experts independently randomly se- +lected 10% of users labeled to evaluate annotation quality. +We obtained 95.4% accuracy of stance and 97.8% accu- +racy of bots on average. This is well above the accuracy +obtained in previously released stance detection datasets +where crowd-sourcing was used (the accuracy reported, in +percentage, ranges from 63.7% to 79.7%) [7]. +In addi- +tion, compared to the 80% and 90.5% accuracy in TwiBot- +20 [17] and TwiBot-22 [15], our 97.8% accuracy of bots has +considerably improved annotation quality. +3.4. Feature Analysis +We randomly selected 2000 labeled users to analyze the +effectiveness of features for detection. We analyzed fea- +tures in different aspects, including creation time, friend +count, name length, etc. Following [9], we use the informa- +tion gain (IG) to measure the informativeness of a feature +to the predicting class. It can be informally defined as the +expected reduction in entropy caused by the knowledge of +the value of a given attribute. +Use Y to denote the user’s category, H(Y ) to repre- +sent the entropy of Y , and y is the value of Y , y ∈ +{y1, y2, . . . , yK}. In stance detection, K is 3, and in bot +detection, K is 2. +H(Y ) = − +K +� +k=1 +p (yk) log2 p (yk) . +(1) +H(Y | X) denotes H(Y ) when the feature X is given +and it can be computed by: +H(Y | X) = − +� +x∈Φ +px +K +� +k=1 +p (yk | x) log2 p (yi | x) , (2) +where x is the value of X, x ∈ Φ. The IG(X; Y ) in- +dicates that the category information increases (uncertainty +decreases) after Y gets feature X: +IG(X; Y ) = H(Y ) − H(Y | X). +(3) +The features with greater IG contain more information +for detection. +According to the type of features, we di- +vide the features into boolean and numeric features, and +the boolean features take the value of True or False. The +numeric features are taken the logarithm except for the cre- +ation time. Then, the data is divided into K intervals uni- +formly according to the value domain, the number of sam- +ples in each interval is counted, and then the IG is calculated +using the discrete values. In this paper, K is set to 51. +User stance features. The features with the same distribu- +tion are first removed, and then the IG of the user’s features +is calculated to obtain the boolean and numerical features +with the top 10 IG for bot detection. The boolean and nu- +merical features are shown in Fig. 1 and Fig. 2, respectively, +in decreasing order of IG. +The boolean and numerical features with the top 3 +IG were analyzed: default profile: Most users with op- +posing stances prefer to use the default profile. +de- +fault profile sidebar border color: Most users with oppos- +ing stances prefer to use the default profile’s sidebar border +color. +default profile sidebar fill color: Most users with +opposing stances prefer to use the default profile’s sidebar +color. created at: Most users with opposing stances have + +been created recently. +statues count: Users with oppos- +ing stances have a larger share of users with lower statuses. +favourites count: Among the users with lower favourites, +those who are opposed are more. +User bot features. Conducting the same processing above +to obtain the boolean and numerical features with the top 10 +IG for stance detection. The boolean and numerical features +are shown in Fig. 3 and Fig. 4, respectively, in decreasing +order of IG. +The boolean and numerical features with the top 3 IG +were analyzed: has url: Most bots have empty URL con- +tent. default profile: Compared to humans, bots tend to +use the default profile. default profile image: Most of the +users with the default background image are bots. follow- +ers friends ratios: Bots usually increase the follower count +by following each other, which leads to a smaller follow- +ers friends ratio. listed count: Bots belong to more public +lists than human users. description length: In order to mas- +querade as a human user, bots tend to fill in the account de- +scription more often and with longer descriptions than hu- +mans. +Our experiments show that the features selected are more +effective than those extracted in previous literature [18, 33, +61], the details are presented in Sec. 7.1. +4. Dataset Construction +4.1. Feature Representation Construction +We concatenate user property features and user tweet +features to serve as user feature representations, r += +[rprop∥rtweet]. The details of the user feature representa- +tions are shown in Tab. 10. +Property features extraction. User property features are +obtained based on the analysis in Sec. 3.4. The selected +numerical features are normalized by Z-score to obtain the +representation of numerical feature rnum. +The selected +boolean features are numericalized, where True and False +are replaced with 1 and 0, respectively, to obtain the rep- +resentation of boolean feature rbool. The representation of +user property features is obtained by concatenating rnum +and rbool, rprop = [rnum∥rbool]. +Tweet features extraction. +The tweets contain 54 lan- +guages, of which English is the most frequent, with a ra- +tio of 73.6%. More details are available in Sec. 7.1, and +the statistics of non-English languages are shown in Fig. 5. +It is not easy to encode multilingual tweets well using a +monolingual pre-trained BERT model. Therefore, we use +LaBSE [13], a multilingual BERT, to extract tweet features. +Specifically, We use LaBSE to encode user tweets. We av- +erage the representation of all tweets to obtain the repre- +sentation of user tweets rtweet. The demonstration of the +effectiveness encoded by LaBSE is shown in Sec. 7.2. +4.2. Relationship graph construction +The complex social graph structure, including multiple +entities such as users, tweets, hashtags, URLs, etc., makes +graph-based account detection a complex problem. Since +the focus of attention in user-level detection is on the user. +The recently proposed state-of-the-art detection methods +based on heterogeneous graphs [4, 14, 18, 49] only use the +relationship between users. Therefore, we simplified the +social network graph by keeping only users as nodes when +constructing the social graph, as shown in Fig. 6. For the +other types of entities, only the relationships between users +are constructed using them. +Explicit relationship extraction. For explicit relationships +such as follower, friend, mention, reply, and quoted, con- +nections between users are constructed directly from their +relationships. The edges constructed based on the above re- +lationships are all directed edges, as shown in Tab. 4. +Implicit relationship construction. We also extracted 2 +implicit relationships between users: URL co-occurrence +and hashtag co-occurrence. Specially, the co-occurrences +relationship between user nodes vi and vj can be deter- +mined by the probability of entities co-occurring, whose +weight is calculated through average Pointwise Mutual In- +formation (PMI): +W (vi, vj) = +1 +��Ψ{i,j} +�� +� +ek∈Ψ{i,j} +log p (vi, ek) p (vj, ek) +p (ek)2 +, +(4) +where Ψ{i,i} denotes the set of entities common to vi +and vj. Use 1/Ni approximates p (vi, ek) when calculate +PMI, where Ni denote the length of entities list of vi. Fi- +nally, we obtain the MGTAB heterogeneous graphs contain- +ing 410,199 nodes and over 100 million edges. +Table 4. Relations in the MGTAB heterogeneous graph. +Relation +Direction +Description +Source +Target +follower +user A +user B +user A is followed by user B +friend +user A +user B +user A follows user B +mention +user A +user B +user A mentions user B in tweets +reply +user A +user B +user A replies to tweet of user B +quote +user A +user B +user A quotes tweet of user B +URL +Undirected +user A and user B have the same URL +hashtag +Undirected +user A and user B have the same hashtag +5. Experiments +5.1. Experiment Settings +Datasets. +In stance detection, we evaluate models on +our proposed benchmark, SemEval-2016 T6 [47], and +SemEval-2019 T7 [25]. +In bot detection, in addition to +our proposed benchmark, we evaluate models on 4 publicly + +netural +against +support +0 +200 +400 +600 +800 +1000 +def_pf +Number + 0 + 1 +netural +against +support +0 +200 +400 +600 +800 +1000 +def_pf_sidebar_border_c +netural +against +support +0 +200 +400 +600 +800 +1000 +def_pf_sidebar_fill_c +netural +against +support +0 +200 +400 +600 +800 +1000 +has_url +netural +against +support +0 +200 +400 +600 +800 +1000 +geo_enabled +netural +against +support +0 +200 +400 +600 +800 +1000 +def_pf_bg_color +netural +against +support +0 +200 +400 +600 +800 +1000 +pf_bg_img_url +netural +against +support +0 +200 +400 +600 +800 +1000 +pf_use_bg_ima +netural +against +support +0 +200 +400 +600 +800 +1000 +def_pf_ima +netural +against +support +0 +200 +400 +600 +800 +1000 +verified +Figure 1. Distribution of boolean features with top 10 IG in stance detection. +���� +���� +���� +���� +���� +���� +��� +��� +��� +��� +��� +���������� +������� +������� +������� +� +� +�� +�� +���� +���� +���� +���� +���� +�������������� +� +� +�� +�� +���� +���� +���� +���� +���������������� +� +�� +�� +�� +��� +��� +��� +��� +������������� +� +� +� +�� +��� +��� +��� +��� +��� +������������ +� +�� +��� +��� +���� +���� +���� +���� +������������������ +� +��� +��� +���� +���� +���� +���� +������������������������ +� +� +�� +�� +�� +���� +���� +���� +���� +���� +��������������� +� +�� +�� +�� +���� +���� +���� +���� +����������� +� +�� +�� +��� +��� +��� +��� +������������������ +������ +Figure 2. Distribution of numerical features with top 10 IG in stance detection. +available bot detection datasets: Cresci-17 [10], Cresci- +15 [9], TwiBot-20 [17], and TwiBot-22 [15]. We use all +the annotated data in experiments. Following [15, 17], we +conduct a 7:2:1 random partition as training, validation, and +test set for all datasets. +Baselines. We use competitive and state-of-the-art stance +dection and bot detection methods include: +Adaboost +Classifier (AB) +[28], Decision Tree (DT) [42], Random +Forest (RF) [6], Support Vector Machines (SVM) [5], +Graph Convolutional Network (GCN) [34], Graph Atten- +tion Network (GAT) [55], Heterogeneous Graph Trans- +former (HGT) [32], Simple Heterogeneous Graph Neu- +ral Network (S-HGN) [43], Bot Detection with Relational +Graph Convolutional Networks (BotRGCN) [18], and Re- +lational Graph Transformers (RGT) [14]. +5.2. Benchmark Performance +We evaluate baselines on datasets and present their +detection accuracy and F1-score in Tab. 5. +All hyper- +parameters are listed in Sec. 7.3 for replication. +We observed that the graph-based methods performed +better than feature-based methods, all top 3 models are +graph-based. +In addition, it is obvious to observe that +heterogeneous GNNs perform better than homogeneous +GNNs. +We speculate that this is because heterogeneous +GNNs are sufficient to capture the multiple relationships +between users. RGT could model the heterogeneous in- +fluence between users, achieving the best performance on +most datasets. Better utilizing weights and directions of the +edge is a potential future research direction. +5.3. Study of Training Set Size +We select each 10% of the labeled users as the test and +validation sets. Then, we utilize different proportions of +labeled users as the training set, increasing from 10% to +80%. The graph-based model performances under different +training sets are shown in Fig. 7. +The heterogeneous GNNs’ performance is better than +homogeneous GNNs under different training sets. This phe- +nomenon is consistent with the results in Sec. 5.2. +As more annotated data is used, all detection models +become more effective. Existing account detection meth- +ods are generally supervised and rely on large amounts of +labeled data. +MGTAB’s large scale contributes to train- +ing better detection models. +In addition, MGTAB pro- +vides 400,000 unlabeled users to support the study of semi- +supervised account detection methods. To the best of our +knowledge, MGTAB has the most unlabeled users in the +account detection field. +5.4. Social Graph Relationship Analysis +In this section, we analyze the impact of using various +relationships in the MGTAB. In addition to single relation- +ships, we also experimented with using multiple relation- + +human +bot +0 +500 +1000 +1500 +has_url +Number + 0 + 1 +human +bot +0 +500 +1000 +1500 +def_pf +human +bot +0 +500 +1000 +1500 +def_pf_ima +human +bot +0 +500 +1000 +1500 +def_pf_sidebar_border_c +human +bot +0 +500 +1000 +1500 +def_pf_sidebar_fill_c +human +bot +0 +500 +1000 +1500 +geo_enabled +human +bot +0 +500 +1000 +1500 +verified +human +bot +0 +500 +1000 +1500 +pf_use_bg_ima +human +bot +0 +500 +1000 +1500 +pf_bg_img_url +human +bot +0 +500 +1000 +1500 +def_pf_bg_color +Figure 3. Distribution of boolean features with top 10 IG in bot detection. +� +��� +��� +���� +���� +���� +���� +����� ������������������ +��� +����� +� +� +�� +�� +��� +��� +��� +��� +��� +������������ +� +�� +��� +��� +���� +���� +���� +���� +������������������ +� +�� +�� +���� +���� +���� +���� +���� +����� ��������� +� +�� +�� +�� +���� +���� +���� +���� +����������� +���� +���� +���� +���� +���� +���� +���� +���� +���� +���� +���� +���������� +� +�� +�� +�� +���� +���� +���� +���� +���� +������������� +� +� +�� +�� +���� +���� +���� +���� +�������������� +� +� +�� +�� +���� +���� +���� +���� +���� +���������������� +� +�� +�� +��� +��� +��� +��� +������������������ +������! +Figure 4. Distribution of numerical features with top 10 IG in bot detection. +Figure 5. Non-English tweets and their percentage. +ships. We randomly conduct a 1:1:8 partition as training, +validation, and test set. This partition is shared across all +experiments in Sec. 7.1 and Sec. 7.2. +Tab. 6 illustrates that graph-based account detection +methods perform better when more relationships are used. +This trend suggests that future research in account detec- +tion should focus on better utilizing multiple relationships +between users. +Besides, we observed that hashtag co- +occurrence has the worst performance of all the relation- +ships. We suspect this is because hashtag co-occurrence is +highly random, and two unrelated users can have hashtag +co-occurrence. Although MGTAB provides edge weights +for URL and hashtag co-occurrence relationships, existing +graph-based account detection models cannot fully exploit +them, leading to bad performance. +6. Conclusion +We presented MGTAB, a large-scale dataset for stance +detection and bot detection. +We used expert annotation +and majority voting to ensure high-quality annotations. To +build the standardized dataset, we selected 20 user features +with the highest information gain, which was experimen- +tally demonstrated most effective. +We extracted 7 types +of relationships between users and simplified the complex +Twitter network. Compared to previous datasets, MGTAB +can better support the study of graph-based account detec- +tion methods. Our experiments found that graph-based ap- +proaches are generally more effective than feature-based ap- +proaches and perform better when introducing multiple re- +lationships. +References +[1] Abeer Aldayel and Walid Magdy. Your stance is exposed! +analysing possible factors for stance detection on social me- +dia. Proceedings of the ACM on Human-Computer Interac- +tion, 3:1–20, 2019. 1, 2 +[2] Seyed Ali Alhosseini, Raad Bin Tareaf, Pejman Najafi, and +Christoph Meinel. Detect me if you can: Spam bot detection +using inductive representation learning. +Companion Pro- + +German:0.014 +-Hindi:0.015 +Spanish:0.031 +Vietnamese:0.042 +Turkish:0.326 +Japanese:0.043 +French:0.058 +Arabic:0.059 +Chinese:0.076 +Others:0.082 +Italian:0.153- +Indonesian:0.101Figure 6. We simplify the original complex heterogeneous graph network (left) and construct a user-level multi-graph network (right). +Black, red, and green denote neutral, against, and support. +Table 5. Performance of baseline methods on datasets. Use the most commonly used follower and friend relationships during evaluation. +Each baseline is conducted five times with different seeds, and we report the average performance and standard deviation. “/” indicates +that the dataset does not contain user relationships to support the grah-based methods. Best and second best results are highlighted in bold +and underline. +Task +Dataset +Metric +Methods +Feature-based +Graph-based +Homogeneous +Heterogeneous +AB +RF +DT +SVM +GCN +GAT +HGT +S-HGN +BotRGCN +RGT +Stance +SemEval-2016 +Acc +74.2±0.4 +72.8±0.3 +72.2±0.3 +76.1±0.3 +/ +/ +/ +/ +/ +/ +F1 +72.1±0.3 +70.3±0.3 +69.2±0.3 +72.3±0.3 +/ +/ +/ +/ +/ +/ +SemEval-2019 +Acc +84.0±0.5 +83.8±0.4 +83.1±0.4 +84.2±0.4 +/ +/ +/ +/ +/ +/ +F1 +64.4±0.4 +63.6±0.3 +64.1±0.4 +65.2±0.4 +/ +/ +/ +/ +/ +/ +MGTAB +Acc +74.6±1.4 +79.6±0.7 +66.9±0.9 +81.2±0.7 +82.4±0.9 +82.2±1.2 +83.2±0.4 +85.3±0.5 +84.7±1.5 +87.8±0.4 +F1 +73.9±1.5 +79.0±0.8 +66.0±0.8 +80.7±0.8 +81.5±0.9 +81.0±1.2 +81.8±0.3 +84.4±0.4 +84.3±1.4 +86.9±0.4 +Bot +Cresci-17 +Acc +91.2±0.2 +89.1±0.2 +86.2±0.2 +84.1±0.3 +/ +/ +/ +/ +/ +/ +F1 +83.4±0.2 +80.9±0.2 +76.4±0.2 +72.8±0.3 +/ +/ +/ +/ +/ +/ +Cresci-15 +Acc +95.9±0.3 +97.0±0.8 +96.2±1.3 +96.6±0.2 +98.2±0.6 +98.1±0.2 +98.4±0.3 +97.5±0.5 +98.5±0.4 +98.6±0.3 +F1 +95.5±0.3 +96.7±0.9 +95.9±1.4 +96.3±0.3 +98.0±0.4 +98.0±0.1 +98.4±0.3 +97.2±0.5 +97.3±0.5 +98.5±0.2 +TwiBot-20 +Acc +85.7±0.4 +85.0±0.5 +80.1±0.5 +85.2±0.3 +77.2±1.2 +83.2±0.4 +85.9±0.6 +85.4±0.3 +86.8±0.5 +86.9±0.3 +F1 +85.6±0.4 +84.9±0.5 +80.0±0.5 +84.8±0.4 +76.6±0.4 +81.9±0.5 +85.6±0.6 +85.3±0.2 +86.6±0.4 +86.7±0.4 +TwiBot-22 +Acc +69.3±0.5 +74.3±0.7 +72.6±0.8 +76.4±0.9 +78.3±1.3 +79.3±0.8 +74.9±1.2 +76.7±1.3 +79.6±0.4 +76.5±0.4 +F1 +34.8±0.5 +30.4±0.6 +51.6±0.6 +54.6±0.8 +54.8±1.0 +55.6±1.1 +39.2±1.6 +45.7±0.5 +57.6±1.4 +43.1±0.5 +MGTAB +Acc +90.1±0.9 +89.5±0.4 +87.1±0.5 +88.7+1.4 +85.8±1.3 +87.0±1.3 +90.3±0.3 +91.4+0.4 +89.6±0.8 +92.1+0.4 +F1 +87.7±1.1 +86.8±0.5 +83.7±0.7 +85.3+1.7 +78.3±1.7 +82.3±2.1 +87.5±0.4 +88.7+0.6 +87.2±0.7 +90.4±0.5 +ceedings of The 2019 World Wide Web Conference, 2019. +3 +[3] Md. 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Each baseline is conducted five times with +different seeds, and we report the average performance and standard deviation. Best results are highlighted in bold. +Task +Method +Relation +Single relation +Multiple relations +1:follower +2:friend +3:mention +4:reply +5:quoted +6:hashtag +7:url +1+2 +3+4+5 +1+2+3+4+5+6 +stance +GCN +76.7±0.6 +76.9±0.6 +76.8±0.7 +75.9±1.0 +76.9±0.6 +60.8±1.2 +76.2±0.4 +78.1±0.6 +77.1±0.5 +79.1±0.3 +GAT +77.0±0.5 +76.7±0.5 +78.0±0.4 +76.8±0.2 +77.3±0.3 +64.4±1.1 +77.1±0.4 +77.8±0.4 +77.3±0.4 +77.9±0.4 +BotRGCN +79.1±0.3 +76.1±0.4 +76.2±0.5 +76.8±0.6 +77.0±0.2 +77.2±0.2 +76.8±0.5 +79.4±0.2 +78.0±0.4 +79.2±0.5 +S-HGN +81.2±0.2 +80.8±0.2 +79.4±0.2 +78.5±0.2 +78.6±0.3 +75.6±0.3 +80.3±0.3 +81.6±0.2 +80.0±0.2 +81.7±0.2 +HGT +79.1±0.1 +79.6±0.2 +77.4±0.2 +77.4±0.2 +77.7±0.2 +78.1±0.3 +77.0±0.2 +79.2±0.2 +77.9±0.2 +78.7±0.1 +bot +GCN +81.2±0.5 +84.1±0.7 +84.6±0.3 +85.2±0.6 +85.5±0.6 +76.3±1.2 +81.5±0.4 +83.6±0.5 +86.2±0.4 +82.5±0.5 +GAT +81.2±1.5 +83.0±1.6 +83.3±2.0 +83.2±0.2 +82.9±0.3 +73.6±1.6 +79.7±0.4 +84.4±0.8 +86.4±0.3 +78.4±0.9 +BotRGCN +83.5±0.5 +83.2±0.3 +82.9±0.2 +83.2±0.3 +82.9±0.5 +83.1±0.3 +82.4±0.3 +86.9±0.2 +86.6±0.3 +87.2±0.2 +S-HGN +87.5±0.3 +87.3±0.3 +87.3±0.3 +86.8±0.3 +86.9±0.2 +86.9±0.3 +86.8±0.2 +87.3±0.4 +87.4±0.3 +87.9±0.2 +HGT +87.1±0.3 +87.4±0.4 +86.5±0.4 +86.4±0.4 +86.6±0.3 +85.6±0.3 +85.7±0.2 +87.5±0.2 +86.4±0.0 +87.2±0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +0.80 +0.82 +0.84 +0.86 +0.88 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +0.76 +0.78 +0.80 +0.82 +0.84 +0.86 +Bot +F1-score +Percentage + GCN + GAT + RGCN + S-HGN + HGT + RGT +Stance +F1-score +Percentage +Figure 7. 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Arming +the public with artificial intelligence to counter social bots. +Human Behavior and Emerging Technologies, 2019. 1, 3 +[61] Kai-Cheng Yang, Onur Varol, Pik-Mai Hui, and Filippo +Menczer. +Scalable and generalizable social bot detection +through data selection. In AAAI, 2020. 3, 5, 12, 13 +[62] Guido Zarrella and Amy Marsh. Mitre at semeval-2016 task +6: Transfer learning for stance detection. In *SEMEVAL, +2016. 2 + +7. Supplementary +7.1. Feature Analysis +Information gain of features. Boolean and numerical fea- +tures with top 10 IG in user stance detection and their IG +are shown in Tab. 7. +Table 7. User stance detection features and IG. Features are listed +in descending order of IG. +Features +IG +Type +def pf +0.044927 +Boolean +def pf sidebar border c +0.041702 +def pf sidebar fill c +0.035676 +has URL +0.019093 +geo enabled +0.018741 +def pf bg color +0.016575 +pf bg img URL +0.016373 +pf use bg ima +0.013501 +def pf ima +0.010923 +verified +0.001304 +created at +0.107146 +Numerical +statuses count +0.101377 +favourites count +0.070515 +friends count +0.064378 +listed count +0.063106 +description length +0.062262 +followers friends ratios +0.056947 +followers count +0.055433 +name length +0.054854 +screen name length +0.015331 +Boolean and numerical features with top 10 IG in bot +detection and their IG are shown in Tab. 8. +Feature effectiveness analysis. The details of the user fea- +ture representations are shown in Tab. 10. Many proposed +works in the literature addressed different features for ac- +count detection. To further demonstrate the effectiveness +of the features extracted in this paper, property features de- +signed from different literatures [18,33,61] are used to com- +pare the performance of different models under the most +commonly used friend and follower relationships [18]. In +the experiment, we only use property features, and the re- +sults are shown in Tab. 11. +7.2. Impact of Different BERT Models +The 54 languages included in the MGTAB dataset are +shown in Table 9. +To demonstrate the effectiveness of +encoding using LaBSE [13], in this section, we adopt +four pre-trained encoding models, LaBSE, RoBERTa [41], +SBERT [51],and BART [37] to encode user tweets. The re- +sults using the above models to encode all tweets of users +are shown in Tab. 12. The detection performance of using +LaBSE is better compared to other models. We infer that +this is because noise will introduced when encoding multi- +lingual text using the English pre-training model. LaBSE +Table 8. Bot detection features and IG. Features are listed in de- +scending order of IG. +Features +IG +Type +has URL +0.064248 +Boolean +def pf +0.025997 +def pf ima +0.025402 +def pf sidebar border c +0.023105 +def pf sidebar fill c +0.022359 +geo enabled +0.019302 +verified +0.010902 +pf use bg ima +0.007877 +pf bg img URL +0.005923 +def pf bg color +0.005841 +followers friends ratios +0.391857 +Numerical +listed count +0.333101 +description length +0.194765 +followers count +0.176186 +name length +0.040335 +created at +0.034079 +friends count +0.031598 +statuses count +0.015544 +favourites count +0.01176 +screen name length +0.007641 +Table 9. Language that appears in MGTAB (ISO 639-1/639-2). +ISO +Name +ISO +Name +af +AFRIKAANS +lv +LATVIAN +ar +ARABIC +mk +MACEDONIAN +bg +BULGARIAN +ml +MALAYALAM +bn +BENGALI +mr +MARATHI +ca +CATALAN +ne +NEPALI +cs +CZECH +nl +DUTCH +cy +WELSH +no +NORWEGIAN +da +DANISH +pa +PUNJABI +de +GERMAN +pl +POLISH +el +GREEK +pt +PORTUGUESE +en +ENGLISH +ro +ROMANIAN +es +SPANISH +ru +RUSSIAN +et +ESTONIAN +sk +SLOVAK +fa +PERSIAN +sl +SLOVENIAN +fi +FINNISH +so +SOMALI +fr +FRENCH +sq +ALBANIAN +gu +GUJARATI +sv +SWEDISH +he +HEBREW +sw +SWAHILI +hi +HINDI +ta +TAMIL +hr +CROATIAN +te +TELUGU +hu +HUNGARIAN +th +THAI +id +INDONESIAN +tl +TAGALOG +it +ITALIAN +tr +TURKISH +ja +JAPANESE +uk +UKRAINIAN +kn +KANNADA +ur +URDU +ko +KOREAN +vi +VIETNAMESE +lt +LITHUANIAN +zh +CHINESE +can encode text in different languages into a shared embed- +ding space, better suited to the multilingual text collected in + +Table 10. Details of user feature representations. +Features +Description +Type +Dim +profile use background image +If profile has background image +Boolean +1 +default profile +If profile is set +Boolean +2 +verified +If profile is verified +Boolean +3 +followers count +Number of uers following this account +Numerical +4 +default profile image +If profile image is default +Boolean +5 +listed count +Public lists that use members of +Numerical +6 +statuses count +Numbers of tweets and retweets +Numerical +7 +friends count +Number of uers this account following +Numerical +8 +geo enabled +Whether to enable geographical location +Boolean +9 +favourites count +Number of this account likes +Numerical +10 +created at +Time when the account was created +Numerical +11 +screen name length +Length of screen name +Numerical +12 +name length +Length of name +Numerical +13 +description length +Length of description +Numerical +14 +followers friends ratios +followers count/friends count +Numerical +15 +default profile background color +If the profile background uses default color +Boolean +16 +default profile sidebar fill color +If the profile sidebar uses default color +Boolean +17 +default profile sidebar border color +If the border of profile sidebar uses default color +Boolean +18 +has URL +If URL is set +Boolean +19 +profile background image URL +If the profile background image has URL +Boolean +20 +tweet features +Averaged 768-dimensional features +Tweet +21-788 +Table 11. The performance of using different features on MGTAB. Best results are highlighted in bold. +Task +Method +Features +[18] +[61] +[33] +Ours +Acc +F1 +Acc +F1 +Acc +F1 +Acc +F1 +Stance +GCN +63.8±0.1 +63.0±0.4 +61.1±0.5 +60.0±0.5 +62.7±0.7 +61.5±1.2 +65.2±0.5 +64.9±0.4 +GAT +70.2±0.2 +69.1±0.5 +70.3±0.1 +69.5±0.3 +70.7±0.5 +69.9±0.5 +70.9±0.6 +70.1±0.5 +RGCN +64.1±0.4 +63.2±0.4 +62.8±0.5 +61.2±0.6 +65.5±0.2 +64.4±0.5 +65.8±0.3 +64.9±0.5 +Bot +GCN +71.1±0.5 +65.8±0.8 +71.7±0.4 +66.8±0.6 +78.0±0.6 +68.9±0.5 +79.5±0.2 +71.1±0.1 +GAT +68.1±0.2 +61.0±0.3 +68.5±0.4 +63.7±0.6 +73.0±0.4 +69.8±0.4 +73.7±0.3 +71.5±0.4 +RGCN +73.5±0.1 +61.0±0.7 +75.0±0.3 +67.7±0.2 +77.2±0.3 +70.1±0.4 +79.4±0.3 +74.0±0.4 +this benchmark. +7.3. Experiment Details +Experiment setup. In this paper, for all GNN models, we +stack 2-layer GNNs and two fully connected layers, and the +input and output dimensions of the middle GNN layers are +consistent, which are 64, 128, or 256. We use ReLU as the +activation function and set the learning rate from 0.0001 to +0.01. In addition, the dropout rate ranges from 0.3 to 0.5. +We set the number of attention heads as 8 in GAT. We set +the number of transformer attention heads and semantic at- +tention heads as 4 in RGT. In S-HGN, β is 0.05, and the rest +remain at the default settings. We trained all GNN models +for 300 epochs using Adam optimizer. For machine learn- +ing models, the number of estimators for AB and RF is set +to 50 and 100, respectively. We ran all experiments on a +server with 9 TITAN RTX GPUs. +Datasets processing. For SemEval-2016 T6 [47], we ex- +tracted 20 largest features of IG: number of positive words, +number of negative words, positive emotion counts, neg- +ative emotion counts, nouns words frequency, pronoun +words frequency, verb words frequency, adjectives words +frequency, number of special symbols, number of question +mark, number of capital words, number of quoted words, +retweet counts, mention counts, number of URLs, entropy +of hastags, number of hashtags, and number of capitalized +hashtags. For SemEval-2019 T7 [25], the feature was ex- +tracted by using RoBERTa [41]. For TwiBot-20 [17], we +follow [18] for dataset processing and feature extraction. +For Cresci-15 [9], Cresci-17 [10], and TwiBot-22 [15], we +follow [15] for dataset processing and feature extraction. + +Table 12. Performance using different encoding models on MGTAB. Best results are highlighted in bold. +Task +Method +pretrain model +RoBERTa +SBERT +BART +Lasbe +Acc +F1 +Acc +F1 +Acc +F1 +Acc +F1 +Stance +SVM +66.1±0.4 +64.7±0.4 +76.2±0.5 +75.3±0.5 +77.5±0.3 +76.8±0.2 +78.2±0.2 +77.7±0.2 +DT +59.4±0.4 +58.5±0.4 +61.1±0.5 +60.4±0.5 +61.4±0.3 +60.7±0.3 +62.0±0.2 +61.0±0.1 +RF +70.8±0.5 +69.7±0.4 +76.4±0.5 +76.1±0.5 +75.9±0.3 +75.2±0.2 +77.0±0.2 +76.2±0.2 +GCN +74.5±0.8 +73.9±0.9 +78.4±0.2 +77.8±0.1 +78.9±0.2 +78.4±0.4 +78.6±0.1 +77.9±0.2 +GAT +75.7±0.4 +75.0±0.6 +77.4±0.2 +77.0±0.2 +77.4±0.2 +77.0±0.3 +77.6±0.3 +77.1±0.4 +BotRGCN +76.4±0.4 +75.5±0.5 +77.9±0.3 +77.5±0.2 +79.0±0.1 +78.6±0.2 +79.2±0.1 +78.7±0.2 +Bot +SVM +82.0±0.4 +74.1±0.4 +86.1±0.4 +81.7±0.3 +85.3±0.2 +81.2±0.2 +86.2±0.2 +82.5±0.1 +DT +83.3±0.4 +78.6±0.3 +82.7±0.4 +78.0±0.3 +84.5±0.2 +80.2±0.2 +85.4±0.2 +81.3±0.2 +RF +83.8±0.4 +77.1±0.3 +84.1±0.5 +78.3±0.4 +85.1±0.2 +81.0±0.1 +87.0±0.3 +82.9±0.3 +GCN +83.4±0.2 +77.5±0.4 +83.8±0.6 +79.2±0.3 +84.2±0.2 +79.0±0.4 +84.9±0.4 +79.5±1.2 +GAT +78.7±0.7 +73.6±0.9 +83.9±0.4 +78.1±0.8 +82.4±1.1 +78.0±1.1 +85.3±0.4 +79.5±1.3 +BotRGCN +84.3±0.4 +79.3±0.7 +86.1±0.1 +81.5±0.1 +85.7±0.1 +80.9±0.3 +87.2±0.1 +83.2±0.3 + diff --git a/d9AzT4oBgHgl3EQfL_tB/content/tmp_files/load_file.txt b/d9AzT4oBgHgl3EQfL_tB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbceb073d1ebb5f68da1a852295dc807d855c3f8 --- /dev/null +++ b/d9AzT4oBgHgl3EQfL_tB/content/tmp_files/load_file.txt @@ -0,0 +1,2060 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf,len=2059 +page_content='MGTAB: A Multi-Relational Graph-Based Twitter Account Detection Benchmark Shuhao Shi1,†, Kai Qiao1,†, Jian Chen1 Shuai Yang1, Jie Yang1, Baojie Song1, Linyuan Wang1, Bin Yan1,∗ 1Henan Key Laboratory of Imaging and Intelligence Processing, PLA strategy support force information engineering university Abstract The development of social media user stance detection and bot detection methods rely heavily on large-scale and high-quality benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' However, in addition to low an- notation quality, existing benchmarks generally have in- complete user relationships, suppressing graph-based ac- count detection research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To address these issues, we pro- pose a Multi-Relational Graph-Based Twitter Account De- tection Benchmark (MGTAB), the first standardized graph- based benchmark for account detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To our knowl- edge, MGTAB was built based on the largest original data in the field, with over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='55 million users and 130 million tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' MGTAB contains 10,199 expert-annotated users and 7 types of relationships, ensuring high-quality annota- tion and diversified relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In MGTAB, we extracted the 20 user property features with the greatest information gain and user tweet features as the user features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In addition, we performed a thorough evaluation of MGTAB and other public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Our experiments found that graph-based approaches are generally more effective than feature-based approaches and perform better when introducing multiple relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' By analyzing experiment results, we identify ef- fective approaches for account detection and provide poten- tial future research directions in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Our benchmark and standardized evaluation procedures are freely available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='com/GraphDetec/MGTAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Introduction With the continuous development of the Internet, social networks have become an essential part of people’s daily social life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Twitter is one of the most visited social net- works worldwide, providing online news and information exchange to billions of users worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Due to the avail- ability, many account detection benchmarks are constructed based on Twitter data [9,15,17,47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Stance detection and bot detection are essential tasks in account detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Stance detection aims at detecting the user’s stance on a topic or claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' It is a critical technique in applications such as fake news detection [25,31], claims validation [1, 27], and analyzing public opinion on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Bot detection is crucial for detecting information manipulation on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Social bots are automated user accounts operated by computer programs [60] and are often used to abuse social media platforms [10, 19] to ma- nipulate public opinion [9–11,60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Most account detection methods use only part of the in- formation in social media, such as posts, registration infor- mation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=', for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Rarely consider the connec- tion between users [24], making it challenging to ensure detection accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In stance detection, silent users often do not post directly but express their stance through be- haviors, such as following others and favoring posts [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' However, most studies focus only on the posting content of active users and ignore silent users [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The features of social graphs need to be used to detect the silent users’ stance better [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In bot detection, since most studies ignore bots’ social graph features, bots can simulate genuine users through complex strategies to evade feature-based detection methods [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Recent work [14, 18, 38] in account detection has fo- cused on exploiting relationships between users, with performance improvement compared to feature-based ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' However, the existing datasets have several draw- backs to supporting graph-based methods, as follows: (a) Low annotation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Previous account detec- tion datasets were mostly annotated by crowdsourc- ing, while crowdworkers’ lack of domain knowledge resulted in significant noise in the annotation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' (b) Incomplete user relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' None of the stance detection datasets explicitly provide the graph structure among users, and only the bot detection datasets Cresci-15 [9], TwiBot-20 [17], and TwiBot- 22 [15] contain explicit graph structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Moreover, Cresci-15 and TwiBot-20 contain only 2 types of user arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='01123v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='CV] 3 Jan 2023 relationships, which is insufficient for graph-based de- tection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' (c) Complex user information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Social media user in- formation is diverse and voluminous, but most infor- mation has little effect on account detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Existing datasets lack the extraction and organization of essen- tial user information, making account detection a dif- ficult problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To address the shortcomings above, we presented Multi- Relational Graph-Based Twitter Account Detection Bench- mark (MGTAB), a large standardized expert-annotated dataset for stance and bot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' MGTAB contains 10,199 users manually annotated by experts and 400,000 closely related unannotated users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Further, MGTAB ex- tracted 20 most effective user property features by calcu- lating the information gain (IG) and user tweet features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Fi- nally, MGTAB simplified the social graph and constructed a user network with 7 types of relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The contribu- tions of this paper are as follows: We presented MGTAB, a large-scale expert-annotated benchmark for stance detection and bot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' All annotations are carried out by experts and improve an- notation quality by cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The annotation quality has been substantially improved compared to the previous dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We released the first standardized dataset containing the property features, user tweet features and 7 types of user relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We constructed a user-level so- cial graph that can be applied to state-of-the-art graph- based account detection methods, making account de- tection simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The release of the MGTAB dataset will facilitate the development of new methods for graph-based account detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To build MGTAB, we collect over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='55 million Twit- ter users and 135 million tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To the best of our knowledge, it is the biggest data in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We carried out meticulous data cleaning, retaining 400,000 closely related unlabeled users, which supports semi- supervised learning to merge with account detection research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Our experiments show that graph-based detection methods are more effective than feature-based meth- ods in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In addition, We found that the per- formance of graph-based approaches improved when multiple relationships were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The results suggest that future research should focus on using mul- tiple relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Stance Detection The existing stance detection methods can be divided into feature-based methods and graph-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Feature-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Previous research works [56, 58, 62] used machine learning algorithms and deep learn- ing methods such as Support Vector Machines (SVM), Re- current Neural Networks (RNNs) [62], and Convolutional Neural Networks (CNNs) to automatically learn latent fea- tures from a large amount of raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Several recent works [31, 39, 40, 45, 57] focused on the use of bidirectional en- coder representations from transformers (BERT) [12] on stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' [20] explored stance detec- tion based on transfer learning, and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' [39] explored BERT-based data augmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Graph-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Most studies on stance detection use text-based features [40,47,62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Some recent work has shown the effectiveness of using user network graphs as features [1, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Graph Neural Networks (GNNs) [34, 55] have become the model of choice in account detection due to their excellent ability to process graph information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' [38] first achieved stance and rumor detection through a GNN-based architecture, which could capture user inter- action characteristics efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Although GNNs perform well in stance mining, the lack of graph structure in exist- ing stance detection datasets constrains the development of graph-based detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Stance Detection Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We summarize the existing Twitter stance detection dataset in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The SemEval- 2016 T6 dataset [47] is the first dataset for Twitter stance detection, which contains topic-tweet pairs annotated by crowdworkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The SemEval-2019 T7 [25] contains ru- mors about various incidents from Reddit posts and tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' COVID-19-Stance [23] consists of manually annotated tweets covering users’ stances towards four targets rele- vant to COVID-19 health mandates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' COVIDLies [30], COVMis-Stance [31] are also COVID-related datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' P- STANCE [40] is a large stance detection dataset in the polit- ical domain collected during the 2020 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Con- forti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' [7] constructed WT-WT, a financial dataset con- taining tweets and annotations carried out by experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Mo- hammad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' [46] presented the Stance Dataset consisting of target pairs annotated for the stance of tweeters toward the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We present MGTAB, the first stance detection dataset with user network graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The large-scale and high-quality annotation of MGTAB will facilitate the development of user stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In addition, MGTAB provides op- portunities for studying graph-based methods in stance de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Dataset Samples Annotation Graph Instance Expert- annotated SemEval-2016 T6 4,870 tweet � � SemEval-2019 T7 7,730 tweet � � COVIDLies 8,937 tweet � � COVID-19-Stance 7,122 tweet � � COVMis-Stance 2,631 tweet � � WT-WT 51,284 tweet � � P-STANCE 21,574 tweet � � Stance Dataset 4,870 tweeter � � MGTAB (ours) 410,199 tweeter � � Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Statistics about our benchmark versus existing stance de- tection datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Compared to other datasets, MGTAB explicitly provide the graph structure among users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Bot Detection The existing bot detection methods can be divided into feature-based methods and graph-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Feature-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Feature-based methods extract and design features from the user’s metadata and then use traditional classifiers for bot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Early works [9, 53] used simple features such as followers count, friends count, tweets count, and creation date, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Some studies have used more complex features, such as features based on social re- lationships [11, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' There are also research using the fea- tures of user tweets [29, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' For extracted user features, many studies [3, 29, 33, 48, 52] use machine learning algo- rithms for bot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Adaboost (AB) [28], Random For- est (RF) [6], Decision Tree (DT) [42], and SVM [5] have all been applied to bot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' However, the bot may change the registration information according to the fea- tures designed for detection to evade feature-based detec- tion methods [10,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Graph-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Graph-based methods are gener- ally more effective than feature-based methods [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' SA- TAR [16] is constructed based on the social graph of Twit- ter users in a feature engineering manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' GNNs could ex- tract latent representation from complex relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Inspired by the success of GNNs, Alhosseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' [2] first attempt to use Graph Convolutional Neural Networks (GCN) [34] for spam bot detection that efficiently exploits the graph- ical structure and relationships of Twitter accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' [26] symmetrically combine BERT and GCN, uti- lizing text and graph-based features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Some recent stud- ies [4,14,18,49] investigate multiple relationships in social graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' BotRGCN [18] constructs a heterogeneous graph through a user network and applies a relational graph convo- lutional network to bot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' RGT [14] uses relational graph transformers to model the interaction between users in the heterogeneous social graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' However, limited by the lack of relationships in bot detection datasets, the previous researches have used only two types of relationships, friend and follower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Using multiple relationships in social graphs for bot detection remains unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Bot detection dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Despite the highest quality of expert annotation, only the Varol-icwsm is fully annotated by ex- perts due to high costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Most of the datasets are annotated by crowdsourcing, while others are created using automated techniques based on account behavior, filters on metadata, or others more sophisticated procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We summarize the existing bot detection datasets, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Dataset Users Semantic Expert- Graph annotated Caverlee 30,316 � � � Varol-icwsm 2,228 � � � Gilani-17 2,484 � � � Midterm-18 50,538 � � � Cresci-stock 13,276 � � � Cresci-rtbust 693 � � � Botometer-feedback 518 � � � Kaiser 1,374 � � � Cresci-17 14,368 � � � Cresci-15 5,301 � � � TwiBot-20 229,580 � � � TwiBot-22 1,000,000 � � � MGTAB (ours) 410,199 � � � Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Statistics about our benchmark versus existing bot detec- tion datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Semantic represents datasets including tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The Caverlee [36] consists of bot accounts lured by hon- eypot accounts, verified human accounts, and their most re- cent tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Varol-icwsm [22] dataset consists of manually labeled Twitter accounts sampled from different Botome- ter score deciles [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In Gilani-17 [21], Twitter accounts were grouped into four categories based on the number of followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Apart from that, Midterm-18 [61], Cresci- 17 [10], Botometer-feedback [60], Cresci-stock [8], Cresci- rtbust [44], Kaiser [50] are also bot detection datasets with various annotation methods and information completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Although there are many bot detection datasets, few have graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Only three publicly available bot detec- tion datasets provide social graphs: Cresci-15 [9], TwiBot- 20 [17], and TwiBot-22 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Cresci-15 and TwiBot-20 contain only two types of relationships, friend and follower, making it difficult to support the research of multi-relational graph-based detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In TwiBot-22, 1,000 manually la- beled accounts are used to train models to get the labels of the remaining accounts, resulting in label deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Our proposed MGTAB is fully expert-annotated and has 7 types of relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Compared with most previous datasets, it has a larger scale, higher quality annotations, and richer re- lationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Dataset Preprocess 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Data Collection and Cleaning We adopt breadth-first search (BFS) to obtain the user network of MGTAB, which is based on the selection of 100 seed accounts that are closely involved in the discussion of an online event in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We collected 10,000 most recent tweets for each user, sufficient for the account detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The collected data contains a total of 1,554,000 users and 135,450,000 tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We first removed the noisy data and outlier nodes to con- struct a compact graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Specifically, users without follow- ers or friends were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We then discarded users that were not closely relevant to the target online event and even- tually preserved 410,199 accounts and more than 40 million tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Expert Annotation We invited 12 experts in bot detection and stance detec- tion with more than ten years of working experience to an- notate the user stance manually and determine if it is a bot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To further improve the annotation quality, each Twitter user was independently labeled by nine annotators, and annota- tions for all users were obtained using majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The stances were labeled into three classes: neutral, against, and support, and the categories were labeled into two types: hu- man and bot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The annotation of the entire dataset took about four months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The distribution of the annotation labels is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Following TwiBot-20, we use the remain- ing 400,000 unlabeled users as the support set for research on semi-supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Distribution of Labels in annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Stance Bot Label Count % Label Count % neutral 3,776 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='02 human 7,451 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='06 against 3,637 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='66 bot 2,748 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='94 support 2,786 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Quality Assessment The remaining three experts independently randomly se- lected 10% of users labeled to evaluate annotation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We obtained 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4% accuracy of stance and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='8% accu- racy of bots on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' This is well above the accuracy obtained in previously released stance detection datasets where crowd-sourcing was used (the accuracy reported, in percentage, ranges from 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='7% to 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='7%) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In addi- tion, compared to the 80% and 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5% accuracy in TwiBot- 20 [17] and TwiBot-22 [15], our 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='8% accuracy of bots has considerably improved annotation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Feature Analysis We randomly selected 2000 labeled users to analyze the effectiveness of features for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We analyzed fea- tures in different aspects, including creation time, friend count, name length, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Following [9], we use the informa- tion gain (IG) to measure the informativeness of a feature to the predicting class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' It can be informally defined as the expected reduction in entropy caused by the knowledge of the value of a given attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Use Y to denote the user’s category, H(Y ) to repre- sent the entropy of Y , and y is the value of Y , y ∈ {y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' , yK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In stance detection, K is 3, and in bot detection, K is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' H(Y ) = − K � k=1 p (yk) log2 p (yk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' (1) H(Y | X) denotes H(Y ) when the feature X is given and it can be computed by: H(Y | X) = − � x∈Φ px K � k=1 p (yk | x) log2 p (yi | x) , (2) where x is the value of X, x ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The IG(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Y ) in- dicates that the category information increases (uncertainty decreases) after Y gets feature X: IG(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Y ) = H(Y ) − H(Y | X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' (3) The features with greater IG contain more information for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' According to the type of features, we di- vide the features into boolean and numeric features, and the boolean features take the value of True or False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The numeric features are taken the logarithm except for the cre- ation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Then, the data is divided into K intervals uni- formly according to the value domain, the number of sam- ples in each interval is counted, and then the IG is calculated using the discrete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In this paper, K is set to 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' User stance features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The features with the same distribu- tion are first removed, and then the IG of the user’s features is calculated to obtain the boolean and numerical features with the top 10 IG for bot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The boolean and nu- merical features are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 2, respectively, in decreasing order of IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The boolean and numerical features with the top 3 IG were analyzed: default profile: Most users with op- posing stances prefer to use the default profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' de- fault profile sidebar border color: Most users with oppos- ing stances prefer to use the default profile’s sidebar border color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' default profile sidebar fill color: Most users with opposing stances prefer to use the default profile’s sidebar color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' created at: Most users with opposing stances have been created recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' statues count: Users with oppos- ing stances have a larger share of users with lower statuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' favourites count: Among the users with lower favourites, those who are opposed are more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' User bot features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Conducting the same processing above to obtain the boolean and numerical features with the top 10 IG for stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The boolean and numerical features are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 4, respectively, in decreasing order of IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The boolean and numerical features with the top 3 IG were analyzed: has url: Most bots have empty URL con- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' default profile: Compared to humans, bots tend to use the default profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' default profile image: Most of the users with the default background image are bots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' follow- ers friends ratios: Bots usually increase the follower count by following each other, which leads to a smaller follow- ers friends ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' listed count: Bots belong to more public lists than human users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' description length: In order to mas- querade as a human user, bots tend to fill in the account de- scription more often and with longer descriptions than hu- mans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Our experiments show that the features selected are more effective than those extracted in previous literature [18, 33, 61], the details are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Dataset Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Feature Representation Construction We concatenate user property features and user tweet features to serve as user feature representations, r = [rprop∥rtweet].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The details of the user feature representa- tions are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Property features extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' User property features are obtained based on the analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The selected numerical features are normalized by Z-score to obtain the representation of numerical feature rnum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The selected boolean features are numericalized, where True and False are replaced with 1 and 0, respectively, to obtain the rep- resentation of boolean feature rbool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The representation of user property features is obtained by concatenating rnum and rbool, rprop = [rnum∥rbool].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Tweet features extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The tweets contain 54 lan- guages, of which English is the most frequent, with a ra- tio of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' More details are available in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1, and the statistics of non-English languages are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' It is not easy to encode multilingual tweets well using a monolingual pre-trained BERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Therefore, we use LaBSE [13], a multilingual BERT, to extract tweet features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Specifically, We use LaBSE to encode user tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We av- erage the representation of all tweets to obtain the repre- sentation of user tweets rtweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The demonstration of the effectiveness encoded by LaBSE is shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Relationship graph construction The complex social graph structure, including multiple entities such as users, tweets, hashtags, URLs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=', makes graph-based account detection a complex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Since the focus of attention in user-level detection is on the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The recently proposed state-of-the-art detection methods based on heterogeneous graphs [4, 14, 18, 49] only use the relationship between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Therefore, we simplified the social network graph by keeping only users as nodes when constructing the social graph, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' For the other types of entities, only the relationships between users are constructed using them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Explicit relationship extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' For explicit relationships such as follower, friend, mention, reply, and quoted, con- nections between users are constructed directly from their relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The edges constructed based on the above re- lationships are all directed edges, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Implicit relationship construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We also extracted 2 implicit relationships between users: URL co-occurrence and hashtag co-occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Specially, the co-occurrences relationship between user nodes vi and vj can be deter- mined by the probability of entities co-occurring, whose weight is calculated through average Pointwise Mutual In- formation (PMI): W (vi, vj) = 1 ��Ψ{i,j} �� � ek∈Ψ{i,j} log p (vi, ek) p (vj, ek) p (ek)2 , (4) where Ψ{i,i} denotes the set of entities common to vi and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Use 1/Ni approximates p (vi, ek) when calculate PMI, where Ni denote the length of entities list of vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Fi- nally, we obtain the MGTAB heterogeneous graphs contain- ing 410,199 nodes and over 100 million edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Relations in the MGTAB heterogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Relation Direction Description Source Target follower user A user B user A is followed by user B friend user A user B user A follows user B mention user A user B user A mentions user B in tweets reply user A user B user A replies to tweet of user B quote user A user B user A quotes tweet of user B URL Undirected user A and user B have the same URL hashtag Undirected user A and user B have the same hashtag 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Experiment Settings Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In stance detection, we evaluate models on our proposed benchmark, SemEval-2016 T6 [47], and SemEval-2019 T7 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In bot detection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' in addition to our proposed benchmark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' we evaluate models on 4 publicly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='netural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='against ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='support ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Distribution of numerical features with top 10 IG in stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' available bot detection datasets: Cresci-17 [10], Cresci- 15 [9], TwiBot-20 [17], and TwiBot-22 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We use all the annotated data in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Following [15, 17], we conduct a 7:2:1 random partition as training, validation, and test set for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We use competitive and state-of-the-art stance dection and bot detection methods include: Adaboost Classifier (AB) [28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Decision Tree (DT) [42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Random Forest (RF) [6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Support Vector Machines (SVM) [5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Graph Convolutional Network (GCN) [34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Graph Atten- tion Network (GAT) [55],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Heterogeneous Graph Trans- former (HGT) [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Simple Heterogeneous Graph Neu- ral Network (S-HGN) [43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Bot Detection with Relational Graph Convolutional Networks (BotRGCN) [18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' and Re- lational Graph Transformers (RGT) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Benchmark Performance We evaluate baselines on datasets and present their detection accuracy and F1-score in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' All hyper- parameters are listed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='3 for replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We observed that the graph-based methods performed better than feature-based methods, all top 3 models are graph-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In addition, it is obvious to observe that heterogeneous GNNs perform better than homogeneous GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We speculate that this is because heterogeneous GNNs are sufficient to capture the multiple relationships between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' RGT could model the heterogeneous in- fluence between users, achieving the best performance on most datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Better utilizing weights and directions of the edge is a potential future research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Study of Training Set Size We select each 10% of the labeled users as the test and validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Then, we utilize different proportions of labeled users as the training set, increasing from 10% to 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The graph-based model performances under different training sets are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The heterogeneous GNNs’ performance is better than homogeneous GNNs under different training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' This phe- nomenon is consistent with the results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' As more annotated data is used, all detection models become more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Existing account detection meth- ods are generally supervised and rely on large amounts of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' MGTAB’s large scale contributes to train- ing better detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In addition, MGTAB pro- vides 400,000 unlabeled users to support the study of semi- supervised account detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To the best of our knowledge, MGTAB has the most unlabeled users in the account detection field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Social Graph Relationship Analysis In this section, we analyze the impact of using various relationships in the MGTAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In addition to single relation- ships,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' we also experimented with using multiple relation- ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Distribution of numerical features with top 10 IG in bot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Non-English tweets and their percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' ships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We randomly conduct a 1:1:8 partition as training, validation, and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' This partition is shared across all experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 6 illustrates that graph-based account detection methods perform better when more relationships are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' This trend suggests that future research in account detec- tion should focus on better utilizing multiple relationships between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Besides, we observed that hashtag co- occurrence has the worst performance of all the relation- ships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We suspect this is because hashtag co-occurrence is highly random, and two unrelated users can have hashtag co-occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Although MGTAB provides edge weights for URL and hashtag co-occurrence relationships, existing graph-based account detection models cannot fully exploit them, leading to bad performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Conclusion We presented MGTAB, a large-scale dataset for stance detection and bot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We used expert annotation and majority voting to ensure high-quality annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To build the standardized dataset, we selected 20 user features with the highest information gain, which was experimen- tally demonstrated most effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We extracted 7 types of relationships between users and simplified the complex Twitter network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Compared to previous datasets, MGTAB can better support the study of graph-based account detec- tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Our experiments found that graph-based ap- proaches are generally more effective than feature-based ap- proaches and perform better when introducing multiple re- lationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' References [1] Abeer Aldayel and Walid Magdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Your stance is exposed!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' analysing possible factors for stance detection on social me- dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Proceedings of the ACM on Human-Computer Interac- tion, 3:1–20, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 1, 2 [2] Seyed Ali Alhosseini, Raad Bin Tareaf, Pejman Najafi, and Christoph Meinel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Detect me if you can: Spam bot detection using inductive representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Companion Pro- German:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='014 Hindi:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='015 Spanish:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='031 Vietnamese:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='042 Turkish:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='326 Japanese:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='043 French:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='058 Arabic:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='059 Chinese:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='076 Others:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='082 Italian:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='153- Indonesian:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='101Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We simplify the original complex heterogeneous graph network (left) and construct a user-level multi-graph network (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Black, red, and green denote neutral, against, and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Performance of baseline methods on datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Use the most commonly used follower and friend relationships during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Each baseline is conducted five times with different seeds, and we report the average performance and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' “/” indicates that the dataset does not contain user relationships to support the grah-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Best and second best results are highlighted in bold and underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Task Dataset Metric Methods Feature-based Graph-based Homogeneous Heterogeneous AB RF DT SVM GCN GAT HGT S-HGN BotRGCN RGT Stance SemEval-2016 Acc 74.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' $fake: Evidence of spam and bot activity in stock microblogs on twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In ICWSM, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 3 [9] Stefano Cresci, Roberto Di Pietro, Marinella Petrocchi, An- gelo Spognardi, and Maurizio Tesconi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Fame for sale: Ef- ficient detection of fake twitter followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Decis.' metadata={'source': 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Wei, Xiao Zhang, Xuqin Liu, Wei Chen, and Tengjiao Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' pkudblab at semeval-2016 task 6 : A specific convo- lutional neural network system for effective stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In *SEMEVAL, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 2 [59] Chao Yang, Robert Chandler Harkreader, and Guofei Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Empirical evaluation and new design for fighting evolving twitter spammers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' IEEE Transactions on Information Foren- sics and Security, 8:1280–1293, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 3 [60] Kai-Cheng Yang, Onur Varol, Clayton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Davis, Emilio Fer- rara, Alessandro Flammini, and Filippo Menczer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Arming the public with artificial intelligence to counter social bots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Human Behavior and Emerging Technologies, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 1, 3 [61] Kai-Cheng Yang, Onur Varol, Pik-Mai Hui, and Filippo Menczer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Scalable and generalizable social bot detection through data selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In AAAI, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 3, 5, 12, 13 [62] Guido Zarrella and Amy Marsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Mitre at semeval-2016 task 6: Transfer learning for stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In *SEMEVAL, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Supplementary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Feature Analysis Information gain of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Boolean and numerical fea- tures with top 10 IG in user stance detection and their IG are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' User stance detection features and IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Features are listed in descending order of IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Features IG Type def pf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='044927 Boolean def pf sidebar border c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='041702 def pf sidebar fill c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='035676 has URL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='019093 geo enabled 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='018741 def pf bg color 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='016575 pf bg img URL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='016373 pf use bg ima 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='013501 def pf ima 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='010923 verified 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='001304 created at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='107146 Numerical statuses count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='101377 favourites count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='070515 friends count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='064378 listed count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='063106 description length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='062262 followers friends ratios 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='056947 followers count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='055433 name length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='054854 screen name length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='015331 Boolean and numerical features with top 10 IG in bot detection and their IG are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Feature effectiveness analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The details of the user fea- ture representations are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Many proposed works in the literature addressed different features for ac- count detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To further demonstrate the effectiveness of the features extracted in this paper, property features de- signed from different literatures [18,33,61] are used to com- pare the performance of different models under the most commonly used friend and follower relationships [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In the experiment, we only use property features, and the re- sults are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Impact of Different BERT Models The 54 languages included in the MGTAB dataset are shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' To demonstrate the effectiveness of encoding using LaBSE [13], in this section, we adopt four pre-trained encoding models, LaBSE, RoBERTa [41], SBERT [51],and BART [37] to encode user tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The re- sults using the above models to encode all tweets of users are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The detection performance of using LaBSE is better compared to other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We infer that this is because noise will introduced when encoding multi- lingual text using the English pre-training model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' LaBSE Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Bot detection features and IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Features are listed in de- scending order of IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Features IG Type has URL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='064248 Boolean def pf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='025997 def pf ima 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='025402 def pf sidebar border c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='023105 def pf sidebar fill c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='022359 geo enabled 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='019302 verified 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='010902 pf use bg ima 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='007877 pf bg img URL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='005923 def pf bg color 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='005841 followers friends ratios 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='391857 Numerical listed count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='333101 description length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='194765 followers count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='176186 name length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='040335 created at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='034079 friends count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='031598 statuses count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='015544 favourites count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='01176 screen name length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='007641 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Language that appears in MGTAB (ISO 639-1/639-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' ' metadata={'source': 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profile image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='If profile image is default ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Boolean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='listed count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Public lists that use members of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Numerical ' metadata={'source': 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uers this account following ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Numerical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='geo enabled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Whether to enable geographical location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Boolean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='9 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Length of name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Numerical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='description length ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Length of description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Numerical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} 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+page_content='default profile sidebar border color ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='If the border of profile sidebar uses default color ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Boolean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='has URL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='If URL is set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Boolean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='profile background image URL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='If the profile background image has URL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Boolean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='tweet features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Averaged 768-dimensional features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Tweet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='21-788 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' The performance of using different features on MGTAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Best results are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Task Method Features [18] [61] [33] Ours Acc F1 Acc F1 Acc F1 Acc F1 Stance GCN 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4 GAT 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5 70.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4 this benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Experiment Details Experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In this paper, for all GNN models, we stack 2-layer GNNs and two fully connected layers, and the input and output dimensions of the middle GNN layers are consistent, which are 64, 128, or 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We use ReLU as the activation function and set the learning rate from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='0001 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In addition, the dropout rate ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We set the number of attention heads as 8 in GAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We set the number of transformer attention heads and semantic at- tention heads as 4 in RGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' In S-HGN, β is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='05, and the rest remain at the default settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We trained all GNN models for 300 epochs using Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' For machine learn- ing models, the number of estimators for AB and RF is set to 50 and 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' We ran all experiments on a server with 9 TITAN RTX GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Datasets processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' For SemEval-2016 T6 [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' we ex- tracted 20 largest features of IG: number of positive words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' number of negative words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' positive emotion counts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' neg- ative emotion counts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' nouns words frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' pronoun words frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' verb words frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' adjectives words frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' number of special symbols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' number of question mark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' number of capital words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' number of quoted words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' retweet counts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' mention counts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' number of URLs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' entropy of hastags,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' number of hashtags,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' and number of capitalized hashtags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' For SemEval-2019 T7 [25], the feature was ex- tracted by using RoBERTa [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' For TwiBot-20 [17], we follow [18] for dataset processing and feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' For Cresci-15 [9], Cresci-17 [10], and TwiBot-22 [15], we follow [15] for dataset processing and feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Performance using different encoding models on MGTAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Best results are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content=' Task Method pretrain model RoBERTa SBERT BART Lasbe Acc F1 Acc F1 Acc F1 Acc F1 Stance SVM 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='3 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfL_tB/content/2301.01123v1.pdf'} +page_content='2 DT 59.' metadata={'source': 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+1,534 @@ +arXiv:2301.13175v1 [math.CO] 30 Jan 2023 +Cops and robbers on P5-free graphs +Maria Chudnovsky1 +Princeton University, Princeton, NJ 08544 +Sergey Norin2 +Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada +Paul Seymour3 +Princeton University, Princeton, NJ 08544 +Jérémie Turcotte4 +Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada +December 19, 2022; revised January 25, 2023 +1Supported by NSF DMS-EPSRC grant DMS-2120644 and AFOSR grant FA9550-22-1-0083. +2Supported by an NSERC Discovery Grant. Supporté par le Programme de subventions à la recherche du +CRSNG. +3Supported by AFOSR grant FA9550-22-1-0234, and by NSF grant DMS-2154169. +4Supported by an NSERC CGS D scholarship. Supporté par une bourse BESC D du CRSNG. + +Abstract +We prove that every connected P5-free graph has cop number at most two, solving a conjecture of +Sivaraman. In order to do so, we first prove that every connected P5-free graph G with independence +number at least three contains a three-vertex induced path with vertices a-b-c in order, such that +every neighbour of c is also adjacent to one of a, b. + +1 +Introduction +We denote the t-vertex path by Pt. There are a number of well-known open questions about P5-free +graphs (a graph G is H-free if no induced subgraph of G is isomorphic to H, and |G| denotes the +number of vertices of G). For instance: +• the Erdős-Hajnal [4] conjecture implies that for some c > 0, every P5-free graph G has a clique +or stable set of size at least |G|c; +• a conjecture of Esperet [5] implies that for some c > 0, every P5-free graph G has chromatic +number at most ω(G)c, where ω(G) is the clique number of G; +• a conjecture of Hoàng et al. [9] says that for some c > 0, there is a function f such that for every +k there is an algorithm deciding whether a P5-free graph G is k-colourable in time f(k)|G|c. +In this paper, we study another conjecture on P5-free graphs, which concerns the game of cops +and robbers. In this game, there are s cops, and each stands on one vertex of the graph, and so +does the robber. In each turn, first each cop moves to a neighbouring vertex, or does not move; and +then the robber moves to a neighbouring vertex, or does not move. The cops win if at some stage, a +cop is standing on the same vertex as the robber. Cops may share vertices and this game is played +with full information. Given a graph G, how few cops suffice? The cop number c(G) is the smallest +number of cops which can capture the robber G. The game played with one cop was initially defined +by Nowakowski and Winkler [15] and Quilliot [17]. The version with multiple cops was introduced +Aigner and Fromme [1]; in particular they proved that the cop number of any connected planar graph +is at most three. +Inspired by Andreae’s result [2] that the cop number of connected graphs forbidding H as a +minor is bounded for every graph H, Joret et al. [11] proved that the cop number of connected +H-free graphs is bounded if and only if H is a disjoint union of paths. In particular, they showed that +the cop number of connected Pt-free graphs is at most t − 2 for t ≥ 3. Sivaraman [18] conjectured +that two cops can win on connected P5-free graphs and more generally that cop number of connected +Pt-free graphs is at most t − 3 for t ≥ 5. +Other questions on the cop number and forbidden induced subgraphs have also been considered, +for instance relating the cop number and the independence number (in other words, tK1-free graphs) +[16] and excluding multiple induced subgraphs [10, 14, 19]. +However, the question of the cop number of P5-free graphs has received the most attention in +this field; various weakenings of the conjecture about P5-free graphs have been studied. Sivaraman +and Testa [20] conjectured the weaker statement that the cop number of connected 2K2-free graphs +is at most two (2K2, also written 2P2, is graph obtained by the disjoint union of two edges; it can +also be seen as the complement of a four-vertex cycle). This was proved by the fourth author of +this paper [21]. Liu [12] proved various partial results for these problems and Gupta, Mishra and +Pradhan [7] have proved the conjecture holds for multiple subclasses of P5-free graphs. Masjoody +[13] has conjectured the weaker statement that even if two cops perhaps cannot capture the robber +on P5-free graphs, they can confine it to a vertex. +In this paper, we prove Sivaraman’s conjecture on the cop number of P5-free graphs. +1.1. If G is a connected P5-free graph, then c(G) ≤ 2. +1 + +The general strategy we employ is similar to the one used by the fourth author of this paper in [21] +to bound the cop number of 2K2-free graphs. First, show that any graph in the class must contain a +possible winning position for two cops, that is vertices a, b ̸= c such that N[c] ⊆ N[a] ∪ N[b]. Then, +consider a minimal graph in the class for which two cops cannot win, and use the minimality to force +the robber to move to c, after which try to eventually move the cops to a, b and show the robber +cannot escape. +The first part of this strategy is accomplished by the following result, which we prove in Section 3. +1.2. If G is connected and P5-free, with α(G) ≥ 3, then there is a three-vertex induced path of G with +vertices a, b, c in order, such that every neighbour of c is also adjacent to one of a, b. +Here, α(G) is the independence number of G, that is the cardinality of the largest stable subset +of V (G). Let a-b-c be the vertices in order of a three-vertex induced path of G. We say that a-b-c is +domineering, or a domineering 3-path, if every neighbour of c is also adjacent to one of a, b. Thus 1.2 +says that every connected P5-free graph with α(G) ≥ 3 has a domineering 3-path. +The condition α(G) ≥ 3 in 1.2 is needed. It is easy to see that every graph G with α(G) = 2 is +P5-free, and it has no domineering 3-path if and only if its complement H has diameter at most two, +which gives plenty of counterexamples to 1.2 with α(G) ≥ 3 omitted. +The relative of 1.2 proved by the fourth author for 2K2-free graphs [21] is the following. +1.3. If G is connected and 2K2-free, with |G| ≥ 3 and G not a cycle of length five, then there exist +distinct vertices a, b, c such that ab and bc are edges (possibly ac is also an edge) and every neighbour +of c is adjacent to one of a, b. +This differs from 1.2 in three ways, two weakenings and a strengthening. First, it of course assumes +that G is 2K2-free, instead of P5-free. Second, ac might be an edge. But third, it does not need the +assumption α(G) ≥ 3. What if we try to modify 1.3, asking for a domineering 3-path in a 2K2-free +graph G? Then again, it is false, but there are not so many counterexamples; every counterexample +G satisfies α(G) ≤ 2, and it is easy to see that the counterexamples are the complements of Moore +graphs of girth five. These are graphs of diameter two, with girth five, in which every vertex has the +same degree d; such a graph exists only when d = 2, 3, 7 and possibly 57. +In order to prove 1.2, we will need the following definition. Let us say a graph G is bijoined if +• for every two nonadjacent vertices u, v of G, there are exactly two vertices adjacent to both +u, v, and they are adjacent to each other, and +• G has no clique of cardinality four. +It is said to be nontrivial if |G| > 1. If a nontrivial bijoined graph has a complement graph that is +connected, then that complement would be a counterexample to 1.2, so we care about bijoined graphs. +Indeed, we will show in Section 3 that any counterexample to 1.2 has a connected induced subgraph +whose complement is a nontrivial bijoined graph; so the whole question boils down to showing that +there is no such graph. That is proved in Section 2. +The second part of the strategy, using 1.2 to capture the robber with two cops, is accomplished +in Section 4. One important difference between the proofs for the 2K2-free case and the P5-free case +is that once the robber is on c, in the former we can ensure the robber never leaves c, which is not +possible in the latter. +2 + +Let us complete this section with some notation. Suppose G is a graph, which we always consider +to be simple and finite. For v ∈ V (G), we write N(v) for the neighbourhood of v (the set of vertices +adjacent to v), N[v] = N(v) ∪ {v} for its closed neighbourhood of v, and M(v) = V (G) \ N[v] for the +set of vertices distinct from and not adjacent to v. A vertex of a graph G is universal if it is adjacent +to every other vertex. +If X, Y ⊆ V (G) are disjoint, we say X is complete to Y if every vertex in X is adjacent to every +vertex in Y , and X is anticomplete to Y if there are no edges between X, Y . If X = {x}, we say x is +complete to Y if {x} is complete to Y , and so on. +If X ⊆ V (G), we write G[X] for the subgraph of G induced on X and G \ X for G[V (G) \ X]. If +X = {v}, we write G \ v for G \ {v}. +We will often write x1-x2- . . . -xk to represent a path with vertices x1, x2, . . . , xk in order. +2 +Bijoined graphs +We first note that bijoined graphs exist; for instance, if H is a graph of girth at least five in which +every two nonadjacent vertices have exactly one common neighbour, and we add a universal vertex +to H, we obtain a bijoined graph. We will show that no graphs are bijoined other than these, and in +particular, no nontrivial bijoined graph has a connected complement graph. +We need the following well-known lemma: +2.1. Let H be a graph with girth at least five, such that for every two nonadjacent vertices u, v there is +exactly one vertex adjacent to both u, v. If the complement of H is connected, then every two vertices +of H have the same degree. +Proof. +Since the complement of H is connected, it suffices to show that every two nonadjacent +vertices of H have the same degree. Thus, let u, w be nonadjacent. Let N = {v1, . . . , vk} be the set +of neighbours of u, and for 1 ≤ i ≤ k let Ni be the set of neighbours of vi different from u. Thus, +the sets {u}, N, N1, . . . , Nk are pairwise disjoint and have union V (H). Let w ∈ Nk say. For all +j ∈ {1, . . . , k − 1}, w has a neighbour in Nj (because it has distance two from vj), and has exactly +one such neighbour (since H has girth at least five); and has exactly one neighbour in N ∪ {u}; and +so w has degree exactly k. This proves 2.1. +We also need some results about strongly regular graphs. A graph is strongly regular with parame- +ters (n, k, a, c) +if it has n vertices, every vertex has degree k, every two adjacent vertices have exactly +a common neighbours, and every two nonadjacent vertices have exactly c common neighbours. Thus, +Moore graphs are the strongly regular graphs that have parameters (n, k, 0, 1) for some n, k. First, +we need a result about Moore graphs mentioned earlier, due to Hoffman and Singleton [8]: +2.2. A strongly regular graph with parameters (n, k, 0, 1) exists only when n = k2 + 1 and k = 2, 3, 7 +and possibly 57. +Second, we need the following (see Lemmas 10.3.2 and 10.3.3 of Godsil and Royle [6]): +2.3. If a strongly regular graph exists with parameters (n, k, a, c), then either 2k = (n − 1)(c − a) or +(a − c)2 + 4(k − c) is a perfect square. +Now we prove: +3 + +2.4. If G is a bijoined and non-null graph, then G has a universal vertex. +Proof. If u, v are distinct, let R(u, v) be the set of all vertices adjacent to both u, v; thus, if u, v are +nonadjacent then |R(u, v)| = 2 and its two members are adjacent. For convenience, we say “R(u, v) +is an edge”. We observe that no induced cycle of G has length four (because for two opposite vertices +u, v of such a cycle, R(u, v) is not an edge). Thus G is C4-free, where C4 is the cycle of length +four. We suppose that G has no universal vertex, for a contradiction. A 4-clique means a clique of +cardinality four. +(1) For each v ∈ V (G), the complement of G[N(v)] is connected. +Suppose not. Then there is a partition (X, Y ) of N(v) with X, Y ̸= ∅, such that X is complete +to Y . Since G is C4-free, one of X, Y is a clique, say X; let x ∈ X. Then x is adjacent to all other +vertices in N(v), so we may assume that X = {x} and Y = N(v) \ {x}. Since G has no 4-clique, +Y is a stable set. Since x is not universal, there exists z ∈ V (G) nonadjacent to x. Consequently +z /∈ N[v]. But then R(v, z) is a subset of Y and so not an edge, a contradiction. This proves (1). +(2) For each v ∈ V (G), all vertices of G[N(v)] have the same degree in G[N(v)]. +G[N(v)] has no cycle of length three, since G is K4-free; and G[N(v)] is C4-free since G is C4-free. +Thus, G[N(v)] has girth at least five. If u, w ∈ N(v) are nonadjacent, then R(u, w) consists of v and +exactly one vertex of N(v); and so u, w have exactly one common neighbour in G[N(v)]. From (1) +and 2.1, this proves (2). +By (2) and since G is bijoined, for every vertex v there exists kv such that G[N(v)] is a Moore +graph with parameters ((kv)2 + 1, kv, 0, 1). In particular, |R(u, v)| = kv for every neighbour u of v. +It follows that ku = kv for every pair of neighbours u and v, and since G is connected, there exists k +such that kv = k for every v. So G is a strongly regular graph with parameters (n, k2 + 1, k, 2) for +some n (which is determined by k but does not matter), where k ∈ {2, 3, 7, 57} by 2.2. Let us apply +2.3, and deduce that one of the following holds: +• 2(k2 + 1) = (n − 1)(2 − k); but this is impossible since k ≥ 2. +• (k − 2)2 + 4((k2 + 1) − 2) = k(5k − 4) is a perfect square; but this is not the case when +k = 2, 3, 7, 57. +This contradiction proves 2.4. +3 +Finding a domineering 3-path +Let us prove 1.2, which we restate as follows. +3.1. If G is a connected P5-free graph with α(G) ≥ 3, then there exists a domineering 3-path in G. +Proof. We assume that G is a counterexample to the theorem with G minimal. Thus, G is connected +and P5-free, with α(G) ≥ 3, and there is no domineering 3-path in G, and no proper induced subgraph +has these properties. We will prove that the complement of G is bijoined, which we will show is +impossible. We begin with: +4 + +(1) No two adjacent vertices u, v satisfy N(u) ⊆ N[v]. +Suppose that there are two such vertices u, v. If there is a vertex w adjacent to v and not to u, then +w-v-u is domineering, a contradiction; so N[u] = N[v]. Let G′ be obtained by deleting u. Then G′ is +connected, P5-free, and satisfies α(G′) ≥ 3, and so from the minimality of G, there is a domineering +3-path a-b-c of G′. This is not domineering in G, and so u is adjacent to c and nonadjacent to a, b. +But then v ̸= a, b, c, and so v is adjacent to c and not to a, b, contradicting that a-b-c is domineering +in G′. This proves (1). +(2) For each v ∈ V (G) and every component C of G[M(v)], no vertex u ∈ N(v) is complete to V (C). +Because if u is such a vertex, choose w ∈ V (C); then v-u-w is domineering, a contradiction. This +proves (2). +(3) For each v ∈ V (G), G[M(v)] is non-null and connected. Moreover, every vertex in N(v) has a +neighbour in M(v). +By (1), M(v) ̸= ∅. Suppose that C1, C2 are distinct components of G[M(v)]. Since G is connected, +for i = 1, 2 there exists ui ∈ N(v) with a neighbour in V (Ci). By (2), for i = 1, 2, ui has a neighbour +and a non-neighbour in V (Ci), and since Ci is connected, there is an edge aibi of Ci such that ui is +adjacent to ai and not to bi. If u1 has a neighbour in V (C2), we may assume that u1 = u2, but then +b1-a1-u1-a2-b2 is a copy of P5, a contradiction. Thus u1 has no neighbour in V (C2), and similarly u2 +has no neighbour in V (C1). If u1, u2 are nonadjacent then b1-a1-u1-v-u2 is a copy of P5, and if u1, u2 +are adjacent then b1-a1-u1-u2-a2 is a copy of P5, in either case a contradiction. This proves the first +assertion. For the second, let u ∈ N(v); then u has a neighbour in M(v) by (1). This proves (3). +(4) For each v ∈ V (G), G[M(v)] has no domineering 3-path. Consequently α(G) = 3. +Suppose that a-b-c is a domineering 3-path of G[M(v)]. We claim that a-b-c is also domineering +in G. To show this, it suffices to show that every neighbour u of c not in M(v) is adjacent to one +of a, b. But if not, then a-b-c-u-v is a copy of P5, a contradiction. This proves the first assertion of +(4). For the second, suppose that α(G) ≥ 4, and choose v ∈ V (G) that belongs to a stable set of size +four. Then G[M(v)] has a stable set of size three, and it is connected by (3), and has no domineering +3-path as we just showed, contrary to the minimality of |G|. This proves (4). +(5) For each edge uv, if w ∈ N(u) \ N[v] and C is a component of G \ (N(u) ∪ N(v)), then w is +complete or anticomplete to V (C). +Suppose not; then there is an edge ab of C such that w is adjacent to a and not to b. But then +v-u-w-a-b is a copy of P5, a contradiction. This proves (5). +(6) If v belongs to a stable set of size three, then for each edge uv, G \ (N(u) ∪ N(v)) has exactly two +components, both complete graphs. +Certainly it has at most two components, since α(G) = 3, and for the same reason, if G\(N(u) ∪ +N(v)) has two components then they are both complete graphs. Thus we just need to show that +G \ (N(u) ∪ N(v)) has at least two components. +Let C = G \ (N(u) ∪ N(v)). Suppose that C has at most one component. By (1), N(u) \ N[v] is +nonempty. Let w ∈ N(u)\N[v]; then since v-u-w is not domineering, it follows that w has a neighbour +5 + +in V (C), and hence is complete to V (C) by (5). So C is non-null, and N(u) \ N[v] is complete to +V (C). Now M(v) = V (C) ∪ (N(u) \ N[v]), and V (C), N(u) \ N[v] are both nonempty. If there is +an induced path a-b-c with a, c ∈ N(u) \ N[v] and b ∈ V (C), it follows that a-b-c is domineering +in G[M(v)], contrary to (4); and similarly there is no induced path a-b-c with b ∈ N(u) \ N[v] and +a, c ∈ V (C). Thus, V (C) ∪ (N(u) \ N[v]) is a clique, contradicting that v belongs to a stable set of +size three. That proves (6). +(7) Every vertex belongs to a stable set of size three; and so for every edge uv, G\(N(u) ∪ N(v)) has +exactly two components, both complete graphs. +Let X be the union of all stable sets of size three. If X ̸= V (G), then since G is connected, there +is an edge uv with u /∈ X and v ∈ X. But then by (6), G\(N(u)∪N(v)) has exactly two components, +both complete graphs, and consequently u belongs to a stable set of size three, a contradiction. This +proves (7). +(8) For every edge uv, G \ (N(u) ∪ N(v)) consists of two nonadjacent vertices. +By (7), G \ (N(u) ∪ N(v)) has exactly two components C1, C2, both complete graphs. +For +i = 1, 2, let Xi be the set of vertices in N(u) ∪ N(v) that have a neighbour in V (Ci). Suppose +that c1, c′ +1 ∈ V (C1) are distinct. From (7) applied to the edge c1c′ +1, it follows that X2 ⊆ X1 (since +otherwise the set of vertices nonadjacent to both c1, c′ +1 induces a connected subgraph). Consequently, +N(u) \ N(v) is complete to V (C1). Also, again by (7) applied to the same edge, N(u) \ N[v] ⊆ X1 +and N(v)\N[u] ⊆ X1 (since for each w ∈ N(u)\N[v], if w /∈ X1 then {v, u, w} induces a three-vertex +path, contrary to (7)). +Suppose that some w ∈ N(u) \ N[v] belongs to X2. Then w is complete to V (C2) by (5), and +the set of vertices nonadjacent to both w, c1 is a subset of N[v] including v (because w is complete +to V (C1 ∪ C2), and c1 is complete to N(u) \ N(v)); and so this subset induces a connected subgraph, +contrary to (7). Thus X2 ⊆ N(u) ∩ N(v). If c2, c′ +2 ∈ V (C2) are distinct, then the set of vertices +nonadjacent to both c2, c′ +2 includes u, v, w (where w ∈ N(u) \ N[v]), and these three vertices induce +a path, contrary to (7). So |C2| = 1, C2 = {c2} say. Choose d ∈ N(u) ∩ N(v) adjacent to c2; then +u-d-c2 is domineering, a contradiction. This proves (8). +From (8) and since α(G) = 3, it follows that the complement of G is bijoined; and so from 2.4, G +has a vertex of degree zero, contradicting that it is connected. This proves 3.1. +4 +Bounding the cop number +In order to prove our main result, we need the following definition and two lemmas regarding it. +We say that a subgraph H of a graph G is P3-connected if H is connected, and for every pair of +edges e, f ∈ E(H) there exists a sequence of edges e = e0, e1, . . . , ek = f such that ei and ei+1 are two +edges of an induced P3 in G for every 0 ≤ i ≤ k − 1. Note that the property of being P3-connected is +not an intrinsic property of H, but depends on G: even though H is not required to be an induced +subgraph of G, the pairs of edges in the definition must form induced paths in G, not just in H. +4.1. If H is a P3-connected subgraph of a P5-free graph G and u, v ∈ V (G) \ V (H) are such that +uv ∈ E(G), u is anticomplete to V (H) and the endpoints of some edge e ∈ E(H) are non-neighbours +of v, then v is anticomplete to V (H). +6 + +Proof. +Suppose for a contradiction that v has a neighbour in V (H). +As H contains at least +two vertices (as it contains e) and is connected, there exists f ∈ E(H) such that its endpoint is +a neighbour of v. Let e = e0, e1, . . . , ek = f be the sequence of edges from the definition of P3- +connectedness (necessarily, k ≥ 1). As v is not adjacent to the endpoints of e0 but is adjacent to at +least one endpoint of ek, there exist ei, ei+1 such that v is not adjacent to the endpoints of ei (say, +x, y) but is adjacent to the other end of ei+1 (say, z). By the choice of the sequence, x is not adjacent +to z. Then, u-v-z-y-x is an induced P5 in G, which is a contradiction. This completes the proof of +4.1. +4.2. If H is a P3-connected subgraph of a connected graph G and v ∈ V (G), then either +(4.2.1) there exists a P3-connected subgraph H′ of G such that H ⊆ H′ and v ∈ V (H′), or +(4.2.2) there exists a P3-connected subgraph H′ of G such that v ∈ V (H′) and some u ∈ V (H′) is +complete to V (H) in H′, or +(4.2.3) v is complete to V (H). +Proof. Let Q be a shortest path in G with one end v and another end in V (H). Let v = v0-v1- . . . -vℓ +be the vertex set of Q, where vℓ ∈ V (H). If ℓ = 0, that is v ∈ V (H), then H′ = H satisfies (4.2.1), +and so we assume ℓ ≥ 1. +Suppose now that vℓ−1 has a non-neighbour in V (H). Since H is connected, we may suppose that +Q, vℓ are chosen such that for some w ∈ V (H) we have vℓw ∈ E(H) and vℓ−1w ̸∈ E(G). We claim +that in this case H′ = Q∪H satisfies (4.2.1); let us verify that H′ is P3-connected. As Q is a shortest +path, it is necessarily induced, and so Q is P3-connected. We also know that H is P3-connected. +As vℓ−1vℓ ∈ E(Q) and vℓw ∈ E(H) are two edges of an induced P3 in G it follows that Q ∪ H is +P3-connected, as claimed. +It remains to consider the case when vℓ−1 is complete to V (H). If ℓ = 1, then v = vℓ−1 and +(4.2.3) holds, so we may assume ℓ ≥ 2. Let u = vℓ−1 and let H′ be a subgraph of G with V (H′) = +V (Q) ∪ V (H) and E(H′) = E(Q) ∪ {uw : w ∈ V (H)}. We claim that H′ satisfies (4.2.2). Indeed, +u is complete to V (H) in H′ and H′ is P3-connected since Q is P3-connected and every edge uw ∈ +E(H′) − E(Q) forms a induced P3 in G with the edge vℓ−2u = vℓ−2vℓ−1 ∈ E(Q). This finishes the +proof of 4.2. +We are now ready to prove 1.1. +Proof. +Suppose for a contradiction that there exists a connected P5-free graph G on which the +robber has a winning strategy to evade two cops, and choose such G with |V (G)| minimum. In a +series of claims we will obtain properties of this graph which will yield the desired contradiction. +(1) α(G) > 2. +To show (1), it suffices to see that if α(G) ≤ 2, there exists a dominating set of size at most two, +on which two cops may start the game and win at the following turn, which is a contradiction. +(2) There exists a domineering path a-b-r in G. +This follows directly from 1.2 and (1). In the rest of the proof, we will always refer to a fixed +domineering path a-b-r. +7 + +(3) For every v ∈ V (G), no vertex of N(v) is complete to a component of G[M(v)]. +Suppose otherwise that there exists u ∈ N(v) and a component C of G[M(v)] such that u is +adjacent to every vertex of C. First note that G \ C is connected as all vertices in V (G) \ C adjacent +to C are in N(v). By minimality of G, there exists a winning strategy for two cops on G \ C. We +will use this strategy to define a winning strategy for two cops on G, which will be a contradiction. +When playing on G, we say the robber’s shadow is on x if the robber is on x ∈ V (G) \ C, and is on +v if the robber is on a vertex of C. In particular, the robber’s shadow is always in G \ C. We show +that any move of the robber yields a valid move for the robber’s shadow in the sense that at every +turn of the game the shadow either stays on its current vertex or moves to an adjacent vertex. At a +given turn, suppose the robber moves from x1 to x2 (in particular, x1x2 ∈ E(G)). +• If x1, x2 ∈ C, then the robber’s shadow stays on v, which is a valid move. +• If x1, x2 /∈ C, then the robber’s shadow also moves from x1 to x2, which is a valid move +• If x1 ∈ C and x2 /∈ C, we note that necessarily x2 ∈ N(v). Hence, the shadow moving from v +to x2 is a valid move. +• If x1 /∈ C and x2 ∈ C, then x1 ∈ N(v) and so the shadow moving from x1 to v is a valid move. +Consider the strategy for the cops on G to follow the winning strategy on G \ C to capture the +robber’s shadow. Once the cops have captured the robber’s shadow, either they have captured the +robber (if the robber and its shadow are on the same vertex) or the robber is in C and its shadow is +on v. In the latter case, having captured the shadow, at least one of the cops is on v. The other cop +may then eventually move to u, and then capture the robber since u is complete to C. Note that in +the meantime, the robber cannot leave C as it would be immediately captured by the cop on v. This +proves (3). +Note that the proof of (3) is a retract (special type of homomorphism) argument which is a +standard tool in the study of the game of cops and robbers. A quite general version, which is close +to the one presented here, was proved by Berarducci and Intriglia [3]. +(4) For every v ∈ V (G), G[M(v)] is connected. +Using (3), this is exactly the first part of (3) in the proof of the existence of a domineering 3-path. +(5) There exists a strategy for two cops on G to guarantee that, in order to avoid capture, the robber +moves to r. +By (2) we have that N[r] ⊆ N[a] ∪ N[b]. Since ab ∈ E(G), this implies that G \ r is a connected +graph. By minimality of G, there exists a winning strategy for two cops on G \ r. The strategy for +cops in (5) is to play this strategy on G as long as the robber has not entered r. If the robber never +enters r then it is eventually captured since this strategy is winning when restricted to G \ r. Hence, +the robber eventually moves to r (or chooses r as its initial position). This proves (5). +In the rest of the proof, we will construct a strategy for two cops to attempt to capture the robber +on G. However, since c(G) > 2, we may assume that the robber has, and is playing, a strategy to +avoid capture. The cops’ strategy will begin by employing the strategy from (5), and let c1, c2 ∈ V (G) +be the positions of the cops after the robber moves to r. Note that it is possible that c1 = c2. +(6) c1, c2 ∈ M(r). +8 + +If (6) did not hold, one of the two cops could capture the robber at the next turn, which would +be a contradiction. +(7) For each i ∈ {1, 2}, if ci ∈ N[a], then c3−i ̸∈ N(b). +Suppose otherwise that for some i ∈ {1, 2}, ci ∈ N[a] and c3−i ∈ N(b). Then the cops can move +in one turn from {c1, c2} to {a, b}. Being on r at the start of the turn, the robber necessarily will be +in N[r] following its turn. As N[r] ⊆ N[a] ∪ N[b] by (2), the cops may then capture the robber at +the following turn. This contradiction proves (7). +We say that a subgraph H of G is a snare if +(H1) H is P3-connected, +(H2) V (H) ⊆ M(r), +(H3) there exists d1d2 ∈ E(H) such that d1 ∈ N[c1] and d2 ∈ N[c2], +(H4) a ∈ V (H), and +(H5) V (H) ∩ M(b) ̸= ∅. +We finish the proof by showing that if a snare exists then the two cops can capture the robber, +and finally that a snare exists. +(8) There is no snare. +Suppose for a contradiction there exists a snare H. The cops are currently on c1, c2. The cops +move to the endpoints of an edge d1d2 ∈ E(H) as in (H3). Since H is connected, has at least one +edge and a ∈ V (H), there is an edge f ∈ E(H) incident with a. By (H1), there exists a sequence of +edges d1d2 = e0, e1, e2, . . . , ek = f of H such that ei and ei+1 are two edges of an induced P3 in G for +every 0 ≤ i ≤ k − 1. We may suppose without loss of generality that none of e0, . . . ek−1 is incident +with a. Over the next k turns, the cops will follow this sequence of edges. In other words, in i turns +(for 0 ≤ i ≤ k) the cops will be on distinct endpoints of edge ei. Finally, the cops will move to {a, b}. +Note that by (H2), b /∈ V (H). We remark that, since the robber is following a strategy which will +avoid capture, it never enters the closed neighbourhood of a cop. +Let r′ be the first vertex visited by the robber in M(b), and let R be the set of all the previous +positions of the robber. Note that R ⊆ N[b]. Such an r′ exists as once one of the cops is on b (which +always happens in the strategy described above), the robber must move to M(b) as it would otherwise +be captured at the next turn. +We claim that R is anticomplete to V (H). Suppose otherwise that at least one vertex of R has +a neighbour in V (H). By (H2) r has no neighbours in V (H), and so there must exists consecutive +positions of the robber r1, r2 ∈ R such that r1 has no neighbours in V (H) (and is not itself in V (H)) +but r2 has at least one neighbour in V (H). Since r2 ∈ N[b], the cops are not on {a, b} when the +robber moves to r2, and so they are positioned on distinct endpoints of an edge of H. These endpoints +are non-neighbours of r2, and so it follows from 4.1 that r2 has no neighbours in V (H). This is a +contradiction, which implies our claim. +If cops are positioned on an edge of H when the robber first moves to r′, then the same argument +yields that r′ has no neighbour in V (H), and because of (H4) we in particular have that r′a /∈ E(G). +Otherwise, the cops are on a, b when the robber moves to r′, and so r′a /∈ E(G). Hence, in all cases +r′a /∈ E(G). +9 + +As N[r] ⊆ N[a] ∪ N[b], it follows that r′r ̸∈ E(G). As G[R] is connected, and r′ has a neighbour +in R there exists an induced path r1-r2-r′ such that r1, r2 ∈ R. Suppose that there exists an edge +xy ∈ E(H) such that r′x ∈ E(G), but r′y /∈ E(G). Then r1-r2-r′-x-y is an induced P5. Hence, no +such edge exists. As H is connected and r′a ̸∈ E(H) (in particular, r′ is not complete to V (H)) +it follows that r′ is anticomplete to V (H). Since ab ∈ E(G) and a ∈ V (H), it follows that b is +not anticomplete to H. However, by (H5) b is not complete to H. Since H is connected, there +exists xy ∈ E(H) such that bx ∈ E(G) but by /∈ E(G). Then r′-r2-b-x-y is an induced P5. This +contradiction finishes the proof of (8). +It remains to show that there exists a snare in G, a contradiction. +(9) There exists a snare. +Let G′ = G[M(r)]. By (4) G′ is connected. Let P be an induced path in G′ with ends c1 and c2. +Then P is P3-connected, and either c1 = c2 or P contains an edge satisfying (H3), as it has at most +three edges. +Suppose first that c1 ∈ M(b). By 4.2 applied to G′ with H = P and v = a, either we find a +subgraph H′ satisfying (4.2.1) or (4.2.2), or c1, c2 ∈ N[a]. In the first and second cases, H′ is a snare; +property (H3) is the only one which takes a little effort to verify. If H′ satisfies (4.2.1) and c1 ̸= c2, +then P contains an edge satisfying (H3) as noted above. If H′ satisfies (4.2.1) and c1 = c2, then +c1x satisfies (H3) for any neighbour x of c1 in H′; note that x exists by connectivity of H′ unless +|V (H′)| = 1, that is a = c1 = c2, which contradicts that c1 ∈ M(b). If H′ satisfies (4.2.2) and +u ∈ V (H′) is complete to V (P) in H′ then uc1 satisfies (H3). In the remaining case, c1, c2 ∈ N[a] +and G′[{c1, a}] is a snare. +Thus we may assume that c1, c2 ∈ N(b). +By (7) we have c1, c2 ̸∈ N[a]. +By (3) there exists +d ∈ V (G′) ∩ M(b). Assume first that {c1, c2, d} is not a clique of size three. By 4.2 applied to G′ +with H = P and v = d, there exists a P3-connected subgraph H∗ of G′ such that c1, c2, d ∈ V (H∗), +and H∗ contains a path with ends c1 and c2 with at most three edges. Indeed, if (4.2.1) or (4.2.2) +holds then H∗ = H′ has the required properties. If (4.2.3) holds then H∗ = G′[{c1, c2, d}] is either +an induced P2 if c1 = c2 or an induced P3, as d is adjacent to c1 and c2 and {c1, c2, d} is not a clique +of size three, and so H∗ is again as required. +We now apply 4.2 to G′ with H = H∗ and v = a. As c1, c2 ̸∈ N[a] and c1, c2 ∈ V (H∗), a is not +complete to V (H∗) and (4.2.3) does not hold. Thus (4.2.1) or (4.2.2) holds, that is there exists a +P3-connected subgraph H′ of G′ such that c1, c2, d, a ∈ V (H′) and H′ still contains a path with ends +c1 and c2 with at most three edges. It is routine to check that H′ is a snare. +It remains to consider the case when C = {c1, c2, d} is a clique of size three. We proceed similarly +to the proof of 4.2. Let Q be a shortest path from a to C with vertices a = v0-v1- . . . -vℓ, where +vℓ ∈ C. As c1, c2 ̸∈ N[a], we have a ̸∈ C and so ℓ ≥ 1. +Assume first that ℓ ≥ 2 and let H be the subgraph of G′ defined by V (H) = V (Q) ∪ C, and +E(H) = E(Q) ∪ {vℓu : u ∈ V (C), vℓ−1u ̸∈ E(G′)} ∪ {vℓ−1u : u ∈ V (C), vℓ−1u ∈ E(G′)}. +As Q is P3-connected, H is connected, and every edge in E(H)−E(Q) forms an induced P3 in G either +with the edge vℓ−2vℓ−1 or vℓ−1vℓ, it follows that H is P3-connected. Note that V (H) ⊆ V (G′) = M(r) +and a, d ∈ V (H). Let us show with what choice of edge (H3) holds. If at least one edge with both +ends in C is in H, then pick such an edge. Otherwise, it follows from the definition of H that vℓ−1 +is necessarily complete to C, and so we can pick c1vℓ−1. Hence, H is a snare. +10 + +It remains to consider the case ℓ = 1. In this case a has a neighbour in C, and so ad is the unique +edge from a to C, as c1, c2 ̸∈ N[a]. Then H = G′[{a, d, c1}] is an induced P3 in G′ and the edge dc1 +satisfies (H3). It follows that H is a snare in this last case. +This completes the proof of (9) and thus of the theorem. +Acknowledgements +This research was partially completed at the Second 2022 Barbados Graph Theory Workshop held +at the Bellairs Research Institute in December 2022 and at the Combinatorics Workshop held at +Mathematisches Forschungsinstitut Oberwolfach in January 2023. +References +[1] M. Aigner and M. Fromme. A game of cops and robbers. Discrete Applied Mathematics, 8(1):1 +– 12, 1984. doi:10.1016/0166-218X(84)90073-8. +[2] T. Andreae. On a pursuit game played on graphs for which a minor is excluded. Journal of +Combinatorial Theory, Series B, 41(1):37–47, Aug. 1986. doi:10.1016/0095-8956(86)90026-2. +[3] A. Berarducci and B. Intrigila. On the Cop Number of a Graph. Advances in Applied Mathe- +matics, 14(4):389–403, 1993. doi:10.1006/aama.1993.1019. +[4] P. Erdős and A. Hajnal. Ramsey-type theorems. 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Discrete Mathematics, 345(1):112660, Jan. +2022. doi:10.1016/j.disc.2021.112660. +12 + diff --git a/eNFPT4oBgHgl3EQfzTXE/content/tmp_files/load_file.txt b/eNFPT4oBgHgl3EQfzTXE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ffe07369b3d5612acd5f8383cf7052dfcb5026a0 --- /dev/null +++ b/eNFPT4oBgHgl3EQfzTXE/content/tmp_files/load_file.txt @@ -0,0 +1,692 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf,len=691 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='13175v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='CO] 30 Jan 2023 Cops and robbers on P5-free graphs Maria Chudnovsky1 Princeton University, Princeton, NJ 08544 Sergey Norin2 Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada Paul Seymour3 Princeton University, Princeton, NJ 08544 Jérémie Turcotte4 Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada December 19, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' revised January 25, 2023 1Supported by NSF DMS-EPSRC grant DMS-2120644 and AFOSR grant FA9550-22-1-0083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 2Supported by an NSERC Discovery Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Supporté par le Programme de subventions à la recherche du CRSNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 3Supported by AFOSR grant FA9550-22-1-0234, and by NSF grant DMS-2154169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 4Supported by an NSERC CGS D scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Supporté par une bourse BESC D du CRSNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Abstract We prove that every connected P5-free graph has cop number at most two, solving a conjecture of Sivaraman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In order to do so, we first prove that every connected P5-free graph G with independence number at least three contains a three-vertex induced path with vertices a-b-c in order, such that every neighbour of c is also adjacent to one of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 1 Introduction We denote the t-vertex path by Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' There are a number of well-known open questions about P5-free graphs (a graph G is H-free if no induced subgraph of G is isomorphic to H, and |G| denotes the number of vertices of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' For instance: the Erdős-Hajnal [4] conjecture implies that for some c > 0, every P5-free graph G has a clique or stable set of size at least |G|c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' a conjecture of Esperet [5] implies that for some c > 0, every P5-free graph G has chromatic number at most ω(G)c, where ω(G) is the clique number of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' a conjecture of Hoàng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' [9] says that for some c > 0, there is a function f such that for every k there is an algorithm deciding whether a P5-free graph G is k-colourable in time f(k)|G|c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In this paper, we study another conjecture on P5-free graphs, which concerns the game of cops and robbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In this game, there are s cops, and each stands on one vertex of the graph, and so does the robber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In each turn, first each cop moves to a neighbouring vertex, or does not move;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' and then the robber moves to a neighbouring vertex, or does not move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The cops win if at some stage, a cop is standing on the same vertex as the robber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Cops may share vertices and this game is played with full information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Given a graph G, how few cops suffice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The cop number c(G) is the smallest number of cops which can capture the robber G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The game played with one cop was initially defined by Nowakowski and Winkler [15] and Quilliot [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The version with multiple cops was introduced Aigner and Fromme [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' in particular they proved that the cop number of any connected planar graph is at most three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Inspired by Andreae’s result [2] that the cop number of connected graphs forbidding H as a minor is bounded for every graph H, Joret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' [11] proved that the cop number of connected H-free graphs is bounded if and only if H is a disjoint union of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In particular, they showed that the cop number of connected Pt-free graphs is at most t − 2 for t ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Sivaraman [18] conjectured that two cops can win on connected P5-free graphs and more generally that cop number of connected Pt-free graphs is at most t − 3 for t ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Other questions on the cop number and forbidden induced subgraphs have also been considered, for instance relating the cop number and the independence number (in other words, tK1-free graphs) [16] and excluding multiple induced subgraphs [10, 14, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' However, the question of the cop number of P5-free graphs has received the most attention in this field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' various weakenings of the conjecture about P5-free graphs have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Sivaraman and Testa [20] conjectured the weaker statement that the cop number of connected 2K2-free graphs is at most two (2K2, also written 2P2, is graph obtained by the disjoint union of two edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' it can also be seen as the complement of a four-vertex cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This was proved by the fourth author of this paper [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Liu [12] proved various partial results for these problems and Gupta, Mishra and Pradhan [7] have proved the conjecture holds for multiple subclasses of P5-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Masjoody [13] has conjectured the weaker statement that even if two cops perhaps cannot capture the robber on P5-free graphs, they can confine it to a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In this paper, we prove Sivaraman’s conjecture on the cop number of P5-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If G is a connected P5-free graph, then c(G) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 1 The general strategy we employ is similar to the one used by the fourth author of this paper in [21] to bound the cop number of 2K2-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' First, show that any graph in the class must contain a possible winning position for two cops, that is vertices a, b ̸= c such that N[c] ⊆ N[a] ∪ N[b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then, consider a minimal graph in the class for which two cops cannot win, and use the minimality to force the robber to move to c, after which try to eventually move the cops to a, b and show the robber cannot escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The first part of this strategy is accomplished by the following result, which we prove in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If G is connected and P5-free, with α(G) ≥ 3, then there is a three-vertex induced path of G with vertices a, b, c in order, such that every neighbour of c is also adjacent to one of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Here, α(G) is the independence number of G, that is the cardinality of the largest stable subset of V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let a-b-c be the vertices in order of a three-vertex induced path of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We say that a-b-c is domineering, or a domineering 3-path, if every neighbour of c is also adjacent to one of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 says that every connected P5-free graph with α(G) ≥ 3 has a domineering 3-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The condition α(G) ≥ 3 in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' It is easy to see that every graph G with α(G) = 2 is P5-free, and it has no domineering 3-path if and only if its complement H has diameter at most two, which gives plenty of counterexamples to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 with α(G) ≥ 3 omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The relative of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 proved by the fourth author for 2K2-free graphs [21] is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If G is connected and 2K2-free, with |G| ≥ 3 and G not a cycle of length five, then there exist distinct vertices a, b, c such that ab and bc are edges (possibly ac is also an edge) and every neighbour of c is adjacent to one of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This differs from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 in three ways, two weakenings and a strengthening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' First, it of course assumes that G is 2K2-free, instead of P5-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Second, ac might be an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' But third, it does not need the assumption α(G) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' What if we try to modify 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3, asking for a domineering 3-path in a 2K2-free graph G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then again, it is false, but there are not so many counterexamples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' every counterexample G satisfies α(G) ≤ 2, and it is easy to see that the counterexamples are the complements of Moore graphs of girth five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' These are graphs of diameter two, with girth five, in which every vertex has the same degree d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' such a graph exists only when d = 2, 3, 7 and possibly 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In order to prove 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2, we will need the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let us say a graph G is bijoined if for every two nonadjacent vertices u, v of G, there are exactly two vertices adjacent to both u, v, and they are adjacent to each other, and G has no clique of cardinality four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' It is said to be nontrivial if |G| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If a nontrivial bijoined graph has a complement graph that is connected, then that complement would be a counterexample to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2, so we care about bijoined graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Indeed, we will show in Section 3 that any counterexample to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 has a connected induced subgraph whose complement is a nontrivial bijoined graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' so the whole question boils down to showing that there is no such graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' That is proved in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The second part of the strategy, using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 to capture the robber with two cops, is accomplished in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' One important difference between the proofs for the 2K2-free case and the P5-free case is that once the robber is on c, in the former we can ensure the robber never leaves c, which is not possible in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 2 Let us complete this section with some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose G is a graph, which we always consider to be simple and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' For v ∈ V (G), we write N(v) for the neighbourhood of v (the set of vertices adjacent to v), N[v] = N(v) ∪ {v} for its closed neighbourhood of v, and M(v) = V (G) \\ N[v] for the set of vertices distinct from and not adjacent to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' A vertex of a graph G is universal if it is adjacent to every other vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If X, Y ⊆ V (G) are disjoint, we say X is complete to Y if every vertex in X is adjacent to every vertex in Y , and X is anticomplete to Y if there are no edges between X, Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If X = {x}, we say x is complete to Y if {x} is complete to Y , and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If X ⊆ V (G), we write G[X] for the subgraph of G induced on X and G \\ X for G[V (G) \\ X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If X = {v}, we write G \\ v for G \\ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We will often write x1-x2- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' -xk to represent a path with vertices x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' , xk in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 2 Bijoined graphs We first note that bijoined graphs exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' for instance, if H is a graph of girth at least five in which every two nonadjacent vertices have exactly one common neighbour, and we add a universal vertex to H, we obtain a bijoined graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We will show that no graphs are bijoined other than these, and in particular, no nontrivial bijoined graph has a connected complement graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We need the following well-known lemma: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let H be a graph with girth at least five, such that for every two nonadjacent vertices u, v there is exactly one vertex adjacent to both u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If the complement of H is connected, then every two vertices of H have the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since the complement of H is connected, it suffices to show that every two nonadjacent vertices of H have the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus, let u, w be nonadjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let N = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' , vk} be the set of neighbours of u, and for 1 ≤ i ≤ k let Ni be the set of neighbours of vi different from u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus, the sets {u}, N, N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' , Nk are pairwise disjoint and have union V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let w ∈ Nk say.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' For all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' , k − 1}, w has a neighbour in Nj (because it has distance two from vj), and has exactly one such neighbour (since H has girth at least five);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' and has exactly one neighbour in N ∪ {u};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' and so w has degree exactly k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We also need some results about strongly regular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' A graph is strongly regular with parame- ters (n, k, a, c) if it has n vertices, every vertex has degree k, every two adjacent vertices have exactly a common neighbours, and every two nonadjacent vertices have exactly c common neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus, Moore graphs are the strongly regular graphs that have parameters (n, k, 0, 1) for some n, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' First, we need a result about Moore graphs mentioned earlier, due to Hoffman and Singleton [8]: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' A strongly regular graph with parameters (n, k, 0, 1) exists only when n = k2 + 1 and k = 2, 3, 7 and possibly 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Second, we need the following (see Lemmas 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3 of Godsil and Royle [6]): 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If a strongly regular graph exists with parameters (n, k, a, c), then either 2k = (n − 1)(c − a) or (a − c)2 + 4(k − c) is a perfect square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Now we prove: 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If G is a bijoined and non-null graph, then G has a universal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If u, v are distinct, let R(u, v) be the set of all vertices adjacent to both u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' thus, if u, v are nonadjacent then |R(u, v)| = 2 and its two members are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' For convenience, we say “R(u, v) is an edge”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We observe that no induced cycle of G has length four (because for two opposite vertices u, v of such a cycle, R(u, v) is not an edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus G is C4-free, where C4 is the cycle of length four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We suppose that G has no universal vertex, for a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' A 4-clique means a clique of cardinality four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (1) For each v ∈ V (G), the complement of G[N(v)] is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then there is a partition (X, Y ) of N(v) with X, Y ̸= ∅, such that X is complete to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since G is C4-free, one of X, Y is a clique, say X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then x is adjacent to all other vertices in N(v), so we may assume that X = {x} and Y = N(v) \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since G has no 4-clique, Y is a stable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since x is not universal, there exists z ∈ V (G) nonadjacent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Consequently z /∈ N[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' But then R(v, z) is a subset of Y and so not an edge, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (2) For each v ∈ V (G), all vertices of G[N(v)] have the same degree in G[N(v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' G[N(v)] has no cycle of length three, since G is K4-free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' and G[N(v)] is C4-free since G is C4-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus, G[N(v)] has girth at least five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If u, w ∈ N(v) are nonadjacent, then R(u, w) consists of v and exactly one vertex of N(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' and so u, w have exactly one common neighbour in G[N(v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' From (1) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1, this proves (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (2) and since G is bijoined, for every vertex v there exists kv such that G[N(v)] is a Moore graph with parameters ((kv)2 + 1, kv, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In particular, |R(u, v)| = kv for every neighbour u of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' It follows that ku = kv for every pair of neighbours u and v, and since G is connected, there exists k such that kv = k for every v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' So G is a strongly regular graph with parameters (n, k2 + 1, k, 2) for some n (which is determined by k but does not matter), where k ∈ {2, 3, 7, 57} by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let us apply 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3, and deduce that one of the following holds: 2(k2 + 1) = (n − 1)(2 − k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' but this is impossible since k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (k − 2)2 + 4((k2 + 1) − 2) = k(5k − 4) is a perfect square;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' but this is not the case when k = 2, 3, 7, 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This contradiction proves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 3 Finding a domineering 3-path Let us prove 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2, which we restate as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If G is a connected P5-free graph with α(G) ≥ 3, then there exists a domineering 3-path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We assume that G is a counterexample to the theorem with G minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus, G is connected and P5-free, with α(G) ≥ 3, and there is no domineering 3-path in G, and no proper induced subgraph has these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We will prove that the complement of G is bijoined, which we will show is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We begin with: 4 (1) No two adjacent vertices u, v satisfy N(u) ⊆ N[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose that there are two such vertices u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If there is a vertex w adjacent to v and not to u, then w-v-u is domineering, a contradiction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' so N[u] = N[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let G′ be obtained by deleting u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then G′ is connected, P5-free, and satisfies α(G′) ≥ 3, and so from the minimality of G, there is a domineering 3-path a-b-c of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This is not domineering in G, and so u is adjacent to c and nonadjacent to a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' But then v ̸= a, b, c, and so v is adjacent to c and not to a, b, contradicting that a-b-c is domineering in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (2) For each v ∈ V (G) and every component C of G[M(v)], no vertex u ∈ N(v) is complete to V (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Because if u is such a vertex, choose w ∈ V (C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' then v-u-w is domineering, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (3) For each v ∈ V (G), G[M(v)] is non-null and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Moreover, every vertex in N(v) has a neighbour in M(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (1), M(v) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose that C1, C2 are distinct components of G[M(v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since G is connected, for i = 1, 2 there exists ui ∈ N(v) with a neighbour in V (Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (2), for i = 1, 2, ui has a neighbour and a non-neighbour in V (Ci), and since Ci is connected, there is an edge aibi of Ci such that ui is adjacent to ai and not to bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If u1 has a neighbour in V (C2), we may assume that u1 = u2, but then b1-a1-u1-a2-b2 is a copy of P5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus u1 has no neighbour in V (C2), and similarly u2 has no neighbour in V (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If u1, u2 are nonadjacent then b1-a1-u1-v-u2 is a copy of P5, and if u1, u2 are adjacent then b1-a1-u1-u2-a2 is a copy of P5, in either case a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' For the second, let u ∈ N(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' then u has a neighbour in M(v) by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (4) For each v ∈ V (G), G[M(v)] has no domineering 3-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Consequently α(G) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose that a-b-c is a domineering 3-path of G[M(v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We claim that a-b-c is also domineering in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' To show this, it suffices to show that every neighbour u of c not in M(v) is adjacent to one of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' But if not, then a-b-c-u-v is a copy of P5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves the first assertion of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' For the second, suppose that α(G) ≥ 4, and choose v ∈ V (G) that belongs to a stable set of size four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then G[M(v)] has a stable set of size three, and it is connected by (3), and has no domineering 3-path as we just showed, contrary to the minimality of |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (5) For each edge uv, if w ∈ N(u) \\ N[v] and C is a component of G \\ (N(u) ∪ N(v)), then w is complete or anticomplete to V (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' then there is an edge ab of C such that w is adjacent to a and not to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' But then v-u-w-a-b is a copy of P5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (6) If v belongs to a stable set of size three, then for each edge uv, G \\ (N(u) ∪ N(v)) has exactly two components, both complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Certainly it has at most two components, since α(G) = 3, and for the same reason, if G\\(N(u) ∪ N(v)) has two components then they are both complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus we just need to show that G \\ (N(u) ∪ N(v)) has at least two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let C = G \\ (N(u) ∪ N(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose that C has at most one component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (1), N(u) \\ N[v] is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let w ∈ N(u)\\N[v];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' then since v-u-w is not domineering, it follows that w has a neighbour 5 in V (C), and hence is complete to V (C) by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' So C is non-null, and N(u) \\ N[v] is complete to V (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Now M(v) = V (C) ∪ (N(u) \\ N[v]), and V (C), N(u) \\ N[v] are both nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If there is an induced path a-b-c with a, c ∈ N(u) \\ N[v] and b ∈ V (C), it follows that a-b-c is domineering in G[M(v)], contrary to (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' and similarly there is no induced path a-b-c with b ∈ N(u) \\ N[v] and a, c ∈ V (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus, V (C) ∪ (N(u) \\ N[v]) is a clique, contradicting that v belongs to a stable set of size three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' That proves (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (7) Every vertex belongs to a stable set of size three;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' and so for every edge uv, G\\(N(u) ∪ N(v)) has exactly two components, both complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let X be the union of all stable sets of size three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If X ̸= V (G), then since G is connected, there is an edge uv with u /∈ X and v ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' But then by (6), G\\(N(u)∪N(v)) has exactly two components, both complete graphs, and consequently u belongs to a stable set of size three, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (8) For every edge uv, G \\ (N(u) ∪ N(v)) consists of two nonadjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (7), G \\ (N(u) ∪ N(v)) has exactly two components C1, C2, both complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' For i = 1, 2, let Xi be the set of vertices in N(u) ∪ N(v) that have a neighbour in V (Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose that c1, c′ 1 ∈ V (C1) are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' From (7) applied to the edge c1c′ 1, it follows that X2 ⊆ X1 (since otherwise the set of vertices nonadjacent to both c1, c′ 1 induces a connected subgraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Consequently, N(u) \\ N(v) is complete to V (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Also, again by (7) applied to the same edge, N(u) \\ N[v] ⊆ X1 and N(v)\\N[u] ⊆ X1 (since for each w ∈ N(u)\\N[v], if w /∈ X1 then {v, u, w} induces a three-vertex path, contrary to (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose that some w ∈ N(u) \\ N[v] belongs to X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then w is complete to V (C2) by (5), and the set of vertices nonadjacent to both w, c1 is a subset of N[v] including v (because w is complete to V (C1 ∪ C2), and c1 is complete to N(u) \\ N(v));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' and so this subset induces a connected subgraph, contrary to (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus X2 ⊆ N(u) ∩ N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If c2, c′ 2 ∈ V (C2) are distinct, then the set of vertices nonadjacent to both c2, c′ 2 includes u, v, w (where w ∈ N(u) \\ N[v]), and these three vertices induce a path, contrary to (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' So |C2| = 1, C2 = {c2} say.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Choose d ∈ N(u) ∩ N(v) adjacent to c2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' then u-d-c2 is domineering, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' From (8) and since α(G) = 3, it follows that the complement of G is bijoined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' and so from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='4, G has a vertex of degree zero, contradicting that it is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 4 Bounding the cop number In order to prove our main result, we need the following definition and two lemmas regarding it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We say that a subgraph H of a graph G is P3-connected if H is connected, and for every pair of edges e, f ∈ E(H) there exists a sequence of edges e = e0, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' , ek = f such that ei and ei+1 are two edges of an induced P3 in G for every 0 ≤ i ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Note that the property of being P3-connected is not an intrinsic property of H, but depends on G: even though H is not required to be an induced subgraph of G, the pairs of edges in the definition must form induced paths in G, not just in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If H is a P3-connected subgraph of a P5-free graph G and u, v ∈ V (G) \\ V (H) are such that uv ∈ E(G), u is anticomplete to V (H) and the endpoints of some edge e ∈ E(H) are non-neighbours of v, then v is anticomplete to V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose for a contradiction that v has a neighbour in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As H contains at least two vertices (as it contains e) and is connected, there exists f ∈ E(H) such that its endpoint is a neighbour of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let e = e0, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' , ek = f be the sequence of edges from the definition of P3- connectedness (necessarily, k ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As v is not adjacent to the endpoints of e0 but is adjacent to at least one endpoint of ek, there exist ei, ei+1 such that v is not adjacent to the endpoints of ei (say, x, y) but is adjacent to the other end of ei+1 (say, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By the choice of the sequence, x is not adjacent to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then, u-v-z-y-x is an induced P5 in G, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This completes the proof of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If H is a P3-connected subgraph of a connected graph G and v ∈ V (G), then either (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1) there exists a P3-connected subgraph H′ of G such that H ⊆ H′ and v ∈ V (H′), or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2) there exists a P3-connected subgraph H′ of G such that v ∈ V (H′) and some u ∈ V (H′) is complete to V (H) in H′, or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3) v is complete to V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let Q be a shortest path in G with one end v and another end in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let v = v0-v1- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' -vℓ be the vertex set of Q, where vℓ ∈ V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If ℓ = 0, that is v ∈ V (H), then H′ = H satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1), and so we assume ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose now that vℓ−1 has a non-neighbour in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since H is connected, we may suppose that Q, vℓ are chosen such that for some w ∈ V (H) we have vℓw ∈ E(H) and vℓ−1w ̸∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We claim that in this case H′ = Q∪H satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' let us verify that H′ is P3-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As Q is a shortest path, it is necessarily induced, and so Q is P3-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We also know that H is P3-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As vℓ−1vℓ ∈ E(Q) and vℓw ∈ E(H) are two edges of an induced P3 in G it follows that Q ∪ H is P3-connected, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' It remains to consider the case when vℓ−1 is complete to V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If ℓ = 1, then v = vℓ−1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3) holds, so we may assume ℓ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let u = vℓ−1 and let H′ be a subgraph of G with V (H′) = V (Q) ∪ V (H) and E(H′) = E(Q) ∪ {uw : w ∈ V (H)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We claim that H′ satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Indeed, u is complete to V (H) in H′ and H′ is P3-connected since Q is P3-connected and every edge uw ∈ E(H′) − E(Q) forms a induced P3 in G with the edge vℓ−2u = vℓ−2vℓ−1 ∈ E(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This finishes the proof of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We are now ready to prove 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose for a contradiction that there exists a connected P5-free graph G on which the robber has a winning strategy to evade two cops, and choose such G with |V (G)| minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In a series of claims we will obtain properties of this graph which will yield the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (1) α(G) > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' To show (1), it suffices to see that if α(G) ≤ 2, there exists a dominating set of size at most two, on which two cops may start the game and win at the following turn, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (2) There exists a domineering path a-b-r in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This follows directly from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 and (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In the rest of the proof, we will always refer to a fixed domineering path a-b-r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 7 (3) For every v ∈ V (G), no vertex of N(v) is complete to a component of G[M(v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose otherwise that there exists u ∈ N(v) and a component C of G[M(v)] such that u is adjacent to every vertex of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' First note that G \\ C is connected as all vertices in V (G) \\ C adjacent to C are in N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By minimality of G, there exists a winning strategy for two cops on G \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We will use this strategy to define a winning strategy for two cops on G, which will be a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' When playing on G, we say the robber’s shadow is on x if the robber is on x ∈ V (G) \\ C, and is on v if the robber is on a vertex of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In particular, the robber’s shadow is always in G \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We show that any move of the robber yields a valid move for the robber’s shadow in the sense that at every turn of the game the shadow either stays on its current vertex or moves to an adjacent vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' At a given turn, suppose the robber moves from x1 to x2 (in particular, x1x2 ∈ E(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If x1, x2 ∈ C, then the robber’s shadow stays on v, which is a valid move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If x1, x2 /∈ C, then the robber’s shadow also moves from x1 to x2, which is a valid move If x1 ∈ C and x2 /∈ C, we note that necessarily x2 ∈ N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Hence, the shadow moving from v to x2 is a valid move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If x1 /∈ C and x2 ∈ C, then x1 ∈ N(v) and so the shadow moving from x1 to v is a valid move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Consider the strategy for the cops on G to follow the winning strategy on G \\ C to capture the robber’s shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Once the cops have captured the robber’s shadow, either they have captured the robber (if the robber and its shadow are on the same vertex) or the robber is in C and its shadow is on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In the latter case, having captured the shadow, at least one of the cops is on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The other cop may then eventually move to u, and then capture the robber since u is complete to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Note that in the meantime, the robber cannot leave C as it would be immediately captured by the cop on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Note that the proof of (3) is a retract (special type of homomorphism) argument which is a standard tool in the study of the game of cops and robbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' A quite general version, which is close to the one presented here, was proved by Berarducci and Intriglia [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (4) For every v ∈ V (G), G[M(v)] is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Using (3), this is exactly the first part of (3) in the proof of the existence of a domineering 3-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (5) There exists a strategy for two cops on G to guarantee that, in order to avoid capture, the robber moves to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (2) we have that N[r] ⊆ N[a] ∪ N[b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since ab ∈ E(G), this implies that G \\ r is a connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By minimality of G, there exists a winning strategy for two cops on G \\ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The strategy for cops in (5) is to play this strategy on G as long as the robber has not entered r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If the robber never enters r then it is eventually captured since this strategy is winning when restricted to G \\ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Hence, the robber eventually moves to r (or chooses r as its initial position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This proves (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In the rest of the proof, we will construct a strategy for two cops to attempt to capture the robber on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' However, since c(G) > 2, we may assume that the robber has, and is playing, a strategy to avoid capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The cops’ strategy will begin by employing the strategy from (5), and let c1, c2 ∈ V (G) be the positions of the cops after the robber moves to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Note that it is possible that c1 = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (6) c1, c2 ∈ M(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 8 If (6) did not hold, one of the two cops could capture the robber at the next turn, which would be a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (7) For each i ∈ {1, 2}, if ci ∈ N[a], then c3−i ̸∈ N(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose otherwise that for some i ∈ {1, 2}, ci ∈ N[a] and c3−i ∈ N(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then the cops can move in one turn from {c1, c2} to {a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Being on r at the start of the turn, the robber necessarily will be in N[r] following its turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As N[r] ⊆ N[a] ∪ N[b] by (2), the cops may then capture the robber at the following turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This contradiction proves (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We say that a subgraph H of G is a snare if (H1) H is P3-connected, (H2) V (H) ⊆ M(r), (H3) there exists d1d2 ∈ E(H) such that d1 ∈ N[c1] and d2 ∈ N[c2], (H4) a ∈ V (H), and (H5) V (H) ∩ M(b) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We finish the proof by showing that if a snare exists then the two cops can capture the robber, and finally that a snare exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (8) There is no snare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose for a contradiction there exists a snare H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The cops are currently on c1, c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' The cops move to the endpoints of an edge d1d2 ∈ E(H) as in (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since H is connected, has at least one edge and a ∈ V (H), there is an edge f ∈ E(H) incident with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (H1), there exists a sequence of edges d1d2 = e0, e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' , ek = f of H such that ei and ei+1 are two edges of an induced P3 in G for every 0 ≤ i ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We may suppose without loss of generality that none of e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' ek−1 is incident with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Over the next k turns, the cops will follow this sequence of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In other words, in i turns (for 0 ≤ i ≤ k) the cops will be on distinct endpoints of edge ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Finally, the cops will move to {a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Note that by (H2), b /∈ V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We remark that, since the robber is following a strategy which will avoid capture, it never enters the closed neighbourhood of a cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let r′ be the first vertex visited by the robber in M(b), and let R be the set of all the previous positions of the robber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Note that R ⊆ N[b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Such an r′ exists as once one of the cops is on b (which always happens in the strategy described above), the robber must move to M(b) as it would otherwise be captured at the next turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We claim that R is anticomplete to V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose otherwise that at least one vertex of R has a neighbour in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (H2) r has no neighbours in V (H), and so there must exists consecutive positions of the robber r1, r2 ∈ R such that r1 has no neighbours in V (H) (and is not itself in V (H)) but r2 has at least one neighbour in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since r2 ∈ N[b], the cops are not on {a, b} when the robber moves to r2, and so they are positioned on distinct endpoints of an edge of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' These endpoints are non-neighbours of r2, and so it follows from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1 that r2 has no neighbours in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This is a contradiction, which implies our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If cops are positioned on an edge of H when the robber first moves to r′, then the same argument yields that r′ has no neighbour in V (H), and because of (H4) we in particular have that r′a /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Otherwise, the cops are on a, b when the robber moves to r′, and so r′a /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Hence, in all cases r′a /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 9 As N[r] ⊆ N[a] ∪ N[b], it follows that r′r ̸∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As G[R] is connected, and r′ has a neighbour in R there exists an induced path r1-r2-r′ such that r1, r2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose that there exists an edge xy ∈ E(H) such that r′x ∈ E(G), but r′y /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then r1-r2-r′-x-y is an induced P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Hence, no such edge exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As H is connected and r′a ̸∈ E(H) (in particular, r′ is not complete to V (H)) it follows that r′ is anticomplete to V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since ab ∈ E(G) and a ∈ V (H), it follows that b is not anticomplete to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' However, by (H5) b is not complete to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Since H is connected, there exists xy ∈ E(H) such that bx ∈ E(G) but by /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then r′-r2-b-x-y is an induced P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This contradiction finishes the proof of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' It remains to show that there exists a snare in G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' (9) There exists a snare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let G′ = G[M(r)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (4) G′ is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let P be an induced path in G′ with ends c1 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then P is P3-connected, and either c1 = c2 or P contains an edge satisfying (H3), as it has at most three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Suppose first that c1 ∈ M(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 applied to G′ with H = P and v = a, either we find a subgraph H′ satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2), or c1, c2 ∈ N[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In the first and second cases, H′ is a snare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' property (H3) is the only one which takes a little effort to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If H′ satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1) and c1 ̸= c2, then P contains an edge satisfying (H3) as noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If H′ satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1) and c1 = c2, then c1x satisfies (H3) for any neighbour x of c1 in H′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' note that x exists by connectivity of H′ unless |V (H′)| = 1, that is a = c1 = c2, which contradicts that c1 ∈ M(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If H′ satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2) and u ∈ V (H′) is complete to V (P) in H′ then uc1 satisfies (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In the remaining case, c1, c2 ∈ N[a] and G′[{c1, a}] is a snare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus we may assume that c1, c2 ∈ N(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (7) we have c1, c2 ̸∈ N[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By (3) there exists d ∈ V (G′) ∩ M(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Assume first that {c1, c2, d} is not a clique of size three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' By 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 applied to G′ with H = P and v = d, there exists a P3-connected subgraph H∗ of G′ such that c1, c2, d ∈ V (H∗), and H∗ contains a path with ends c1 and c2 with at most three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Indeed, if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2) holds then H∗ = H′ has the required properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3) holds then H∗ = G′[{c1, c2, d}] is either an induced P2 if c1 = c2 or an induced P3, as d is adjacent to c1 and c2 and {c1, c2, d} is not a clique of size three, and so H∗ is again as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We now apply 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2 to G′ with H = H∗ and v = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As c1, c2 ̸∈ N[a] and c1, c2 ∈ V (H∗), a is not complete to V (H∗) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='3) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2) holds, that is there exists a P3-connected subgraph H′ of G′ such that c1, c2, d, a ∈ V (H′) and H′ still contains a path with ends c1 and c2 with at most three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' It is routine to check that H′ is a snare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' It remains to consider the case when C = {c1, c2, d} is a clique of size three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' We proceed similarly to the proof of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let Q be a shortest path from a to C with vertices a = v0-v1- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' -vℓ, where vℓ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As c1, c2 ̸∈ N[a], we have a ̸∈ C and so ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Assume first that ℓ ≥ 2 and let H be the subgraph of G′ defined by V (H) = V (Q) ∪ C, and E(H) = E(Q) ∪ {vℓu : u ∈ V (C), vℓ−1u ̸∈ E(G′)} ∪ {vℓ−1u : u ∈ V (C), vℓ−1u ∈ E(G′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' As Q is P3-connected, H is connected, and every edge in E(H)−E(Q) forms an induced P3 in G either with the edge vℓ−2vℓ−1 or vℓ−1vℓ, it follows that H is P3-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Note that V (H) ⊆ V (G′) = M(r) and a, d ∈ V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Let us show with what choice of edge (H3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' If at least one edge with both ends in C is in H, then pick such an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Otherwise, it follows from the definition of H that vℓ−1 is necessarily complete to C, and so we can pick c1vℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Hence, H is a snare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 10 It remains to consider the case ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' In this case a has a neighbour in C, and so ad is the unique edge from a to C, as c1, c2 ̸∈ N[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Then H = G′[{a, d, c1}] is an induced P3 in G′ and the edge dc1 satisfies (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' It follows that H is a snare in this last case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' This completes the proof of (9) and thus of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Acknowledgements This research was partially completed at the Second 2022 Barbados Graph Theory Workshop held at the Bellairs Research Institute in December 2022 and at the Combinatorics Workshop held at Mathematisches Forschungsinstitut Oberwolfach in January 2023.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='org/abs/1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='11484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Turcotte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Cops and robbers on 2K2-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' Discrete Mathematics, 345(1):112660, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content='112660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFPT4oBgHgl3EQfzTXE/content/2301.13175v1.pdf'} +page_content=' 12' metadata={'source': 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For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access +M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey + +1 +Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000. +Digital Object Identifier 10.1109/ACCESS.2017.Doi Number +Tactile based Intelligence Touch Technology in IoT +configured WCN in B5G/6G-A Survey +Mantisha Gupta1, Student Member, IEEE, Rakesh Kumar Jha2, Senior Member, IEEE, +Sanjeev Jain3, Member IEEE +1School of Electronics and Communication Engineering (SoECE), Shri Mata Vaishno Devi University, Katra, Jammu, J&K, India, 182320. +2Associate Prof., Department of Electronics and Communication Engineering, Indian Institute of Information Technology, Design and Manufacturing, +Jabalpur (IIITDM Jabalpur). (e-mail: jharakesh.45@gmail.com). +3Prof. Computer Science Department, Central University Jammu, J&K, India. (e-mail: dr_sanjeevjain@yahoo.com ) +Corresponding author: (e-mail: jharakesh.45@gmail.com). +This work was supported by the 5G and IoT Lab, SoECE, TBIC, TEQIP-III at Shri Mata Vaishno Devi University, Katra, Jammu +ABSTRACT Touch enabled sensation and actuation is expected to be one of the most promising, +straightforward and important uses of the next generation communication networks. In light of the next +generation (B5G/6G) system’s need for low latency, the infrastructure should be reconfigurable and +intelligent in order to be able to work in real time and interoperable with the existing wireless network. It +has a drastic impact on the society due to its high precision, accuracy, reliability and efficiency as well as +the ability to connect a user from far away or remote areas. Such a touch-enabled interaction is primarily +concerned with the real time transmission of the tactile based haptic information over the internet, in +addition to the usual audio, visual and data traffic, thus enabling a paradigm shift towards establishing a real +time control and steering communication system. Due to the existing system’s latency and overhead, it +creates delays and limits the usability of the future applications. In light of the aforementioned concerns, +this study proposes an intelligent touch-enabled system for B5G/6G and IoT based wireless communication +network that incorporates the AR/VR technologies. The tactile internet and network slicing serve as the +backbone of the touch technology which incorporates intelligence from techniques such as artificial +intelligence and machine/deep learning. The survey also introduces a layered and interfacing architecture +complete with its E2E solution for the intelligent touch based wireless communication system. It is +anticipated for the next generation system to provide numerous opportunities for various sectors utilizing +AR/VR technology in robotics and healthcare facilities, all with the intension of helping in addressing +severe problems faced by the society. Conclusively the article presents a few use cases concerning the +deployment of touch infrastructure in automation and robotics as well as in intelligent healthcare systems, +assisting in the diagnosis and treatment of the prevailing covid-19 cases. The paper concludes with some +considerable future research aspects of the proposed system with few of the ongoing projects pertaining to +the development in the incorporation of the next generation (6G) system. +INDEX TERMS 6G, AI, AR, intelligence, IoT, ML, network slicing, tactile internet, VR. + +I. +INTRODUCTION +Mobile and wireless communications have been playing +a decisive role in the current economy with the technologies +like 2G,3G,4G,5G,GPRS,EDGE that successfully satisfy +the user end with a significant role in the business, +education, +logistics +and +other +primary +industrial +applications, effectively connecting the majority of the +world’s population. These, in the present day are proficient +enough to connect to the devices and people for an +unprecedented exchange of multimedia and data content, +enjoying its fastest growth in the history due to its enabling +technologies, encouraging its widespread deployment, +further intensifying the communication and the industrial +sector[1]. +As per the Cisco Visual Networking Index (VNI), there +is an effective forecasting of the impact of the visual +networking applications on global networks, incrementing +from about 11.5 exabytes in 2017 to an expected surge of +77 exabytes towards 2023. The compound annual growth +rate (CAGR) is expected to be about 74 percent of the +present mobile traffic, 66 percent of the cellular (Wi-Fi) +traffic, accompanied by the smart phones, dominating more +than 90 percent of the mobile data traffic in the coming few +years[2]. + +IEEEAccesSThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access +M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey + +2 +BASE STATION +PARK +NAN( Neighborhood Area Network) +Smart home +HOTSPOT +HOTSPOT +HOTSPOT + +Smart Parking +Smart grid +D2D Pair +BUS STOP +Smart classroom +Smart traffic +monitoring +Smart City +E Health +Smart +Education +Smart Home +Car security +Smart parking +Surveillance +and security +Smart railways +Smart airways +V2V +communication + +FIGURE 1: General architecture of existing wireless and IoT based scenario. + +The Internet of Things (IoT) has therefore been a novel +paradigm that is swiftly accelerating in the modern wireless +communication scenario, connecting billions of devices +with a seamless access of internet and applications, +altogether forming an internet of everything (IoE). +Conclusively the main strength of the IoT lies on the +significant impact it has on variety of facets of everyday +life and user behavior [3]. Since then, there has been a +worldwide increase in the development of cellular network +over the past decade. Table I hereby provides a list of all +the commonly used acronyms throughout the paper for +better understanding. +Many technical challenges have instantiated the +designing of a robust wireless network, capable of +delivering the necessary performance to support the +emerging applications. The previous generations have seen +a paradigm shift in the cellular technology and unlike these, +the B5G/6G is said to be an integrative structure of the +present 5G air interface spectrum. This in combination with +the LTE and Wi-Fi provides a seamless user experience, +accompanied by the universally high coverage rate. The +existing 5G core network inculcates an unprecedented +flexibility and intelligence in the upcoming 6G system with +an improved spectrum regulation, along with the energy +and cost efficiencies. +Along the lines, the system also introduces extreme base +stations with high device densities and an unparalleled +number of antennas, as well as high carrier frequencies with +large bandwidths [4], [5]. From the onset of 2020 onwards +and till this date, the existing wireless communication +networks have been standardized and deployed globally, +with eMBB, MMTC and (URLLC) being the key 5G/B5G +communication scenarios [6]–[8].The widely researched, +IoT-enabled wireless network architecture, compatible with +the B5G/6G system has been effectively illustrated in Fig 1. +The figure represents more or less every possible IoT +configured services and applications, requiring high data +rate and low latency, interacting with the surrounding +environment. All these together constitute a smart system. +This smart system entails all the possible smart devices, +gadgets and sensors, all in all actuating a smart and +automatized infrastructure. This infrastructure therefore +highlights the utilization of the D2D communication +network[9]–[11], the massive MIMO[12], small cell access +points (SCA)[13], the IoT[14] with the network cloud [15], +[16], all in all forming a part of the said 5G/B5G cellular +framework[17], [18]. +All these along with the number of other sensor based +interactions, make the existing system automatic enough to +ease the human effort and save time, all the while +considering the exponential surge in the data traffic, +operating the millions of devices that are connected to the +internet. The key technologies in this framework +incorporates the spectrum sharing [19] with the cognitive +radio [20], the interference management [21], the ultra +dense network (UDN) [22], the mm-wave [23], 5G/B5G +cloud and RAN [24], [25] and SDNs [26]–[28]. The +existing 5G network therefore rides on the coattails of an +explicit New Radio (NR) interface accompanied by a +considerable number of virtualization technologies like the + +IEEEAccesSThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access +M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey + +3 +TABLE I + LIST OF COMMONLY USED ACRONYMS +Abb. +Definition +Abb. +Definition +Abb. +Definition +Abb. +Definition +Abb. +Definition +1G,2G, +3G,4G +First, Second, +Third, Fourth +Generation +DL +Deep Learning +LS +Local Server +NTN +Non Terrestrial +Network +UL +Unsupervised +Learning +3GPP +3G Partnership +Project +DS +Delay Spread +LTE +Long Term +Evolution +OFDMA +Orthogonal +Frequency +Division +Multiple Access +UMTS +Universal Mobile +Telecommunication +System +5G +Fifth Generation +EDG +E +Enhanced Data +Rate for GSM +Evolution +LTE-A +LTE- +Advanced +PD- +NOMA +Power Domain +NOMA +URLLC Ultra Reliable and +Low Latency +Communication +5GPP +5G Partnership +Project +E2E +End to End +MAC +Media Access +Control +QoE +Quality of +Experience +VR +Virtual Reality +6G +Sixth Generation +eMB +B +Enhanced Mobile +Broadband +MANO +Management +and +Orchestration +QoS +Quality of +Service +V2V +Vehicle to Vehicle +AI +Artificial +Intelligence +EVD +O +Evolution Data +Optimized +MC- +CDMA +Multicarrier +CDMA +RAN +Radio Access +Network +V2X +Vehicle to +everything +AMPS +Advanced Mobile +Phone Service +E- +UTR +A +Evolved Universal +Terrestrial Radio +Access +MIoT +Massive IoT +RAS +Robotic +Autonomous +system +W- +CDMA +Wideband Code +Division Multiple +Access +ANN +Artificial Neural +Network +FDM +A +Frequency +Division Multiple +Access +MIMO +Multiple Input +Multiple +Output +RL +Reinforcement +Learning +WCN +Wireless +Communication +Network +AR +Augmented +Reality +GSM +Global System for +Mobile +communication +m- +MIMO +Massive +MIMO +RMS +Root Mean +Square +Wi- +Max +Worldwide +Interoperability for +Microwave Access +AS +Angle Spread +GPR +S +General Packet +Radio Service +M2M +Machine to +Machine +SC- +FDMA +Scalable +FDMA +XR +Extended Reality +B5G +Beyond 5G +GW +Gateway +ML +Machine +Learning +SDN +Software +Defined +Network + + +BS +Base Station +HD +High Definition +MTC +Machine Type +Communication +SD-RAN +Software +Defined Radio +Access +Network + + +CAGR +Compound +Annual Growth +Rate +H2M +Human to +Machine +mMTC +Massive +Machine Type +Communication +SL +Supervised +Learning + + +CDMA +Code Division +Multiple Access +HSI +Human System +Interface +MR +Mixed Reality +SLA +Service Level +Agreement + + +CDMA- +2000 +Code Division +Multiple Access- +2000 +HSP +A +High Speed +Packet Access +NCS +Networked +Controlled +System +TDMA +Time Division +Multiple +Access + + +CD- +NOMA +Code Domain +NOMA +IIoT +Industrial IoT +NaaS +Network Slice +as a Service +TI +Tactile Internet + + +CN +Core Network +IoE +Internet of +Everything +NFV +Network +Function +Virtualization +UDN +Ultra Dense +Network + + +CNN +Convolutional +Neural Networks +IoT +Internet of Things +NOMA +Non +Orthogonal +Multiple +Access +UE +User +Equipment + + +D2D +Device to Device +ITU +International +Telecommunication +Union +NR +New Radio +UHD +Ultra High +Definition + + +Network Function Virtualization (NFV), Software Defined +Networking (SDN) and Software Defined RAN (SD- +RAN)[29]–[32]. The +market +driven +allocation +and +reallocation of bandwidth are the few efficacious +parameters in 5G/B5G system [33], [34]. +The smart applications constitute the entire automated +smart city, comprising of the smart infrastructures in our +day to day lives, ranging from a smart traffic monitoring +system, with an efficient V2V/V2X interactions, providing +a competent sensor based collision/accident detection +system to an equally competent smart parking system. The +other applications may vary from the smart health and +education facilities like remote health counseling, online +smart classroom with the teacher-student interaction, the +smart grid system and the smart homes, collectively +forming the smart neighborhood area network. +The virtualization facilitates an advanced computation +of the network resources and their allocation. These, +reasoned with their indispensable applications, facilitate a +profitable proposition like network slicing [35]–[37]. The + +IEEEAccesSThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + + +M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +4 +virtualization in the core network (CN) [38]–[40] with the +billions of miscellaneous IoT devices [14], [41], improves +the integration of the past and present cellular and Wi-Fi +standards. It therefore provides a ubiquitous high rate, low +latency, giving a smooth experience to all the users in the +network. +The prime objective of the existing communication +standard has always been to fulfill the demands of increase +in capacity, the improvement in the data rate, the reduction +in the latency, to provide a better quality of service and +experience (QoS and QoE). Therefore to meet these +requirements, drastic improvements have been made and +are still ongoing, in order to update the existing cellular +architecture. Hence the next generation network is said to +be agile enough to revert back the intensified network +complexity, regardless of handling diverse scenarios. +The existing 5G/B5G networks thus have to rely on the +self organization and virtualization approaches, to deal with +the disproportionate heterogeneousness and complexity of +the network, associated with the massive amount of devices +[42]–[44].In the forthcoming years, intelligence in the +system is required to yield with such massive connectivity +of IoT devices in the existing network with a minimum +latency and network complexity. The B5G infrastructure +thus has to focus on the considerable scaling and +enhancement of the mobile network by incorporating open +interfaces to support vertical segments in the network [45]. +Such vertical segments are most often the third parties +that do not own a particular network infrastructure but +require the networking services with their specific +requirements, along with their latest business solutions. +Automotive manufacturing has been one of the most +notable vertical segments in the existing communication +system, requiring competent networking capabilities +combined with the IoT and edge-cloud services. This in +turn helps in the progress of a number of applications like +autonomous driving, bird eye view, real-time assessment of +road conditions, to name a few[46]. +The mobile internet in B5G/6G will thus make +provisions for human to human (H2H) interactions with the +primary goal of connecting the machines and gadgets to +construct an IoT interface which is often built on D2D and +H2H interactions. Therefore, to solve the drawbacks such +as latency, poor data rate and compatibility, high +complexity, privacy and security, the next generation +reconfigurable IoT allows for a real time control labeled as +the ‘Tactile Internet’ (TI). According to the ITU1, it is a +network that combines extremely low latency with a high +degree of reliability, scalability, and security[47]. +TI here provides an improved and virtualized +environment which is most likely feasible for the +commercial applications like tele-operation with haptic +communication like remote surgery. It has therefore been +an onset towards revolutionizing every fragment of the +society right from the education and healthcare with their + +1 ITU: International Telecommunication Union +prospected future applications varying from inculcating +sensations and sentiments using the intelligent robotic +applications to the smart health facilities like remote +surgery to the virtual shopping experience at the user end. +The applications evolving the modernized WSN like the +smart and automated homes and appliances, vehicles, +factories, remote sensing and monitoring, augmented and +virtual reality(AR/VR) and quantum computing based +applications have an IoT as a common backbone, altogether +forming an Internet of Everything (IoE)[48], [49]. The +touch enabled sensation and actuation is expected to be one +of the most fundamental applications of the B5G/6G +communication technology due to its potential, simplicity +and convenience, taking into consideration the real time +scenario. +For this reason, the ultra-responsive internet thus helps +enable a real-time control of the physical tactile-based +haptic devices, bringing in a paradigm shift toward an +intelligent and touch-enabled technology. ‘Why do we need +a touch technology?’ is the most anticipated question here, +taking into consideration all the previously-known aspects. +The TI although being the most researched domain of the +B5G/6G framework, its exploration towards the onset of an +intelligent touch based technology is still in the infancy and +has thus been emphasized upon in this survey. +A. SCOPE OF THIS SURVEY +The opening gambit for such technological advancement +takes into account the subsequent technical update of the +wireless communication network with the contemporarily +researched 6G system [50], [51]. It therefore encourages a +real time interaction of humans with their environment, +with few instances like the actuation of sensors causing the +tactile sensation and the real time control/interface in our +body system resulting from touching such surfaces. +It therefore defines a new human-machine(H2M) +interaction framework enabling a physiological latency of +human beings to build a real-time interactive system, with +their applications ranging from robotics to healthcare to the +autonomous driving including the use of virtual/augmented +reality (AR/VR/XR/MR)[52]. For this reason the ‘Tactile +Internet’ is regarded as an impetus and a cornerstone for the +deployment of the touch technology and is expected to +influence the development, innovation as well as the +revolution of the healthcare, education, entertainment, +manufacture, automation and smart grids. +This survey therefore paves a concrete path towards the +initiation of the intelligent and touch enabled technology in +the B5G/6G and IoT based wireless communication +network. +B. MOTIVATION AND CONTRIBUTION OF THIS +SURVEY +The main purpose of this paper is to present a +comprehensive and the state of art proposal motivated +towards deploying an intelligent and touch enabled +technology interface in the B5G/6G and IoT based wireless + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3148473, IEEE Access + + +M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +5 +TABLE II +COMPARATIVE ANALYSIS OF PROPOSED SURVEY WITH THE EXISTING SURVEYS. +Ref no. +Year +Key contribution +Technology used +Communication system used +Intelligence technique +used. +Touch +technology +Issues addressed +[68] +2021 +6G applications like holographic +telepresence, e health and in body +networks. +6G +B5G/6G based mobile +communication. +AI,ML and edge +intelligence +No +An in-depth look at the latest 6G innovations. +[67] +2021 +Space-air-ground-sea integrated +communication network. +6G +Network Slicing and Tactile +Internet in 6G +AI +No +Addressing the coverage requirements in terrestrial and +NTN2 like satellite and UAV with high data rates and +network security. +[66] +2020 +A +novel +architecture +that +employs +computing resources in a cross-layered +infrastructure to enable network computing +SDN,NFV,TI +Tactile internet, transport and +cross-layered protocols. +- +No +Leverages transport and network layers to increase +network effectiveness in regards to congestion control +and reliability. +[65] +2020 +To +provide +virtual +and +logically +independent slices for obtaining and +deliver +slice +services +from +the +infrastructure provider to the customers. +ML,DL, 5G network +slicing +Network slicing with +intelligence and virtualization +ML,DL, SMDP3, +N3AC4 +No +Performance optimization. +[64] +2020 +Design and operation of B5G wireless +network using AI/ML technologies. +AI,ML,B5G +Network slicing and +intelligence +AI,ML +No +Overview of ML/AI algorithms with channel modeling +and +estimation +with +network +management and +optimization in B5G wireless network. +[63] +2019 +ML application in the 5G/B5G WCN. +ML,DL +- +ML,DL,DNN5 +No +MAC layer based resource management, networking +and mobility management, and localization in the +application layer using ML. +[62] +2019 +Integration of robotics, human-computer +interactions +and +virtual +control +environment. +AI, Edge Computing +Network slicing with Tactile +internet and AI +AI +No +An insight on the Role of Tactile internet in industrial +systems as well as an enabling factor of the Industrial +Revolution (4.0) with RAS6 and Virtual control +networks. +[58] +2018 +Incorporation +of +deep +reinforcement +learning (RL) to handle cognitive smart +cities +services +and +improves +their +performance. +ML, DNN +- +Deep RL +No +Incorporate ML with a high level intelligence in the +smart city services. +[56] +2018 +Role of the Network slice Orchestration +and Management, Network slice Broker in +5G/B5G network. +5G/B5G, NFV, SDN, +Cloud and Edge +Computing. +Network slicing with +virtualized fog/edge computing +- +No +5G network slicing use cases with the E2E slice +orchestration and management in eMBB, MIoT, eV2X, +URLLC networks. +[60] +2017 +Two types of feedback: Kinesthetic (based +on force, torque, velocity, position) and +Tactile (based on texture, friction, touch) +AI and predictive +analysis +AI based predictive system +AI +No +Enabling touch transmission and actuation in real time +using TI, having a control on the real and virtual objects +i.e., H2H and MTC interface. +[47] +2016 +The E2E Tactile architecture in real time +has 3 domains: Master, Network and +Control domain, in TI and 5G network. +HSI, SDN, NFV +Tactile Internet based haptic +communication +AI +No +Touch transmission in real time using robotics, haptic +equipment, by means of a communication network +combining the TI and the 5G network with their +applications +in +industry, +automation, +healthcare, +VR/AR. +Our paper +2021 +2021 +To propose an intelligent touch based +system in B5G/IoT system incorporating +AR/VR +B5G /6G network +slicing, TI, IoT, ML +TI and intelligence based +Touch communication system +in B5G/6G +AI,ML,DL, Hybrid +model +Yes +Layered and interfacing architecture for intelligent +touch system for E2E solution. + +2 NTN: Non Terrestrial Network +3 SMDP: Semi Markov Decision Process +4 N3AC: Network slicing Neural Network Admission Control +5 DNN: Dense Neural Network +6 RAS: Robotic Autonomous System + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +6 +network. We first throw light on the next generation 6G +WCN describing its key requirements and applications. The +survey then subsequently emphasizes on the role of +network slicing in B5G/6G network to sustain a high +connectivity of massive number of devices without any +latency and buffering at the user end. +The network may be sliced into Cloud and RAN services +and applications at the customer end. It therefore enables an +easy access, storage facility and virtualization, with each +slice serving different applications on the same physical +network. As a result the network slicing and the tactile +internet in the B5G/6G are therefore the backbone for +incorporating a touch enabled infrastructure with an +induced intelligence by the AI/ML/DL implementation in +the proposed system. +Through this paper we propose here an intelligent touch +configured system that would be applicable in the B5G/6G +WCN. We hence summarize our contributions as follows: +1) We describe the well researched B5G/6G +communication system while throwing light on +some of its key parameters and prominent use +cases with their applicability in the next generation +networks. +2) We provide a discussion on the Tactile Internet +along with its applications in the B5G/6G +networks as an enabler of intelligent touch based +system. +3) We introduce an intelligence in the B5G/6G +network with a descriptive mention of few of the +intelligence learning techniques in the wireless +communication. +4) We further provide a backdrop of the intelligent +touch +configured +technology +involving +the +orchestration of the network slicing with the tactile +internet allied with the intelligence and IoT +connectivity in the B5G/6G wireless domain. +5) We propose a layered and an end to end (E2E) +interfacing +architecture +enabling +the +touch +technology interface in the B5G/6G domain. +6) We further discuss the research challenges and +further explore the research areas of the next +generation (6G) deployment and ultra low latency +tactile based applications in future. +C. COMPARATIVE ANALYSIS WITH THE EXISTING +SURVEYS +The Table II indicates a complete summary of the +existing survey papers on the network slicing and the tactile +internet with some of the intelligent technologies like +ML/DL +implemented +in +the +wireless +cellular +communication network in 5G/B5G scenarios. In contrast +to the other surveys, our survey provides a comprehensive +overview of the proposed intelligent touch enabled +technology in the B5G/6G and IoT configured WCN. +Farris et al. [6], describes the essentiality of the mobile +edge computing (MEC) for supporting a wide range of user +centric applications. These have an important role in the +smart city scenarios presented in Taleb et al. [53] where a +smart MEC based architecture is significant for reducing +the core network traffic while guaranteeing an ultra short +latency for the existing network. +Thus MEC here acts as a key factor in enhancing QoS +and attaining the 1ms requisite latency for the 5G/B5G +mobile systems. It is accompanied by a considerable +number of virtualization technologies like NFV, SDN and +SD-RAN. The virtualization in turn promotes an advanced +computation and allocation of the network resources. +Myriad existing studies have provided a comprehensive +overview of the technical challenges and applications +associated with the network slicing, TI and the emerging +intelligence in B5G/IoT systems. +The works [39], [45], [54] provide comprehensive +reviews of literature on network slicing in B5G networks. +Richart et al.[39] discusses resource slicing and their +allocation in virtual networks powered by SDN and NFV, +as well as how these can be distributed appropriately to +network slices without impairing the efficiency of other +slices. While Afolabi et al. [45]describes the state of art +network slice life cycle architecture operating across the +multiple domains thereby enabling an effective network +programmability +and +flexibility +with +the +creation, +management and orchestration of the network slices, +utilizing the massive IoT and multimedia broadband +connectivity. +Foukas et al. [54] reviews the concept of network slicing +and proposes a generalized layered architecture consisting +of an infrastructure layer, a network function layer, and a +service layer, along with their associated benefits and +challenges. The existing progress in the B5G network +slicing with its key trends along with their corresponding +potential challenges is presented in [55]. The authors in. +[56] throw light on the various concepts of network slicing +and softwarization7 in the B5G technology with their +applicability across the RAN and core network, altogether +establishing an E2E slicing infrastructure. +The intelligent tools like AI/ML/DL play significant role +in outlining automation, deployment and disposition of +different applications the existing as well as the next +generation networks (B5G/6G) [57]. The work of +Mohammadi and Fuqaha [58] provide an intensive facet of +the most prevalent deep reinforcement learning in +structuring of the cognitive smart cities and its applicability +concerning the energy consumption with garnering of water +and agricultural utilization. +The Kafle et al. [59] successfully highlights the various +ways of applying AI/ML techniques for the automation of +network functions in different configurations, ranging from +development organizations to industrial forums. One such +intelligent application of the B5G/6G requiring the 1ms +latency is Tactile Internet (TI) which is quite efficient in +establishing a bilateral communication between the humans + +7Network softwarization is the notion of designing, architecting, deploying and +administrating network components, largely based on software programmability +properties. It enables flexibility, adaptability, and even total reconfiguration of a +network on the spot[37]. + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +7 + +FIGURE 2: Structural organization of paper. +and machines, forming a HSI, which ultimately enables a +haptic communication system. +The term ‘Tactile Internet’ coined by Gerhard Fettweis in +[46], has been a key enabler in fulfilling the need for a +higher data space, essentially resulting in a continuous +increase in the storage and computation among various +cellular devices connected to the internet. Therefore the TI +is centered on H2M interactions with the devices using the +B5G and IoT connectivity, enabling haptic and tactile +sensations at both the transmitting and receiving end +forming a bilateral communication and feedback network. +The works of Maier et al. [52] highlight the commonalities +and various subtle differences between the TI, IoT and the +B5G vision. +The authors in Simsek et al. [47] highlight the critical +requirements and architectural approaches for TI, as well as +the technical issues and challenges associated with the +resource management, core networking and edge cloud/ AI +capabilities. The Aijaz et al. [60] helps to examine a +number of the stringent design challenges to revolutionize +the tactile internet, providing an enhanced haptic perception +with a 1ms round trip delay. Authors in Antonakoglou et al. +[61] explain the evaluation of methodology and technology +assessments for the necessary haptic communication +infrastructure. +The examination of the advancements in tele-operation +over long distances has an effect on haptic communication, +while using the Tactile Internet. Therefore as per [62], the +Tactile internet is a key enabler for realizing industrial +revolution(4.0) by users and devices in real time. The work +by Y.Sun et al. [63] explains how machine learning may +help with resource control at the MAC layer, network and +mobility management in the network layer, and application +layer localization. +While C.X Wang et al. [64] gives an overview of ML/AI +technologies while addressing issues like channel modeling, +estimation, network management and optimization in B5G +wireless communication network. Following which D.Bega +et al. [65] encourages exercising ML approach towards the +market optimization while maximizing infrastructure +provider monetization. Hence to thoroughly optimize the +availability of the computing resources, authors in [66] +present a novel tactile based flexible next generation +internet architecture (FlexNGIA) that capitalizes on the +coexistence of transport and network layers to provide +better congestion control and reliability services via cross- +layered network computing. +You et al. [67] gives an insight of the next generation +6G communication highlighting its parameters specifically +emphasizing its coverage requirements for the functionality +in terrestrial as well as non terrestrial environments. The +paradigm aspect of this work is the integration of the space- +air-ground and sea based communication network. +C.De.Alwis et al. [68] discusses several 6G use cases, +including holographic telepresence, e-health, and in-body +networks that require extremely high data rates, ultra-low +latency and high reliability. As a result, the continuous + +IEEEAccessSec.l-A: Scope ofthis survey +Sec.I-B: Motivation and +contribution of this survey +Sec.l-C: Comparative analysis +with the existing surveys +Se.I-A: Transition from 5G +to B5G/6G +Sec.I-B: An Overview of the +B5G/6G Communication System +Sec.II-C:6G enabled service cases +metrics and driving trends +Sec.II-A: Slicing Process in +B5G/6G Network +Sec.IV-A: Introduction +Sec.IV-B: Tactile Internet +Architecture +Sec.V-A: ML in B5G/6G +Wireless Communication System +Sec.V-B: Generalized Work Flow +proces for Machine Learning +Sec.V-C: MLtechniques in +B5G/6G communication system +Sec.VI-A: Intelligent touch +based system +Sec.VI-A: Appiation Layer +Sec.VII-B: Data +Accumulation/Storage Layer +Sec.VI-C: Edge/Fog Computing +Layer +Sec.VI-D: Connectivity Layer +Sec.VII-E: Perception Layer +Sec.VIII-A: Touch Based +Transmitter System +Sec.VII-B: Touch Based +Receiver System +Sec.VII-C: Touch Based +Middleware System +SeX-A:Architectureoftouch +based IoT Middleware +Sec.IX-B: Existing IoT +Middleware Platfoms +Sec.XI-A: Robotic Interaction +Sec.XI-B: AR/VR based +Enterainment/Shopping +Sec.XI-C:Tactile and Haptic +Sensation Based Tele-diagnosis +forcontact feeCovid-19cases +examination.This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +8 + +FIGURE 3: Timeline of wireless communication technology evolution from 1G to 6G + +penetration of mobile platforms, robotics, human-computer +interaction, and autonomous agents in virtual environments +will distinguish future communication and industrial +systems. +The remainder of the paper follows the structure +depicted in Fig 2. In Section II and III we discuss the next- +generation 6G system in detail with its main parameters, +service and applications along with network slicing and +virtualization concepts in 6G system. Section IV delves into +the architecture and state-of-art uses of the tactile internet, +which is a crucial component of the touch interface system. +Section V introduces the intelligence that is to be +incorporated into the next-generation wireless network. +Section VI explains the proposed intelligent touch +technology, including its layered and E2E architecture in +Section VII and VIII. Section IX discusses some basic +major components of the proposed system, while Section X +presents an interfacing architecture with the existing +network. Section XI contributes some potential and +applicable use-cases of the proposed system, while Section +XII highlights some of the research challenges and +upcoming future aspects. Section XIII summarizes the +recent research and few of the ongoing projects on 6G +before concluding in Section XIV. +II. +INTRODUCTION TO NEXT GENERATION +TECHNOLOGY (6G) +The 6G communication era anticipates how humans will +engage with digital virtual worlds beyond 2030, as the +projected digital transition with the existing B5G networks +have already begun and will continue to evolve over the +next decade. Although 5G/B5G is recognized for network +cloudification via micro service architecture, the next +generation 6G network is strongly linked to the intelligent +network orchestration and management. New digital virtual +worlds with the connected intelligence must have novel +technologies that support these communication and +networking challenges beyond 2030. +The 6th generation wireless communication network is +anticipated to consolidate the terrestrial, aerial, and +maritime communication into a robust network that would +be more reliable, faster, and capable of supporting a large +number of devices with ultra-low latency requirements +while remaining cost-effective. Due to the exponential +growth in the number of IoT devices, the next generation +systems must achieve high spectral and energy efficiency +(SEE), low latency, and massive connectivity to provide for +services like smart traffic monitoring, VR navigation, +telemedicine at the user end, along with digital sensing +using a full HD video transmission in connected +autonomous devices like drones and robots[69]. +The timeline in Fig 3 shows the evolution of the wireless +communication network from 1G to the recent 6G. The +next section explains why the shift from 5G to 6G is +necessary. +A. TRANSITION FROM 5G/B5G TO 6G +As 5G/B5G networks are consistently deployed, the +inherent limitations of this system are being exposed, in +comparison to its original assertion as a platform for IoE +applications. It is gradually more difficult for the existing +multiple access techniques, to cope up with the +exponentially growing IoT devices. As a result, the 5G +communication systems, already being implemented in the +world today are incapable of supporting these many IoT + +IEEEAccess1980 +1G +•AdvancedMobile Phone Service (AMPS) +3G/3.5G/3.75G +•TotalAccessCommunicationSystem(TACS) +: FDMA, Analog voice +·CDMA-2000.UMTS +• WCDMA, HSPA +•Digitalvoice+Data +·Packet switching,Broadband +2G/2.5G/2.75G +2000 +1990 +•GSM, GPRS,EDGE +: TDMA/CDMA +2010 +3.95G/4G +• Digital Voice +LTE, LTE-A, Wi-Max +·OFDMA/SC-FDMA +4G/4.5G +·Packet switching +• LTE-A. Wi-Max +2016 +• MIMO +•OFDMA/MC-CDMA +.Wirelessbroadband +•MIMO, m-MIMO +•Mobile Internet +•D2D,Hetnet +HDVideo streaming +· All IP (IPV4/IPV6) with unified +LANWAN/PANandWLAN +5G/B5G +2020 +B5G/6G +•NOMA,Hybrid(OMA+NOMA) +• SM-MIMO, THz comm. +·mm-Wave,Beamforming +·Cloudization,Softwarization +•IMT-2020.5G-NR +Virtualization,Slicing +•mMTC,eMBB, URLLC +2030 +•Intelligence(Al/ML/DL) +·Cloudization (Cloud/Fog/Edge) +·Quantum Computing, Blockchain(DLT) +·Softwarization,Virtualization, +TactileInternet,Fullyautomated +Slicing (SDN/NFV) +vehicles, Holographic verticles, +·loT/loE,AR/VR, V2X, UHD and +Digital sensing, and underwater +360°vide0s,Smartcities, +communication +Tele-medicine and wearable +devicesThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +9 +TABLE III: +COMPARISON OF PERFORMANCE ATTRIBUTES BETWEEN 5G, B5G AND 6G COMMUNICATION SYSTEMS. +Performance +attributes +5G +B5G +6G +Application types + +eMBB + +URLLC + +mMTC + +Reliable eMBB (ReMBB) + +URLLC + +mMTC + +Hybrid( URLLC + eMBB) + +MBRLLC + +mURLLC + +HCS + +MPS +Architecture + +Dense small base stations +operating at sub-6GHz in +conjunction +with +umbrella +macro base stations. + +Mm-wave small cells with a +range of approximately 100m +for fixed access. + +Denser +small +cells +operating at sub-6 GHz +with umbrella macro base +stations + +Mm-wave cells with a +diameter of less than +100m. + +For mobile and fixed access, cell- +free smart surfaces with high +frequency are supported by mm- +wave small cells. + +Drone-carried base stations and +tethered +balloons +provide +temporary hotspots. +Frequency bands + +Sub 6 GHz + +Mm-Wave for fixed network +accessibility. + +Sub 6 GHz + +Mm-Wave for fixed +network accessibility. + +Sub 6GHz + +Mm-wave for mobile network +accessibility. + +High frequency and THz bands +above 300GHz are investigated. + +Non RF technologies like VLC, +Optical fiber communication etc +Spectral and +Energy Efficiency +(SEE) +10x in bps/Hz/m2/Joules +100x in bps/Hz/m2/Joules +1000x in bps/Hz/m2/Joules +Data rate +1Gb/s +100Gb/s +1Tb/s +E2E delay +5ms +1ms +<1ms +Radio-only delay +100ns +100ns +10ns +Processing delay +100ns +50ns +10ns +E2E reliability +requirement +99.999% +99.9999% +99.99999% +Interoperable +devices + +Smart phones + +Sensors + +Drones + +Smart phones + +Sensors + +Drones + +XR equipment. + +Sensors and DLT devices + +CRAS + +Smart implant system + +XR and BCI equipment +devices. The requirement for faster data rates has fueled the +evolution of wireless networks, which has necessitated a +continuous 1000-fold increase in network capacity[70]. +As the demand for wireless capacity continues to surge, +the emerging IoT system, which connects millions of +people to billions of machines, has resulted in a radical +paradigm shift from the rate centric eMBB services from +the previous eras towards URLLC and intensified mMTC +services, as per the 3GPP, which is working on the +implementation of 5G/B5G standard[71]. Although it can +be asserted that the evolutionary aspects of the existing 5G +supporting the data hungry eMBB services have gained a +significant momentum, while the promised revolutionary +disposition systems, operating exclusively at high mm- +wave frequencies have yet to materialize. +Despite the fact that today’s linked 5G systems are +easily capable of supporting the most fundamental IoE and +URLLC services (such as factory automation), it is still +debatable whether or not they will be able to deliver the +smart city services based IoE applications in future. +Conversely, the initial and the existing B5G deployments +are most likely to rely on the low frequencies (sub 6GHz) to +support mobile data transmissions. While on the other hand, +an enormous influx of new IoT services such as XR +(including AR/VR/MR), flying vehicles, and connected +autonomous systems would most likely derail 5G’s original +purpose of supporting small packet and sensing-based +URLLC applications[72]. +Thus, in order to successfully operate these IoE +services, a wireless system must simultaneously provide a +high level of reliability, low latency, and a high data rate for +a wide range of heterogeneous devices. These new services +necessitate the resolution of novel and distinct challenges, +unprecedented in terms of their complexity including the +tradeoff between latency, throughput and reliability. Not +only do these services help entail new approaches for an +effective regulation and handling of performance and +challenges but also aid in exploration of frequencies beyond +6GHz range in order to create a self sustaining and +intelligent wireless network. +This aforementioned network is capable of provisioning +and orchestrating communication, computing, control, + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +10 +TABLE IV +COMPARISON OF VARIOUS 6G SERVICE CASES WITH PARAMETERS AND APPLICATION AREAS. +S.No +6G service cases +Performance Attributes +Application Areas +1. +FeMBB (Further +enhanced Mobile +Broadband) + +Enhanced broadband in densely populated areas. + +Operates in THz communication. + +Enhanced multimedia applications like 4D video gaming, mobile +TV, connected wearables and sensors. + +Public transportation + +High speed trains + +Smart cities +2. +umMTC ( Ultra massive +Machine Type +Communication) + +Reliable connectivity with massive scale (trillions of devices) of +connected devices. + +Improves connection density. + +Enables IoE with ultra dense cellular IoT networks. + +SigFox and LoRa as potential technologies for enhanced +connectivity and network coverage in 6G. + +Internet of Industrial +Smart Things (IIsoT) + +Smart buildings + +Internet enabled supply +chains, +logistics, +fleet +management and water +quality monitoring. + +Natural/wildlife sensing + +Forest monitoring +3. +ERLLC (Enhanced +Reliable and Low +Latency +Communication) + +End to End fast turnout time. + +Intelligent +framing +and +coding +with +efficient +resource +management. + +Intelligent UL/DL communication + +Remote robotic surgery using smart surfaces and intelligence. + +Telemedicine + +Internet +of +Healthcare +(IoH) + +Remote Robotic Surgery + +XR +4. +ELPC (Extremely Low +Power Communication) + +Uses +Intelligent +Reflecting +Surfaces +(IRS) +known +as +Reconfigurable Intelligent Surfaces (RIS). + +Reduces hardware dependency and Tx-Rx complexities. + +Reduced energy consumption with passive array transmission. + +Smart homes + +Smart cars + +UAVs +5. +LDHMC (Long Distance +and High Mobility +Communication) + +High mobility and seamless communication for long distances +(>1000km). + +Accurate channel estimation. + +Use of FBMC and UFMC as alternative to OFDM. + +Deep sea tourism + +High speed transportation + +Space sightseeing +6. +MBBLL (Mobile Broad +Bandwidth and Low +latency) + +MEC to attain end to end low latency. + +Low complexity mechanism for VR experience by user. + +Mobile AR,VR +7. +MLLMT (Massive Low +Latency Machine Type) + +Data availability, ultra scalability and low latency. + +Time critical applications where decision making takes fraction of +seconds. + +Automation, controlling and monitoring of industrial 4.0 use cases. + +Home and building +automation + +UAVs + +IoT enabled Healthcare +8. +MBBMT (Massive +Broadband Machine +Type) + +Touch based experience with high data rates. + +Massive IoT connectivity in densely populated areas. + +Tactile sensations captured by sensors/devices converts to digital +data. + +Tactile Internet + +localization and sensing of the scenarios that are best suited +for IoT needs[73]. So to address these issues, a game- +changing 6G wireless system is required, with a design that +is organically tuned to the performance requirements of IoE +applications and associated technical advancements. +B. AN OVERVIEW OF THE B5G/ 6G +COMMUNICATION SYSTEM +The 6G wireless communication network, which is +currently being researched, is expected to integrate +terrestrial, aerial, and maritime communication systems into +a robust network that is more reliable, fast, and capable of +supporting a large number of devices with ultra-reliable and +low-latency requirements. The AI, ML/FL, quantum +communication, blockchain/DLT, beyond 6GHz and +towards Terahertz communication, TI, swarm UAVs, Zero +touch network and service management (ZSM), large +intelligent surfaces (LIS), NTN and 3D networking, VLC, +compressive sensing with an efficient energy transfer and +harvesting are just few of the currently proposed by the +ongoing researches worldwide[74]. +Owing to a massive growth in the number of IoT devices, +realization of the advanced services like smart traffic +monitoring, VR based navigation, smart medical facilities +like tele-medicine and HD video transmission in drones and +robots is possible. Hence the B5G/6G communication +systems aim to achieve high SEE, low latency, and massive +connectivity. The ever-increasing number of IoT devices +makes it difficult for the existing multiple access strategies +to handle such a huge number of devices, therefore +requiring a more extensive network in order to make use of + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +11 + +FIGURE 4: Network slice functionality in B5G/6G network + +the massive bandwidth capabilities offered by B5G/6G +communication systems. +According to the work by Zhang et al. [75], a speculated +vision and functionality of 6G networking scenario +provides a technological framework and requirements for +industries in the future generation communication system +with cell less architecture, decentralized networking, and +resource allocation with 3D radio interoperability. The next +generation wireless network comprises of a large number of +linked devices with numerous base stations (BSs) and +access points (APs), each of which will serve multiple +devices at the same time, forming a coordinated multipoint +(CoMP)[76]. +Hence more data would be transmitted via future wireless +communication networks, since most of the value-added +apps and services rely significantly on the data exchanges. +Therefore large devices will generate a massive quantity of +data, which will need high-performance processing units +and backhauling connections. The following subsections go +through the new 6G enabled driving trends, metrics, and +use service cases. +C. 6G ENABLED SERVICE CASES, METRICS AND +DRIVING TRENDS. +With the new performance metrics, new technological +trends step up to redefine the prevalent B5G applications by +morphing the classical URLLC, eMBB, and mMTC into +something entirely new and innovative. As a result, Table +III gives a comparison of some of the key performance +attributes of the 5G, B5G, and 6G wireless communication +systems[73]. Various countries have initiated projects +aiming at the research and deployment of B5G/6G +communication networks, as discussed in Section XIII. The +research on 6G is accelerating, and has been documented in +recent works like [77]–[79]. +According to the recent research, a variety of possible 6G +applications have been classified as mobile broad +bandwidth and low latency (MBBLL)[80], massive broad +bandwidth machine type (mBBMT)[78], massive low +latency machine type (mLLMT)[81], further-enhanced +mobile broadband (FeMBB)[82], extremely reliable and +low-latency communications (ERLLC)[83], ultra-massive +machine-type +communications +(umMTC)[44], +long- +distance and high-mobility communications (LDHMC)[84] +and extremely low power communications (ELPC)[85]. +Detailed descriptions of each of these are provided in the +preceding Table IV, along with their parameters and +application areas. +III. +NETWORK SLICING IN B5G/6G BASED IOT +NETWORKS +The forthcoming B5G/6G networks aim to serve a wide +range of applications, thus recognizing the 5G epoch as the +century of mobile telecom networks, all the while +promoting dedicated use-cases and endowing unequivocal +services, to meet diverse user requirements. Hence the +technologies like UHD, multimedia, AR/VR/XR therefore +needs a faster speed and a relatively higher capacity and +connectivity, compared to the mission-critical applications +like IoT/MIoT and autonomous systems require an ultra- +low latency and ultra-reliable operation. + +IEEEAccessSERVICE LIFECYCLE +E2E SERVICE OPERATION ANDMANAGEMENT +MANAGEMENTLOOP +SERVICELAYER +V2X +COMMUNICATION +AMF +LSMF +SYSTEM +BASE STATION +UPF +ASSURANCE +NETWORK LAYER +FULFILLMENT +SMART CITY AND +UPF +AMF +SMF +CONNECTIVITY +ORCHESTRATION +BASE STATION +DATA ACQUISITION, PROCESSING, ABSTRACTION AND DISTRIBUTION +() +(g) +RESOURCES AND FUNCTIONS MANAGEMENT +ANDORCHESTRATION (MANO) +RAN +CORE += +ORCHESTRATION +ORCHESTRATION +() +( +FOGNODE +ROUTER +GATEWAY +TRANSPORT +NFV&MEC +EDGECLOUD +WIRELESSAND +CORE/CENTRAL +ORCHESTRATION +ORCHESTRATION +FIXEDACCESSNETWORKS +CLOUD +RESOURCES AND FUNCTION LAYER +INFRASTRUCTUREORCHESTRATIONThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +12 + +FIGURE 5: Network slice orchestration and management + +The 6G cellular framework formulation anticipates to be +accomplished on the existing and researched 5G/B5G +technology, thus supporting a surplus of network services +with +miscellaneous +performance +requirements. +The +advancement of cellular networks and resulting generation- +wide improvements is motivated primarily by the desire to +enable better data-based services. A variety of aspects +render 5G important, including the mm wave spectrum +distribution and reallocation of bandwidth. Virtualization +with a billion individual networks in the CN and IoT +facilitates the convergence between previous and present +cellular and Wi-Fi requirements[83]. +This in turn offers a pervasive high rate and low latency +experience for network customers. The most significant +part of the B5G/6G infrastructure comprises of the network +service and its development platform. It is highly capable of +improving the network scalability while fulfilling the user +requirements by utilizing existing services. The network +slicing functionality in B5G/6G domain has been depicted +in Fig.4. The virtualized infrastructure here has provisions +for slice instances, and collectively functions with the +infrastructure resources from another slice instance[37]. +A set of homogenous APIs are made available for +creating an abstraction layer to facilitate with the slice +management while controlling its virtual resources during +its operation. These slices can therefore be accessed by +different tenants or third parties using these APIs. Here +SLA act as the slice blueprints, using which the tenant +specifies its requisite slice characteristics ranging from +topology, management, control and so on. The slice +lifecycle is regulated by the service lifecycle management +loop, openly accessed by all the functioning slices[45]. The +management +and +orchestration +(MANO) +offers +an +integrated and a holistic approach towards the regulation of +network slicing and the NFV management. +It offers a standardized level of data abstraction followed +by the adapt specification of its network infrastructure +together with its service management and implementation +process[31]. This section discusses the concept of network +slicing in B5G/6G communication networks with its +functionality, management and orchestration in RAN and +CN and finally its application in the 6G with the proposed +architecture. +A. SLICING PROCESS IN THE B5G/6G NETWORK. +The existing B5G is most likely to consider a variety of +business and service quality requirements like the enhanced +capacity coupled with the intelligent traffic and offloading +techniques accompanied with a highly complex and +heterogeneous network. All of these fulfill the required +performance criterion together with an autonomous +network management. A high data rate guarantees a high +level of end user service quality with an unlimited mobile +broadband connectivity in the jam-packed areas like +stadiums, concerts and shopping centers, by means of the +terminals having the AI capabilities [86]. +The reduced latency with high data rates are capable of +supporting the UHD streaming from the cloud technology +and improvised VR devices and other wearable computing +gadgets. It therefore provides a faster web downloading +while enabling a premium user experience in services like +YouTube streaming, Netflix and so on with a high video +resolution. The network slicing is a fundamental key for the +B5G/6G technology. Thus the network slices here are an +end to end concept of the next generation technology where +the slice operator supports a massive amount of customers. +Here each of them in the long run requires a multiple end to + +IEEEAccess- +B +SMART HOME +EMBB +AR/VR/XR +wiE +[ +UHD VIDEO STREAMING +Service +Layer +Network +Orchestrator + SMARI CITY +MMTC +NetworkSlices ++ +Resource +Sensor +IoT +Application Service +Allocation +Orchestrator +Infrastructure Layer +X +AUTONOMOUS VEHICLES +(IIoT) +REMOTE SURGERY +SMART CLASSROOM +Application/Business LayerThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +13 + +FIGURE 6: Network slicing in next generation networks (B5G/6G). +end individual and logical networks referred as network +slices. They are categorized into three components, each of +them governing the RAN, core and transport domain[56]. +The RAN and core slices consist of the application +context and personalities respectively with the transport +slices being connectivity between the RAN and core. Each +individual domain has a controller, i.e. the RAN controller, +core controller as well as the transport controller, all of +these supported by an end to end orchestrator. To realize +the B5G/6G networks, 5G network slicing plays a crucial +role in guiding about slice utilization for automation, +assurance and optimization of transport slices involving +various low latency and high reliability applications. +These applications may range from automated vehicles, +tactile applications, smart devices and so forth[87]. The +Fig.5 diagrammatically illustrates the network slice +orchestration process applicable in B5G/6G networks. At +the back end, the resource allocation takes place in the +infrastructure layer where resources are provided to the +individual slices. The slicing applications are managed by +the network orchestrator in the service layer, so as to enable +an +effective +application +slice +management +and +orchestration. It can be undoubtedly claimed that the most +defining feature and in other words the ‘secret sauce’, for +the 5G/B5G success is the E2E network slicing, which will +be applicable in 6G networks as well. +Hence the network slicing is responsible for the optimal +resource efficiency and flexibility in the network. It +therefore enables the implementation of new business +models as NaaS, supporting various mission critical use +cases +including +the +industrial +automation(4.0) +and +availment of remote health facilities[50], [88]. The network +slicing architecture pertaining to the B5G/6G system has +been illustrated in Fig 6. The slicing process is therefore +described as the three sub processes, interlinked with each +other: the intelligent cloud slicing, the RAN slicing and the +application slicing, all of which are functional in the +B5G/IoT enabled networks. +Commencing from the lowermost stratum we have +various applications that furnished at the consumer end +ranging from the real time online gaming with a UHD +streaming to the live online classroom teaching sessions, +efficiently making use of the AR/VR technology to deliver +the required information. The other applications may also +involve the use of robots in real time actuations and tactile +and touch based haptic communications, which may be put +to a practical use in industrial operations, automation in the +robotics and machinery, vehicles and UAV fleet to +accomplish the services requested by the users at the +customer end in the form of application slices. +The succeeding layer is that of RAN slicing, taking place +in between the CN and RAN, i.e., at the backend of the +network, routing the clients with the final applications. It is +capable of enabling an effective resource and spectrum +allocation with power and energy efficient cognitive radio +network system. All of this is in the form of RAN slices and +successfully connects the edge devices with the cloud +network. The cloud computing and storage enables +intelligent cloud slicing technique, where the cloud enabled +applications are accessed in the form of cloud slices. These +support different edge devices at the user end while +facilitating the requested services. +The following section describes the tactile mode of +communication in B5G/6G system that is to be +incorporated with the aforementioned slicing techniques, so +as to deploy and configure the proposed touch interfacing +wireless system with incorporated intelligence which is +discussed in detail in the subsequent sections. +IV. +TACTILE INTERNET IN B5G/6G SYSTEM +The internet initially was designed and indented to be a +reliable and interoperable means of communication across +the globe. With time, not only has it evolved to convey a +large amount of content, but it also helps enhance the real + +IEEEAccessCS1 +CLOUDSERVICES +CS2 +INTELLIGENT +CLOUD SLICING +FOGNODES +CSN +SMART HOME +W +WEARABLE +AIML +DEVICES +EDGEDEVICES +RS! +RS2 +GREENCOMMUNICATION +COGNTTIVERADIOTRANSMISSION +RAN SLICING +SERVINGGATEWAY +PACKET DATA GATEWAY +RSN +TACTILE SUPPORT +UAWNETWORK +AUTONOMOUSVEHICLES +VIRTUALCLASSROOM +AS2 +AR/VR/XR +AS3 +Tube +HAPTICCOMMUNICATION +fa +ROBOTICS +UHD VIDEOSTREAMING +TELESURGERY +APPLICATION SLICING +Y +ASN +SMARTCITY +(rom)This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +14 + +FIGURE 7: Tactile internet E2E architecture with bilateral feedback system + +world experience of the users. The tactile internet thus +improves remote real-time physical engagement with +existing and virtual items. Whereas integrating TI with +touch interface will provide a two way interactive +experience that blurs the boundaries between the real and +the virtual world. The works in [89]–[92] examine the +technical requirements of TI and its ability to support future +smart city applications. It therefore provisions for a real- +time control with the physical and haptic sensations to be +experienced by the users remotely. +Unlike the traditional internet applications, the new +tactile internet (TI) intends to serve as a medium for +remote, real-time physical interactions with actual, physical +objects, including humans, machines, and processors. The +tactile internet enables enhanced virtualized remote +classroom instruction with the participants in collaboration +with the remote environment collectively amid the presence +of remote and virtual resources. Thus, the data rates of +wireless communications have been increasing, which is +primarily due to innovation in electronics and the latest +communications technology, including text messaging, +video streaming, emails, and file sharing. +The TI is centered on the H2M interactions with the +haptic devices. To facilitate haptic communication, the +transmission of data via tactile internet creates a network +that is both extremely reliable and extremely responsive. +Haptic interaction is a type of interaction that includes the +use of remote touch. It refers to the kinesthetic perception +of information conveyed by the muscles and joints of the +body via force, torque, position, and velocity, as well as the +tactile perception of information conveyed by the +mechanoreceptors of the human skin via surface texture and +friction. Sharing information through kinesthetic mode +facilitates a global control loop with stringent latency +requirements, while containing a feedback of generally +audio/video type[93]. +Now with the enabling of haptic data, the TI enables a +networked control system (NCS) supporting the connected +sensors and actuators while controlling highly dynamic +processes. Therefore it helps in digitally transmitting the +sense of touch from one place to another, facilitating the +URLLC network in the B5G/6G system. In other words, the +tactile internet’s purpose is to provide a remote and +dynamic way for people to experience physical haptic or +the touch based control, while exchanging closed-loop +information between the virtual and physical worlds[90]. +Wireless communications can thus be a medium for +controlling and directing real and virtual objects using such +a platform. This revolutionary technology continues to +transform healthcare, transportation, education, logistics, +smart grid systems and many more, hence covering a major +portion of the economy sector in the society. It thus +provides sub-millisecond connectivity for the healthcare +applications like remote surgery. +This section draws attention to one of the most popular +applications of the B5G/6G communication system: +‘Tactile Internet’ and helps review its parameters, +applications and the basic architecture, while focusing on +its application in the intelligent 6G and IoT systems. +A. INTRODUCTION +The ‘Tactile Internet’, as coined by Gerhard P. Fettweis, +has been a catalyst for the economic development and +creativity and in bringing a new stage of maturity in +adapting technologies for a changing global environment. +[46]. Given that cellular communications connects a vast +majority of people worldwide, it is therefore imperative to +connect the technology as well. According to IEEE P1918.1 +working group, the Tactile Internet may be defined as a + +IEEEAccessCommand signals +Serving gateway + Packet data gateway +ngin +gNB +Tactile suppor +engine +Human operator and HSI +Teleoperator (Slave robot) +Slave Domain +Master Domain +gNB +Serving gateway +Packet data gateway +Haptic feedback +Bilateral ControlThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +15 +network of networks that can be accessed, perceived, and +manipulated by people or machines on a remote, real or +virtual basis in real-time perceptions[94]. +The tactile internet application offers the standard +required latency needed to guide and control real and +virtual +objects +without +causing +cyber +sickness, +revolutionizing +education, +accessibility +and +traffic, +healthcare, athletics, culture, games, and the smart grids, +thereby profoundly shaping our community. The i-phone +for instance has an astounding haptic interface, provided by +the gyroscope and the modern touch screen technology, +which has been a welcome step that could drastically +transform the way we connect. +Additionally, we have an instinctive (or innate) sense of +our surroundings, which has a tactile understanding of the +real time connection with our world. The tactile feedback +and a phone give our hands the sensations, which in turn +enable our whole system to be modulated with its +proximity. Whereas inside the vehicle it modulates several +sensors and controls elucidating a real time human-machine +communication interfaces. Thus all TI needs is a highly +responsive, smart, and reliable connectivity in order to +provide a medium for intelligent, real-time touch, control, +sensing and operation. +With high availability of TI, accompanied with its very +fast reaction time and reliability, the human interaction with +the machines enables a new dimension by creating an +interactive, real-time system, which revolutionizes almost +every segment of the society[94]. Taking into account the +industrial dimension, TI is an interconnected system of +specialized components and applications used in industrial +environments to monitor and operate the physical +equipment. Hence the works in [62], [89], [92] contribute to +the role of TI in industrial systems by examining its +potential in emerging and future industrial sectors. +Against this backdrop, the goal of this study is to identify +and address the cutting-edge challenges to implement an +intelligent touch enabled system via tactile internet in the +perspectives +of +B5G/6G +based +wireless +and +IoT +communications networks. Thus, tactile internet serves as a +key to realizing the vision of “Touch Technology”. The TI +has an expected potential and future scope through the +increasing penetration of mobile and cell platforms with the +robotic-human and computer interactions in the virtual +control +environments, +for +interconnecting +people, +machines, appliances and processes in real time. +In order to accomplish this, we present our discussion +with an overview of haptic communication via tactile +internet architecture, with an emphasis on its potential +development in the touch-enabled framework in 6G and +IoT mobile networks. +B. TACTILE INTERNET ARCHITECTURE +The haptic or the touch sensation helps ascertain a +connection +between +the +humans +and +peripheral +environment in a way analogous to the auditory and visual +senses. It therefore occurs bilaterally as a touch, sensed by +imposing a motion or a movement in an environment by +some reacting force. The haptic communication thus +provides an additional dimension and advantage over the +traditional audio visual communication for a real time +control and accessibility in the distant and remote +environment. The tactile internet architecture consists of a +radio access network (RAN) and a core network (CN), both +of which are expected to meet critical requirements for TI +functionality realization. +Each of the three domains in the tactile E2E architecture +can be separated into three sections: the master domain, the +network domain, and the controlled (slave) domain. The +master framework comprises of an operator, either human +or machine, and an operating system interface. Using +various coding techniques, this interface acts as a master +robot or as a controlling device, converting the operator’s +input into tactile input. If the controlling device is haptic in +nature, it lets human interact with objects in virtual or real +environments through physical means such as touching, +feeling, manipulating, or controlling. +It is primarily responsible for the controlled domain’s +operation. In case of a network controlled system, the +master domain includes a controller that issues command +signals to the sensor or actuator system. A domain, where +both robotic machines and other objects at distant locations +in a controlled environment, and directly accessed by the +master domain via command signals is referred to as a +controlled domain. When remote operations are carried out +via haptic feedback, energy is transferred between the +master and the controlled domains and the global control +loop is completed. +As apparent from Fig.7, the components in the E2E +tactile internet architecture are explained as below: +1) MASTER DOMAIN: Generally, an HSI/HMI8 is a +robot where the user may touch, feel, and move +virtual and real-world items while directing the +actions in the slave domain. +2) SLAVE DOMAIN: +The slave domain is controlled by a tele-operator, +through different command signals from the +master domain, interacting with its surroundings. +3) THE NETWORK OR CONTROLLER: +The network domain kinesthetically integrates the +person +with +its +surroundings +and +distant +environment. +4) TACTILE SUPPORT ENGINE: Being on the edge +network, it effectively offers AI capabilities that +are crucial in system stabilization. +5) HAPTIC +DEVICE: +Enables +the +tactile +communication, which means a user may touch, +feel, and engage with virtual or real-world things. +As a result, the most common design for a tactile haptic +device is a linkage-based system that consists of a robotic +arm connected to a stylus and capable of applying force on +its tip. Thus, the growing degree of freedom (DOF) is an +essential phenomenon for the envisaged applications that + +8 HSI/HMI: Human System Interface or Human Machine Interface + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +16 +integrate the network interface in a direct and indirect +connection with a cellular network. Since they both include +sensations rather than conventional multimedia, it is +essential to differentiate between the tactile internet and the +haptic communication, which are comparable to traditional +multimedia like speech, data, and video. +To summarize, haptic communication networks include +communications across the wired and mobile internet, as +well as applications that operate on the tactile internet. This +means that the haptic communications and the tactile +internet have a service and medium connection with each +other, respectively. +V. +NEED FOR INTELLIGENCE +The existing and the future B5G/6G wireless network +are expected to endow the users with an improved coverage +and high data rates with a better cost efficiency, resource +utilization, scalability, adaptability and security. Hence the +B5G/6G wireless communication system is anticipated to +be a backbone of the digital revolution in the next +generation network providing a ubiquitous reliable and +practically an instantaneous connectivity for the humans +and machines. The Artificial Intelligence (AI) may be +regarded as the ‘processing and simulation of the human +intelligence by machines’ and therefore has a great +potential +in +working +out +several +intractable +and +unstructured problems containing large amount of data[84]. +In other words AI may be defined as a science of +constructing computers that are capable of performing tasks +requiring human-like cognitive intellect[95]. All the while +AI has therefore been a widened approach for the machines +to be able to smartly carry out assigned tasks. ML on the +other hand is presently the widely accepted application of +AI, empowering the machines to train and learn from large +datasets and perform tasks without any need for explicit +programming. Next in order is deep learning (DL), a subset +of ML that analyses the artificial neural networks (ANNs). +These have more number of hidden layers to replicate the +human brain, making it one of the most widely used ML +techniques. DL therefore has a lucrative application in +fields like computer vision, bioinformatics and speech +recognition. +Such +induced +intelligence +in +the +wireless +communication network not only reduces the manual effort +in network deployment, configuring and management but +also helps in an improved system performance. It also +asserts the adaptability and reliability of the communication +network by taking robust decisions in real time according to +the prediction and behavior of the users and network. +Hence due to the recent advances and research in the +intelligence (AI/ML), a wide range of novel technologies +like self driving cars, voice assistants, holographic +telepresence, e-health and wellness applications, pervasive +connectivity +in +smart +environments, +industry +4.0 +applications, massive robotics with the unmanned mobility +in 3D, AR/VR have become possible. +6G wireless networks thus offer a broadband network, +which is fast, instantaneous and safe, in order to enable +mass data exchange at various frequencies using a wide +range of technologies. In addition, these technologies are +moving towards intelligent devices in IoT that will demand +a more reliable, stable, efficient, and a secure connectivity. +Hence the complex connected devices therefore require a +dynamic communication network to address their inherent +complexity. The future wireless networks will eventually +need a self-organizing and configuring capability alongside +their cooperation and coordination between the different +nodes and communication layers. +It enables us to effectively meet challenges like +coverage, spectrum, and energy efficiency. The continuous +acceleration of the machine type communication (MTC) +devices adds to the existing ultra dense network’s +complexity. Therefore many such applications supported by +B5G network must attain a short transit time and low +latency with high reliability, availability and security. The +majority of these are resource constrained and unable to +rely on their bounded resources and thus call for an +uninterrupted and safe operation as its main concern. +Consequently these applications owing to their delay and +bandwidth constraints cannot be moved to the cloud or +network controller[96]. +Furthermore these devices generate diverse range of +datasets with a large scale of erroneous or missing values. +Many wearables with VR/AR, intelligent products and +support systems, and other data hungry use cases have a +built-in end infrastructure to afford and deliver content +based services. Thus in order to incorporate an intelligent +system, an intelligent and content aware approach must be +implemented for the planning, design, analysis and +optimization of such network. This necessitates integration +of the network systems with their data sources, decision +making and cyber physical infrastructures, as well as +sensing and communication networks [97]–[99]. +Conclusively +the +favorable +conditions +for +the +implementation of the intelligent learning techniques in 6G +wireless networks range from: +1) Network interoperability with the distribution, +affordability and accessibility of computing +resources. +2) The +predictive +nature +of +the +network +characteristics. +3) Accessibility of a considerable amount of data. +Therefore a densely integrated wireless network may be +engineered +by +adopting +artificial +intelligence +(AI) +principles combined with the incorporation of machine +learning (ML) techniques with reasoning and decision +making mechanisms. Accordingly the development of an +intelligent touch based system calls for a promising +development +in +facilitating +the +efficient +resource +distribution in the cloud, fog and edge nodes, aiming to put +together system intelligence and data processing abilities in +close proximity with the original data source. + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +17 +A. MACHINE LEARNING IN B5G/6G WCN +ML as a member of the quintessential AI technology has +nowadays become a key component of 6G wireless +communication network. ML is said to be of a plausible +advantage in the communication system owing to fact that +the dynamic nature of the wireless communication channels +complicates the channel interference models in the B5G +scenarios. Therefore ML techniques are capable of +extracting information from the unknown channel by +learning from the communication data while taking into +account previous learning experience. Furthermore, the +rapidly growing number of wireless access points +necessitates a global optimization of communication +resources as well as a fine-tuning of the system design[73]. +However the existing approaches together with the +massive amount of resources complicate the tasks +concerning the optimization and correlation of the system +parameters. In contrast, advanced ML techniques like as +deep learning and probabilistic learning can represent +highly nonlinear relationships and aid in the determination +of optimal system parameters. Sequentially, ML aids in the +realization and instillation of learning-based adaptive +network configurations by identifying and evaluating their +behavioral patterns in advance rather than reacting to +unanticipated outcomes[63]. +As a result, the current cellular networks that were built +and managed based on the preceding premise may be +unable to keep up with the growing complexity of data +produced and therefore fail to provide the necessary +capacity, dependability, and flexibility. Now as response, +the network may be unable to respond fast enough to +expected occurrences, thereby compromising real-time +communication services. Because the majority of AI/ML +algorithms +are +not +purpose-built +for +the +wireless +communication networks, it is difficult to apply them +directly to the B5G/6G networks. +All the above arguments call for an intelligent +communication interface in the real time, facilitating a +stable and efficient connectivity within the network. The +6G wireless communication system guarantees a wide +range of frequency bands, including sub-6GHz, mm-wave, +THz, and optical bands, while also increasing the +computational complexity in the channel model. As a +consequence, comprehending new channel characteristics +for modeling new channel scenarios is a lengthy process. +Owing to the significantly high computational channel +complexity in many situations, traditional techniques may +aid in certain approximations and assumptions to help +simplify +the +channel +modeling +and +processing +methodology. This ensures the balance between accuracy +and complexity tradeoffs of both channel modeling and +processing methods is beneficial. +B. GENERALIZED WORK FLOW PROCESS FOR +MACHINE LEARNING +For most prevailing and functional machine learning +algorithms, the generalized work flow process in basic steps +is described and shown in Fig. 8 [100]: +1) PROBLEM FORMULATION: +Since the ML training process is time consuming, +it is critical that the problem be appropriately +formulated at the start of the process. Moreover +there should be a strong correlation between the +problem +and +the +information +gathered. +Classification, clustering, and decision making are +three important types of machine learning. The +proposed model should also be considered within +these three categories to aid in the identification of +learning model as well as data collection. Improper +problem formulation results in an unsatisfactory +learning model and performance. +2) DATA COLLECTION: +There are two types of data collection: offline and +online data collection, where data collected in real- +time is used as model feedback in online data +collection and is also used for re-training of +models. In contrast, offline data may be retrieved +from the source without an Internet connection +[101]. By utilizing monitoring and measurement +tools, online and offline data can be collected +efficiently, +securely +and +stored +for +model +adaptation. Data collection marks the beginning of +training and learning. Validation and testing are set +in motion after that. +3) DATA ANALYSIS: +It is divided into two types: preprocessing and +feature extraction where preprocessing is used to +reduce noise from the gathered data. The data’s +features are then extracted, which is a prerequisite +for learning and training [102]. The types of +characteristics that can be extracted from the +network include packet level features and flow +level features. The packet size, mean, root, and +variance are extracted at the packet level and mean +flow length and mean number of packet flow +features extracted at the flow level. +4) MODEL CONSTRUCTION: +While iterative process selection, training, and +tuning are all necessary aspects of the model +selection process, they are applied differently and +a suitable learning model must be chosen +depending on the dataset size. The training stage +entails training the model with the dataset that will +be collected at the start of the stage, whereas the +tuning stage have the model learn for itself by +comparing it to the trained data. +5) MODEL VALIDATION: +Cross validation of the testing process is used to +check the model’s accuracy, which aids in +optimizing the model and preserving the overall +efficiency of the system. +6) DEPLOYMENT AND INFERENCE: +Throughout the deployment and inference stages, +the model’s trade-offs and stability are monitored +to ensure accuracy and to determine the optimal +sequence of steps to be taken. + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +18 +TABLE V +MACHINE LEARNING TECHNIQUES AND THEIR APPLICATIONS IN B5G/6G SYSTEM. +ML +Techniques +Definition +Type +Principle +Applications in 5G /B5G communication system +m- +MIMO +Small +Cells +D2D Hetnets +Small +cell,D2D, +Hetnet +clustering +Spectrum +sensing +and +allocation +Resource +allocation +Anomaly/ +fault +detection +QoS +requirement +Outage +mgmt. +SINR +improvement +Channel +estimation/ +detection +CSI +Behavioral +learning +Cognitive +radio +Energy +harvesting +Smart +grid +SVM9 +Data point +separation using a +hyper plane or +kernel functions. +SL +Classifier +function: Linear +/non linear. + + + + + + + + + + + + + + + + + +KNN10 +Test point decision +by voting of the k +nearest neighbors. +SL +Non parametric +lazy learning +algorithm for +classification and +regression. + + + + + + + + + + + + + + + + + +K-Means +Clustering +Segregation of n data +points into K +clusters, each +associating to cluster +with nearest mean. +UL +Iterative +refinement with +cluster allocation +to the data point +with least ED11 +from it. + + + + + + + + + + + + + + + + + +Bayesian +Learning +Data points trained +by GM12, EM13, +HMM 14 models. +SL +A posteriori +probability +distribution. + + + + + + + + + + + + + + + + + +PCA15 +Relevant information +extraction from large +data sets using +orthogonal +transformation. +UL +Data sets reduction +into principal +components. + + + + + + + + + + + + + + + + + +Q Learning +A model free RL +where an agent has an +access to a set of +possible states and +environment, with no +concern of rewards or +transition between +them. +RL +Off policy RL to get +best action for the +current state. + + + + + + + + + + + + + + + + + +MAB16 +Multiple agents, +sequentially taking +actions receive +random reward to +achieve a steady +state. +RL +Trade off +between the best +action and +information to +achieve a larger +reward in future. + + + + + + + + + + + + + + + + + +MDP17 +A discrete time +stochastic control +state transitioning +process. +RL +A single agent +with partly +random and +partly controlled +states. + + + + + + + + + + + + + + + + + + +9 SVM: Support Vector Machine +10 KNN: K Nearest Neighbor +11 ED: Euclidean Distance +12 GM: Gaussian Mixture +13 EM: Expectation Maximization +14 HMM: Hidden Markov Model +15 PCA: Principal Component Analysis +16 MAB: Multi Armed Bandits +17 MDP: Markov Decision Process + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +19 + +FIGURE 8:Machine learning basic workflow process[100] +C. MACHINE LEARNING TECHNIQUES IN B5G/6G +WIRELESS COMMUNICATION SYSTEM. +A few existing works containing a comprehensive +overview of the existing intelligent techniques with their +application in the B5G/IoT wireless communication +systems are discussed here. Authors in [99], [101] talk +about different learning techniques in IoT applicable +scenarios while taking into consideration their input +information and computational complexity. While the +works in [103], [104] provide fundamental concepts about +the state of art AI based technologies applied in the existing +network, making the system competent enough to +accomplish self configuration, self optimization and self +healing of the concerned system. +Additionally authors in [58] discuss framework and +application of the popular deep reinforcement learning (RL) +technique that is vital in the engineering of the cognitive +smart cities. Furthermore the articles in [105], [106] further +provide a comprehensive survey of the existing RL +algorithms and their stability and behavioral adaptability +with the other learning agents in the implemented network. +Speech recognition, bioinformatics, and computer vision +are just a few of the applications for machine learning that +make efficient use of the technology. +Machine learning is primarily used for prediction and +classification, but it also plays a role in performance +prediction and intrusion detection in networking systems. +Therefore to make decisions directly, machine learning +constructs models that can learn from data without adhering +to a set of rules[107]. ML hence allows a model to enter a +self-learning mode without having to be explicitly trained. +To learn system characteristics that cannot be represented +by an explicit mathematical model, ML models therefore +are used as computing systems. +These models are employed in tasks including +categorization, +regression, +and +intelligent +agent- +environment +interactions. +Using +basic +arithmetic +calculations, the model can efficiently complete the task +once the system characteristics are learned[108]. Models +may be trained by providing them with data sets, and when +they are exposed to fresh data, they are able to learn, +forecast, and develop. There are three types of machine +learning algorithms: supervised learning, unsupervised +learning, and reinforcement learning. +In supervised learning, a model is trained on labeled +datasets and then learns on its own by comparing the +training dataset to the predicted output. This method is +commonly used for classification and regression issues. +Unsupervised learning is a type of machine learning that +uses an unlabeled dataset to detect patterns and +relationships in the data. It is mostly used to solve +clustering and association problems. During reinforcement +learning, an agent interacts with a system set and learns +how to map all information about action, without any +training data[109]. +This section provides a brief overview of some of the +most commonly used artificial intelligence and machine +learning techniques, with Table V illustratively defining +and highlighting various machine learning techniques and +their functionality in B5G and 6G wireless communication +systems. The works in [110]–[113] encompass few of the +researched investigations concerning the ML application on +the B5G/6G wireless communication channel. However +most of the existing works exercise intelligence on a limited +part of the existing wireless communication channel. Some +of the wireless channel characteristics influenced by +intelligent learning techniques are discussed below: +1) CHANNEL MODELING USING ML: +The ML helps deal with the channel modeling +problem by implementing the model based +approach, extracting the wireless channel features +from the existing data. ML has an efficacy in +predicting channel feature, estimating channel +parameter, +CIR18 +modeling, +MPC19 +and +classification of scenarios pertaining to the +concerning channel environment, derived out from +the above cited works. +2) CHANNEL MEASUREMENT: +A channel model based on a feed forward neural +network(FNN20) and RBFNN21 is shown in [114] +which +is +functional +in +predicting +channel +properties like received power, RMS delay and +angle spread(DS/AS).Therefore in addition to the +transmitter and the receiver coordinates, their input + +18 CIR: Channel Impulse Response +19 MPC: Multipath Component Clustering +20 FNN: Feed Forward Neural Network +21 RBFNN: Radial Basis Function Neural Network + +IEEEAccessStep:1 +Problem Formulation +(Prediction, Regression, +Clustering,Decision making) +Step:2 +Data Collection +No +(Realtimedatacollection +formodeltraining) +Step:3 +Step:5 +Requirement +Model Validation +Data Analysis +satisfied? +(Cross Validation, ErrorAnalysis) +(Preprocessing, +FeatureExtraction +Yes +个 +Step:4 +Step:6 +Model Construction +Deployment and Inference +(Offline training and Tuning) +(Tradeoffonspeed,memory,stabilityandinference +accuracy)This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +20 +parameters also influence the distance and +frequency of the Tx-Rx link. +3) NOISE : +ML in [115] makes use of ANN to remove noise +from CIR, while [116] makes use of CNN to +identify the relevant wireless channel. In CNN and +wireless +communication +channels, +MPC +characteristics like as amplitude, latency, and +Doppler frequency serve as input and output +parameters. +4) CHANNEL ESTIMATION: +In order to obtain accurate channel estimation, the +work in [117] uses a 2D non linear complex +support vector regression (SVR) in a rapidly +fading and time varying multipath channel. On the +other hand, the work in [118] refers to a deep +learning-based channel estimation method for +beam space mm-wave massive MIMO systems +that can learn the channel structure from a huge +number of training datasets. +5) MASSIVE RADIO INTERFACE: +ML algorithms help in analyzing the enormous +amount of data produced by the massive MIMO +arrays, where the conventional channel estimation +and detection algorithms are rendered incapable. +Deep +learning +methods +and +techniques, +particularly image processing and video analytics, +provide +the +most +exciting +algorithmic +approaches[119]. +6) WIRELESS NETWORK LOCALIZATION: +The continuous development and updating of +wireless channel locations has been made feasible +via automated learning from crowd-sourced data +employing a large number of mobile devices, +yielding precise localization results. In exchange, +it enables consumers to benefit from improved +location-based services. +7) NETWORK MANAGEMENT: +Machine learning and artificial intelligence has the +ability to optimize a variety of tasks such as fault +detection and user tracking over a wirelessly +linked network. +8) RESOURCE MANAGEMENT: +The resource management mechanism is only able +to function once the system has memorized the +states and conditions of the network users and their +real time wireless environment. This therefore +helps improve the system performance with time +and in turn helps the system incorporate an +intelligent +and +dynamic +decision +making +phenomenon. +Following consideration of all relevant factors, we intend +to incorporate intelligence into the tactile internet +infrastructure in order to achieve a complete automation of +the systems that surround us, taking into account the +feasibility, interoperability, and functionality of the 6G and +IoT-based wireless communication networks. An intelligent +touch-infused technological framework is proposed in the +next section, which incorporates intelligence from existing +smart IoT infrastructure and interfaces it with next- +generation systems by creating a tactile/haptic feeling as we +interact. +VI. +PROPOSED TOUCH TECHNOLOGY IN B5G/6G +AND IOT NETWORK +The +proposed +and +futuristic +anticipated +system +infrastructure is expected to encompass an intelligent and a +reconfigurable touch enabled system that is pertinent in an +IoT interfaced B5G/6G systems. The intelligent Touch- +based IoT paradigm can be made up of a variety of +functional elements that help smart objects perform various +functions such as sensing, actuation, identification, +management, and communication. The touch based IoT +system’s functional elements[3] can be summarized as +follows: +1) SMART DEVICES: +The primary components of the IoT based Touch +System, performing sensing, actuation, and control +functions are capable of sharing data with other +applications and servers. To connect to other smart +devices, each IoT device has to be prepared with +numerous interfaces including the internet access, +I/O interfaces for sensors, audio and video, storage +and memory interfaces. +2) FUNCTIONALITY: +The device functionality ranges from smart- +watches, +wearable +sensors, +automatic +cars, +industrial machines, LED lights and so on. From +office automation and household appliances to +manufacturing lines and commodities tracking, +intelligent IoT techniques are used in a wide range +of applications. As a result, IoT services must be +used to improve IoT application development and +accelerate installation. +3) SERVICES: +These are dedicated to identity and device +modeling and are commonly grouped under the +umbrella +term +identity +services. +Additional +subcategories include information aggregation, +discovery, +control, +collaborative +awareness, +ubiquitous services, and data analytics and +publishing. +4) REMOTE ACCESS: +As opposed to devices that use mechanical +switches or buttons to remotely manage, IoT +devices have either no human involvement or can +be remotely managed without the need for human +intervention. +5) SECURITY: +Taking into consideration the security aspect, as +far as the data on wireless networks is concerned, +especially with regards to denial of service (DoS), +spoofing, and eavesdropping, the information is +vulnerable to an array of attacks. Thus, IoT +systems use many security features, such as +privacy, +authorization, +authentication, +data + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +21 +security, content integrity, and message integrity, +to attempt to thwart these attacks. +6) IOT APPLICATION: +In order to provide IoT users with interfaces, the +application layer supplies IoT users with various +interfaces that enable them to monitor and control +the various aspects of IoT applications. +Thus the projected system is to be most likely based on +the IoT functionality, and is to be expected to be +implemented in the B5G/6G networks, where the system +intelligence is of an utmost need so as to be compatible +with the accelerating high data rate and in turn satisfy the +low +latency +requirements +of +the +next +generation +system[120], [121]. +A. INTELLIGENT TOUCH BASED SYSTEM +The problem statement here describes the need for the +intelligence in B5G system, and therefore can be elaborated +as per the works in [122]–[125] stating that the ever since +the evolution of the wireless network from 1G to the +existing 5G/B5G and subsequently the 6G networks, have +led to a tremendous ascend in the billions of connected +devices bound together by IoT, forming an integrated IoE +network. The increased traffic, due to growing number of +devices, therefore requires high energy efficiency and lower +latency. +The growing new use cases in the evolving B5G network +incorporates the AR/VR/XR based smart systems including +the smart road monitoring, the smart cities, consisting of the +smart homes and IoT governed smart appliances. These +systems are externally controlled and therefore greatly lack +in intelligence. Hence for the efficient functioning of these +devices in the next generation 6G network, an intelligent +system is required to govern the existing AR/VR based +5G/B5G network effectively. +It therefore requires an interfacing mechanism with the +existing network infrastructure. To implement this in real +time, the B5G/6G enabled tactile internet proves to be a +promising technology, which in turn can provide a new +dimension to human to machine interaction by enabling +haptic +sensations +and +therefore +a +touch +enabled +communication interface, in real time. The touch-governed +IoT system is expected to permeate many facets of the +contemporary daily living, including the ability to sense, +process, analyze, and infer environmental parameters from +natural resources and ecologies to human environments. +The main aspect of this touch-based IoT network is its +ability to intelligently connect to the other existing and +future networks, to all the resources that are utilized by +these networks, and to help accomplish that through the +advancement of the prior networks and communication +protocols. For this vision to succeed, we will need to +advance +beyond +conventional +mobile +computing +technologies and design an IoT system in which everything +we can touch is connected and capable of acting as a smart +and intelligent extension of ourselves. +An in-depth knowledge of IoT issues span from an +awareness and a better understanding of the IoT concerns +and the complexities involving the size and depth of the +pervasive communication network, software architectures, +and data transfer and processing. This knowledge is used to +create autonomous and intelligent devices in IoT systems. +The primary goal is to set up a global network of connected +smart objects and devices, all of which can connect to each +other without human intervention. +Each object that has been embedded with a smart +interface and connected to the user platforms and digital +environments is assigned a virtual identity and interfacing, +allowing it to connect and communicate with other +embedded objects[3]. As we build our IoT network, our +physical and virtual entities turn into virtual things in a +cyber world, each with specific abilities that all IoT entities +can use to realize personalization, specialization, and +autonomy based on the communications protocols used to +make the smart entities unique and provide them with +virtual personalities. +The combination of B5G/6G network slicing technology +and the TI application will thus prove to be a driving factor +in the realization of this suggested reconfigurable and +intelligent touch enabled framework, whose research is still +in its infancy. Consequently, it can be concluded that the +main goal of the IoT-based Touch technology system is to +improve the lives of people, where all objects around us +have the ability to figure out what we want, what we +require, and what we like, as well as serve it accordingly +without us having to explicitly command them. +The Fig.9 presents the layered architecture of the +proposed +intelligent +touch +technology +in +B5G/IoT +networks. Hence it requires an AI/ML technology, coupled +with the B5G/6G based Network Slicing and Tactile +Internet; to implement and interface the intelligent touch +based system. Hence the network slicing and tactile internet +explained in the previous sections proves to be the +cornerstone in the incorporation of a reconfigurable +intelligent touch system. +VII. +LAYERED ARCHITECTURE FOR TOUCH +TECHNOLOGY +The Internet of Things connects millions of smart objects, +increasing data traffic and necessitating the use of large +processors and storage systems. This emergence of IoT +system together with the rising demand for the wireless +capacity ultimately paves way for the futuristic intelligent +Touch Technology system. Many factors, including as +interoperability, scalability, QoS, and reliability, must be +considered while designing such an IoT based intelligent +infrastructure. As a result, the required intelligent IoT +architecture based on touch technology should have the +following characteristics: +1) DISTRIBUTIVENESS: +The IoT model for the proposed system should +enable data collection from various sources and +their processing by various smart entities in a +distributed manner. +2) INTEROPERABILITY: +IoT +devices +from +different +vendors +must +communicate to achieve common goals. + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +22 + +FIGURE 9: Proposed layered architecture for intelligent touch technology22 + +22The architecture presents the IoT Layered Infrastructure comprising of : Perception, Connectivity, Edge computing(network), Cloud Storage(transport) and Application layers, providing tactile-based +communication and device interaction, as well as efficient network slicing between the connectivity and application levels in order to transfer and completely support requested services and user requirements. + +IEEEAccessApplication Slicing +Controller Domain +Touch based sensors and Home Automation +Touch based Healthfacilities like +Tele-surgery, Remote medical consultancy +Tactile Haptic communication system +Touch/Tactile based robotics and automated vehicles +灭！ +Automated vehicles +Tactile based robotics +Smart classroom ++ +Response +W +Remote industries(HloT) +Haptic communication +lot sensors and devices +Ultra low latency of Ims +Data Accumulation (storage) +layer +Edge (Fog)Computing layer +@actile suppor +Tactile support +Foy node +engine +I/MLAlgorithmfor +touchimplementatior +Touch based applications +Cognitive (smart antenna) +Basestation +Point +Power optimization in Receiving antennas +0 +Spectrum sharing +Rout +Router +Gateway +Comnectivity layer +RAN Slicing +Command +Human controller +Program control system +lio/video +withHST +feedback system +Perception layer +Master Domain with HSI (edge devices)This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +23 +Additionally, protocols and systems must be +designed in such a way that smart devices from a +variety of manufacturers can exchange sensed data +in an interoperable manner. +3) SCALABILITY: +Any such IoT network is expected to have billions +of objects connected at any time. Because these +platforms run many applications and systems, +hence such applications should be able to handle +and process a huge amount of data. +4) LIMITED RESOURCES: +Both computing units and energy are considered +highly rare units in regards with the limited +resource availability. +5) SECURITY: +The feelings of helplessness and vulnerability that +users have when they are under the control and +dominance of an unknown external device could +be detrimental to the given system deployment. +This section presents the proposed layered architecture for +the intelligent and touch enabled technology which is to be +functionally implemented in the B5G/6G and IoT +configured +wireless +communication +network. +This +architecture, which is defined by the tactile internet and +network slicing as its cornerstone components, can thus be +broadly classified into master, controller, and network +domains. +The master domain is comprised of user-facing edge +devices such as a robotic arm, gesture-based glove, or a +gaming console for real-time online gaming, which +collectively form the HSI network. Thus, this layer contains +various end devices and sensors, which aid in the touch- +based sensory data transmission. All of this is located on +the perception layer as per the layered architecture +presented before. The perception layer is responsible for +transmitting the generated sensory data to its required +destination, over the network. +The reliable and timely transmission of data from the +perception layer devices to the edge computing layer is +ensured by the efficient connectivity between the layers. +Connectivity layer is therefore responsible for processing +and +communication +between +the +master +devices +accompanied with their effective routing and switching, +enabling a reliable delivery of information across the +network. This layer also ensures the security of the +network. +The third layer is the edge computing or the Fog layer, +where the data evaluation is done and is processed at higher +levels. The information is then processed and transferred by +the tactile support from the end user to the base station. +Thus the fog and edge network therefore facilitates a +distributed computing, storage; control and communication +of the network functions with reduction in the latency, +system response and cloud workload of the transmitted +data. It is further connected to the RAN layer which is +responsible for introducing intelligence in the system +through radio network. +The +B5G/6G +network +slicing, +virtualization +and +fragmentation takes place in this layer, the details of which +are explained in the earlier sections. Hence to further avail +the tactile facilities, the RAN slicing is accomplished, +fragmenting it into the isolated frameworks, each of which +is further meant to furnish various high data rate +applications simultaneously, while maintaining the latency +of the requisite ≤1ms. The routed network endows with a +provision for network slicing and storage of data, enabling +access to the various cloud services, for effective data +storage and acquisition. +The data acquisition process, before further furnishing the +requisite applications, follows the data accumulation and +abstraction. Data accumulation involves the data capture +and storage, to be used by the applications later. It also +deals with the query based data processing. Data abstraction +further virtualizes and consolidates the data at a place +before its slicing and cloud storage. The setup of the +signaling procedure and protocol stack along with the +physical layer optimization allows the accessible devices to +direct and control the power of the sensors, edge, fog, and +cloud-based platforms. +The application layer further furnishes the applications +due to application slicing. Here the applications required +are entirely based on the TI and are intelligent and touch +enabled. These may vary from the haptic communication by +the haptic and IoT sensors, the automated remote industrial +operation and monitoring, which is an example of the +industrial revolution 4.0. It has a great potential in the +evolution of healthcare, education, entertainment as well as +edutainment. Since the application layer is in charge of +monitoring, controlling, and analyzing the data, it must be +at the heart of all systems. +The business layer in the end provides these services at +the consumer end i.e. at the controller domain at the +receiver at the destination. For an easy understanding, let's +go over each layer of the architecture with their associated +protocols and functionality. Following the bottom up +approach, the five layered IoT configured architecture of +the proposed touch technology comprises of the following +layers: +1) Perception layer. +2) Connectivity (Data Link) layer +3) Fog/Edge computing (Network) layer +4) Data Accumulation (Transport) layer +5) Application layer +As a result, we will go over each of these layers one by +one, explaining their functionality and interoperability as +we go. This section describes each layer one by one with its +functionality and interoperability beginning with the +application layer upto the perception layer following a top +down approach [126] in Fig 9. +A. APPLICATION LAYER +As the front end of the IoT architecture, the application +layer is where the majority of the IoT potential will be +realized. This is because the application layer provides IoT +developers with the user interfaces, platforms, and tools +that are required to implement IoT applications such as + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +24 +smart homes, intelligent transportation, smart health, and +smart cities, among others. Furthermore, it is in charge of +receiving the data that has been processed from the network +layer. +The IoT application layer provides appropriate protocols +and services needed to transmit messages at application +level. When choosing a communication protocol for a +certain application, many elements should be considered, +including power consumption, necessary BW, transfer and +connection time, delivery guarantee, data security, and +packet size. A description of a few protocols that are +commonly used at the application layer is provided below +and summarized in the Table VI: +1) MQTT: It uses middleware and apps to enable +communication between IoT devices and the +network in a variety of ways, including M2M, +server to server, and machine to server, and it runs +on the TCP/IP protocol. Additionally, it supports +communication over low-bandwidth and unreliable +links and is used for publishing and subscribing to +lightweight message exchanges[127]. +2) XMPP: It allows for low latency communication +and minimal message delivery, making it ideal for +video calls, instant messaging, and chats, as well +as publish/subscribe systems, gaming, and IoT +applications. Because of its simplicity and +adaptability, it makes it possible to communicate +between heterogeneous systems [127]. +3) REPRESENTATIONAL +STATE +TRANSFER +(RESTFUL): The REST protocol is a collection of +best practices, rules, and constraints for developing +web services that enable data exchange and +communication between various devices, as well +as for developing distributed hypermedia systems +with desirable properties such as modifiability and +scalability. RESTful is a request-response and +client-server architecture that allows clients to +access server resources in IoT contexts. It is based +on the HTTP protocol. Because they are +lightweight +and +straightforward +protocols, +RESTful APIs are considered to be a good choice +for a variety of IoT applications[128]. +4) CONSTRAINED +APPLICATION +PROTOCOL +(COAP): In IoT applications, it enables resource- +constrained +and +unsynchronized +devices +to +communicate while providing flow control, +reliable delivery, and simple congestion control. It +uses multicast and unicast requests to enable +publish-subscribe communication strategy. CoAP +uses UDP due to its small message size, low code +footprint, and lack of TCP handshake overhead +before transmission[129]. +5) AMQP: It is widely used in commercial and +business domains because it offers reliable and +secure communication between heterogeneous +devices +and +supports +publish +subscribe +architecture based on an efficient and reliable +messaging queue. In addition, it uses the TCP +protocol for increased reliability. The message +queue and exchange queue models are used to +transfer data over AMQP. In the message queue +model, messages are retained until they are +transmitted to the receiver, whereas in the +exchange queue model, messages are routed in a +proper order[129]. +TABLE VI: +SOME COMMONLY USED APPLICATION LAYER PROTOCOLS +Protocol +MQTT +XMPP +REST ful +CoAP +AMQP +Standard +OASIS23, Eclipse +Foundations +IETF +IETF +IETF, Eclipse Foundation +OASIS, ISO/IEC24 +TCP/UDP +TCP +TCP +TCP +UDP +TCP +Architecture +Publish/Subscribe +Publish/Subscribe; +Request/Response +Request/Response +Publish/Subscribe +Publish/Subscribe +QoS: + + + +Confirmable, Non-confirmable, +Acknowledge, Reset + +Semantics +Connect, +Disconnect, +Publish, +Subscribe, +Unsubscribe, +Close +Get, Post, Put, Set, Result +Post, Put, Delete, Cut +Post, Put. Delete, Get-CON +(Confirmable), Non (non- +confirmable). ACK +(acknowledgement),RST(Reset) +Consume, Deliver, +Publish, Get, Select, +Ack, Delete, Nack, +Recover, Reject, Open, +Close +Security +TLS/SSL25 +TLS/SASL +TLS/SSL +DTLS/IPSec +TLS/SSL, IPSec, SASL +Features + For high latency +and low BW +networks. + For resource +constrained +devices + Decentralization by server +as there is no central +master server. + Flexibility + Open standards + High network overhead + Scalability + Easy implementation +and interaction + Flexibility + Unsuitable for +distributed networks. + Reliability + Lost packets retransmission + Multicast support + Resource monitoring + Low overhead + Scalability + Reliability + Performance security + Heavy protocol as it +requires memory and +power resources. + +23 OASIS: Organization for Advancement of Structured Information Standards +24 ISO/IEC: International Organization for Standardization/ International Electrochemical Commission +25 SSL: Secure Sockets Layer + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +25 +TABLE VII +SOME COMMONLY USED DATA ACCUMULATION LAYER PROTOCOLS +Protocol +TCP +UDP +DCCP +STCP +TLS +DTLS +RSVP +Packet size +20-40 bytes +header +8 bytes header +12-16 bytes +header +12 bytes header +5 byte header +224-1 bytes +(handshake +message) +16 bytes header +Packet +transport +form +Segment +Datagram +Datagram +Datagram +Segment +Datagram +Datagram +Flow +control + + + + + + + +Congestion +control + + + + + + + +Error +detection + + + + + + + +Reliability + + + + + + + +Features + Performance, +robustness and +network capacity +improvement. + Orderly data +delivery between +hosts. + Transmission not +possible if +sequenced packet +is not +acknowledged. + No guarantee on +packet delivery + High packet loss + Packets may +arrive out of +order. + Retransmission +needed when +data is corrupted. + Eliminates delay +in out of order +waiting data +packets. + Supports strict, +partial and +unordered +delivery modes. + Multi-homing +support + Congestion +control. + Flexibility for +VoIP +applications + Multi-homing +method + Minimizes DoS +attacks + Dynamic IP +addressing + Prevent +tampering by +intruders + Prevent +passive +listening by +attackers + Adds more +latency + Security and +reliability for +handshake message +transmission. + Unordered message +queuing + Retransmission +timer to reduce +packet loss +probability. + Slower than STCP + Data integrity. + Error reporting + Multicast +comm. among +heterogeneous +devices. + QoS routing + Scalability +issue. + +B. DATA ACCUMULATION/STORAGE LAYER: +It interacts with the application layer to send and receive +data without mistakes in the typical way. The transmitting +side here is in charge of breaking down messages received +from the application layer into segments and sending them +to the network layer. Following that, the segmented +messages received will be reassembled into messages that +will be passed directly into the application layer by the +receiving side of the communication channel. +This layer is responsible for ensuring the integrity and +reliability of transmitted data by providing features such as +packet delivery order, congestion avoidance, multiplexing, +byte orientation, and data integrity. Hence known as the +routing layer because it is in charge of routing data packets +through the network area, where its protocols are in charge +of packet ordering, error detection, and correction. This +section provides an overview of a few protocols that are +commonly used at the data accumulation layer. The +protocols are summarized in the Table VII: +1) TRANSPORT CONTROL PROTOCOL (TCP): It is +a heavyweight, connection-oriented protocol, +where the connection is not established until all the +required data has been sent between each end +device. TCP is suited for reliable communication +since it requires acknowledgement messages to +ensure each sending and receiving procedure, as +well as support for retransmission of lost or +corrupted packets and a flow control mechanism. +Hence, +this +protocol’s +packet +overhead +is +extremely high, resulting in increased power +consumption from devices and thus making it +impossible to operate on power-constrained +devices. TCP divides the data packet into multiple +packets, each with an ordering number and source +and destination IPs[130]. +2) USER DATAGRAM PROTOCOL (UDP): It is a +connectionless protocol that attempts to give +protocols and applications that run over IP, an +unreliable, minimum message queuing, message +passing, and best effort transport. There is no +requirement for end-to-end connectivity between +communicating entities, which enables efficient +communication for some applications that require +real-time performance with low latency, such as +video and voice. UDP does not provide a port with +any attribute for addressing the source and +destination functions and provides a data integrity +check [130]. +3) DATAGRAM +CONGESTION +CONTROL +PROTOCOL +(DCCP): +It +establishes +unicast +bidirectional connections for datagrams with +unreliable dynamic congestion control. These +characteristics make DCCP ideal for applications +that transmit large amounts of data and require a +trade-off between reliability and timeliness, such +as VoIP and media streaming. Due to its +unreliability and absence of a receiving window, +the flow rate of DCCP can be progressively +increased[130]. +4) STREAM +CONTROL +TRANSMISSION +PROTOCOL (SCTP): It is a connectionless, +message-oriented, IP transport layer protocol + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +26 +similar to UDP that enables SCTP-based peer-to- +peer +(P2P) +communication +and +reliable +transmission for applications communicating over +an IP network. As a result, it inherits the majority +of TCP’s functionality, such as packet recovery +and congestion management[130]. +5) TRANSPORT LAYER SECURITY (TLS): It was +created to provide security channels among +communicating peers and to give authentication, +data secrecy, data integrity, and encryption to +applications +by +preventing +eavesdropping, +message forging, and tampering. It runs on top of +several transport layer protocols. It is composed of +two components: the handshaking protocol, which +is responsible for authenticating communication +ends, agreeing on shared keys, and negotiating +cryptographic parameters and modes; and the +record protocol, which divides the traffic into +multiple records and protects them using the traffic +keys[131]. +6) DATAGRAM TRANSPORT LAYER SECURITY +(DTLS): It was created to secure datagram +applications that do not require or provide +dependable data delivery, such as datagram online +gaming, internet telephony, and media streaming, +which are deemed delay sensitive. DTLS is an +enhancement of the TLS protocol that prevents +message forgery, tampering, and eavesdropping +when transmitting data streams. Therefore, it +should be able to cope with and resolve a variety +of datagram difficulties, such as packet loss, +packet reordering, and delay[132]. +7) RESOURCE +RESERVATION +PROTOCOL +(RSVP): It is a multicast and unicast control +transmission protocol that was created to enable +data stream transmission with a flexible, robust, +scalable, and heterogeneous resource reservation +setup at each router. There is also support for +resource reservations in each node along the data +path, multipoint to multipoint communication +paradigm, cache management routers, and receiver +initiated reservation[133]. +C. EDGE/FOG COMPUTING LAYER: +It is responsible for providing data with routing paths so +that the packets can be transmitted across the network area. +This layer creates logical connections, sends out error +messages, and maintains the data transmission routing path. +Furthermore, this layer contains all network devices such as +switches, firewalls, bridges, and routers that are required to +work with appropriate communication protocols such as +3G-4G-5G, Wi-Fi, infrared, ZigBee, and Fibre to the X. +This layer is in charge of forming, addressing, and routing +data packets, as it receives datagram packets from the +transport layer and converts them to data packets before +transmitting them to the destination side. This section gives +an overview of some protocols commonly used in the +edge/fog computing layer with the protocols summarized in +Table VIII: +1) +ROUTING PROTOCOL FOR LOW POWER AND +LOSSY NETWORK (RPL): It is a tree-based, IPv6 +proactive +distance +vector +routing +protocol +developed by the routing-over-low-power and +lossy networks working group to run commercial +appliance networks with insecure connectivity, +poor data rates, and substantial losses. It has a +storing and non-storing mode to reduce memory +requirements and eliminate loops in low-resource +applications. It is prone to high packet loss owing +to congestion, has a long delay, and is vulnerable +to assaults since it lacks end-to-end encryption. +Overhead packets for control are flooded into the +networks as a result[134]. +2) +COGNITIVE ROUTING PROTOCOL FOR LOW +POWER NETWORK (CORPL): It is an extension +of the RPL protocol, which was designed to be +compatible with cognitive networks in order to +improve performance. This feature ensures high +packet delivery ratio, and keeps the network from +colliding. The CORPL routing mechanism takes +advantage of an opportunity to select the most +effective forwarding next hop from a pool of +eligible neighboring nodes. There are minimum +collisions and duplication of data packets[135]. +3) +CHANNEL +AWARE +ROUTING +PROTOCOL +(CARP): It is a distributed protocol with a +lightweight data package that was created for +underwater +and +IoT +applications. +Network +initialization and data forwarding are two steps in +the routing operation performed by CARP. The +receiving node updates its distance to the sink +node, broadcasting the welcome messages with +their ID and hop count. In data transmission, the +sender sends a ping message to its neighbors, +determining the best relaying node based on the +link quality and information received from their +ping messages, and then forwards data. When +selecting a relaying node, residual energy, network +quality, and buffer space are all taken into +account[136]. +4) +COLLECTION TREE PROTOCOL (CTP): A tree- +based routing system that was created to give the +greatest effort for any cast communication in +networks with low energy demands. An early form +of networking is where a source node (sink node) +announces itself as the root node, where minimal +cost is paid to deliver data to the root. Other nodes +connect to the root tree via lightning ads and then +send their collected data into the sink node with +the minimum amount available. CTP, on the other +hand, does not permit reverse routing from the +sink to the sensors[137]. +5) +LIGHTWEIGHT +ON-DEMAND +AD +HOC +DISTANCE VECTOR ROUTING PROTOCOL- +NEXT +GENERATION +(LOAD-NG): +When +compared to the on demand distance vector +(AoDV) protocol, it is a more lightweight distance + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +27 +vector and reactive protocol that is designed to +provide a secure, scalable, and efficient routing in +lossy and low power networks. As a reactive +protocol, LOAD-ng generates on-demand route +requests to discover a path to the target node and +when data has to be delivered, the receiving +unicast replies hop by hop from the destination +node back to the sender node. If a route is found to +be broken, attempts to fix it are made, or an error +message is sent to the requested node[138]. +6) +AD-HOC ON DEMAND MULTIPATH VECTOR +FOR IOT (AOMDV-IOT): It seeks to discover and +establish connections between nodes and the +internet. For each node, AoMDV-IoT generates +two routing tables: an internet connecting table +(ICT) and a routing table. In addition, it converts +IP addresses into internet linking addresses (ILA) +and when a node wants to be connected to the +internet, the IP associated with the desired internet +is connected to ILA so that the search function can +be utilized. If there is no internet node in ICT, the +source node will broadcast the requested packet to +update both tables until an optimal route to an +internet node is discovered[139]. +D. CONNECTIVITY LAYER: +The IoT connectivity layer in the touch system +architecture is comprised of a variety of communication +protocols that are primarily responsible for providing +services to the network layer. Hence, it is in charge of +connecting and transmitting signals from end devices to +higher layers via routers and gateways. +The IoT connectivity layer consists of a variety of +communication +protocols +that, +depending +on +the +transmission range and coverage area, provide services to +the network layer. Some of the most commonly used +protocols are reviewed further down this section. +1) +NFC PROTOCOL: A short-range protocol that +allows mobile objects to communicate with one +another over a few cm of distance and allows data +to be transferred in seconds between the connected +NFC devices that are in close proximity to one +another. It is RFID-based and thus uses an +alternate magnetic field to connect devices that are +either active or passive. In active mode, all of the + +TABLE VIII: +SOME COMMONLY USED EDGE/FOG COMPUTING LAYER PROTOCOLS +Protocol +RPL +CORPL +CARP +CTP +LOAD-ng +AOMDV-IoT +Network +Topology +Mesh, Hierarchical +Cognitive M2M +networks, Mesh +- +Tree-based topology, +Mesh +Grid +Dynamic IoT +network +Routing +metrics +Connectivity, Link +quality, BW, +Reliability. +Reliability, +Collision risk +End to end packet +latency, energy +consumption per bit, +buffer spaces, packet +delivery ratio. +ETX26 of neighbors +Hop-count +Lifetime Hop +count +Network +metrics +MP2P,P2MP27 +communication +P2P, MP2P, +P2MP +MP2P, P2MP,P2P +MP2P,P2MP +P2P +P2P,P2MP +Data rates +Low data rates +Low data rates +Low data rates +Low traffic rates +- +- +Latency +High Latency +Delay of +sensitive +applications +Low Latency +High Latency +High Latency +Low Latency +Algorithm +Distance vector +Distance vector +Link state +Distance vector +Distance vector +Distance vector +Scalability + + + + + + +Security + + + + +Uses integrity +check values. + +Network +mobility + + + + + + +Applications  Home automation. + Industrial +automation + Building automation + Smart grid and +Smart cities. + Smart Grid. + Underwater WSNs +application + Commercial products + Industrial WSNs + Teaching + Research + Home applications + Industrial +application + Mobile IoT +applications + +26 ETX: Expected Transmitted Count +27 MP2P: Multipoint to Point Communication ;P2MP: Point to Multipoint Communication + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +28 +TABLE IX: +SOME COMMONLY USED SHORT RANGED CONNECTIVITY LAYER PROTOCOLS. +Protocol +NFC +6LowPAN +BLE +Zigbee +Z-Wave +Network type +P2P +Star, Mesh +Star +Star, Tree, Mesh, Cluster, +Hybrid +Mesh +Frequency +band +13.56MHz +2.4GHz +(2.402-2.481) GHz +2.4GHz, 915MHz, 868MHz +868(Europe), +908(USA), 900ISM +Transmission +range +10cm +(10-100)m +100m +(10-100)m +30m +Data rate +106Kbps-424Kbps +(20,40,60 )kbps +125kbps (12 Mbps) +250Kbps +(9.6, 40, 200)kbps +No. of nodes +2 +65000 +65535 +65000 +232 +Power +consumption +15mA +- +15mA +30mA +5mW +Routing +protocols +NFC possesses routing +features +RPL, AoDV +RPL, 6LoWPAN +Zigbee, RPL, AoDV, Zigbee +network routing (ZBR) +AoDV, DSR +Applications + P2P data transfer + Payment and ticketing +applications + Smart home + Smart agriculture + Industrial IoT + Healthcare +applications + Mobile phones + Smart homes + Wearables and PC + Security and privacy + Healthcare + Sports and fitness etc + Smart home + Medical monitoring + AI sensors + Consumer electronics. + Home automation + Smart lighting + +connecting devices generate a magnetic field, +whereas in passive mode, one device generates a +magnetic field while the others use load +modulation to transmit data. Passive mode is +energy saving and is widely used in today’s smart +phones[140], [141]. +2) +6LOWPAN: It allows smart devices to connect to +the internet via the IPv6 protocol while also taking +into account the nature of wireless IoT networks +by creating a very small header message format. It +also removes obstacles to using IPv6 addressing +protocol in IoT devices with limited processing +power, data rate, and power[142]–[144]. +3) +BLUETOOTH LOW ENERGY (BLE) PROTOCOL: +BLE is a low-power alternative to short-range +wireless +communication +developed +by +the +Bluetooth Special Interest Group. Additionally, it +allows for fast data packets to be transmitted at +speeds up to 2Mbps in the ISM band[145]. +4) +ZIGBEE: The objective is to develop a low-cost, +scalable and power-sipping wireless connectivity +that is suitable for a wide range of controlling and +monitoring purposes. With intelligent routing and +setup procedures, this protocol builds on IEEE +802.15.4’s features to enable high failure resilience +and easy installation. It also works well with other +wireless communication technologies due to its +strict security and listening techniques[146]. +5) +Z-WAVE: Smart light controllers and other sensors +in home devices use this low-power wireless +communication technology. With low latency +transmissions and data rates of up to 200kbps, this +technology +operates +over +900 +MHz +ISM +bands[147], [148]. +6) +LOW +POWER +WIDE +AREA +NETWORKS +(LPWAN) PROTOCOLS: LPWAN protocols are +low-power, low-bandwidth, and low-cost protocols +that are particularly useful for long-distance +communications. Furthermore, the devices that +implement these protocols have transmission +ranges ranging from 1m to 50 km. The general +characteristics of LPWAN protocols are as +follows, followed by a brief discussion of each +protocol: +a. +These +LPWAN +protocols +are +implemented by low-power devices. +b. These protocols are limited to the +transmission of small smart packets, +typically 100 bytes or less. +c. +Devices +that +implement +LPWAN +protocols are composed of extremely +low-cost components, typically costing +less than a few dollars. +d. Within and outside their domains, these +devices are designed to provide good +coverage and reliability. +7) +LONG +RANGE +WIDE +AREA +NETWORK +(LORAWAN): It is a physical layer communication +protocol that uses very less power and has a +battery life of up to ten years. LoRaWAN +specializes in M2M, industrial and smart city +applications, +which +require +long-range +communication, ranging from 2 to 5 km in urban +areas and up to 15km in suburban areas. The +process of communication through large networks, +which contain billions of intelligent devices, also +promotes the data rates of this protocol in the +complete duplex wireless medium, from 0.3 to +50Kbps[149]. + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +29 +TABLE X: +COMPARISON OF LPWAN PROTOCOLS +Protocols +LoRaWAN +SigFox +NB-IoT +Topology +Star of stars, +Mesh +Star +Star +Modulation +CSS28 +BPSK +QPSK +Frequency +Unlicensed ISM +bands: 868 +MHz (Europe), +915 MHz (N. +America), 433 +MHz (Asia). +Unlicensed ISM +bands: 868 +MHz (Europe), +915MHz (N. +America), 433 +MHz (Asia) +Licensed LTE +frequency +bands +Transmission +range +5 km (urban), +20 km (rural) +10 km (urban), +40 km (rural) +1 km +(urban),10 km +(rural) +Data rate +250bps-50kbps +100bps +200kbps +No. of nodes +1000 +100 +5500 +Power/ +Current +consumption +50mW +(3-50)µA +500mW-4W/ +(19-49)mA +Handover +No end devices +will be +associated with +a single base +station. +No end devices +will be +associated with +a single base +station. +End devices are +associated with +a single base +station. +Applications + Smart city + Smart logistical +and +transportation + Industrial +application + Real time +monitoring + Video +surveillance + Smart farming + Status +monitoring + Smart building + Asset tracking +and logistics + Electric +metering + Manufacturing + Automation + Smart city +8) +NB-IOT: It is a narrowband radio technology that +was developed and standardized by the 3GPP to +support IoT applications with low data rates and +high complexity. It proposes a new radio access +method based on LTE standards but with less +capabilities in order to lower the power +consumption +of +IoT +devices +with +limited +resources[150]. +9) +SIGFOX: It is a technology of narrowband and +ultra narrowband for connecting a large number of +power-controlled devices. In order to operate, the +protocol must operate on a frequency band of +868MHz, where the spectrum is split into 400 +channels of 100Hz. Rural areas can receive signals +from IoT devices that can transmit up to 140 +packets per day at a data rate of up to 100bps, and + +28 CSS: Chirp Spread Spectrum +urban areas can receive signals that can reach +distances of (30-50)km in rural areas and (3-10)km +in urban areas[149]. +The Table X compares the characteristics of the LPWAN +protocols discussed in this section[149]: +E. PERCEPTION LAYER +The major goal of this layer is to feel the physical +characteristics of the entities that surround us and within the +dominating IoT network, which relies on sensing +technologies like RFID, WSN, and GPS. It’s also in charge +of translating sensory data into digital signals that may be +transmitted via a network. Indeed, embedded intelligence +and nanotechnology play an important role in this layer, as +they improve the processing capabilities of any object by +inserting small chips or microcontrollers into everyday +smart devices. This layer consists of all user end devices +(smart devices, wearables, sensors, actuators etc.) that are +connected to the IoT and cellular network and capable of +accessing and transmitting tactile sensations over the +network. +Additionally, there are some fundamental attributes that +are an integral part of the end devices, as detailed in the +Table XI [151]. For the system to enable an intelligent and +reconfigurable touch based system, both the master domain +and the controller domain, need to have a bilateral +connectivity with the intelligent RAN network. It is further +governed by the AI/ML algorithms, responsible for +intelligent resource allocation and data transmission. Hence +motivation for the proposed model lies entirely in the fact +that to incorporate an entirely intelligent system in the 6G +and IoT framework requires the AI/ML technology +implemented on the B5G network slicing and its +application, TI as the building blocks of the touch system. +TABLE XI: +FUNDAMENTAL ATTRIBUTES OF CONNECTED USER END DEVICES. +S.No +Attributes +Description +1. +Identification +(ID) +Each of the connected objects is assigned an +ID based on conventional parameters such as +universal product code, Media Access +Control (MAC-ID), and IPv6 ID. +2. +Meta +information +It contains device model, ID, revision, +hardware, serial number, and manufacture +data for each IoT device. +3. +Security +controls +Allows the device owner to employ security +settings to limit the types of devices that can +connect to the user's device. +4. +Relationship +management +It enables each IoT device to establish, +update, and terminate relationships with +other devices. +5. +Service +composition +This component allows smart objects to +interact, with the goal of providing the +optimum integrated services to users and is +also in charge of processing the data +obtained from different objects to provide +user with best solution. + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +30 +One of the most promising aspects of this proposal is that +it is based on the assumption that 6G applications will have +a high data rate and latency requirement of less than 1ms in +the majority of cases. The application of the TI based +intelligent systems are still in the nascent stage and +therefore require a well defined technology to deploy the +touch enabled technology in the next generation networks, +combining the network slicing with the TI. +Thus an IoT enabled interface acts as a backbone of the +proposed reconfigurable and intelligent tactile based touch +communication +wireless +system. +Summarizing +the +important requisites for the proposed touch based +technology can therefore be listed below as: +1) An interfacing architecture with the existing +B5G/IoT framework is required that can link the +present B5G network with the next generation 6G +network through IoT based intelligent and sensory +devices and sensors. +2) The network slicing and TI prove to be a strong +backbone in implementing the given framework. +Also the intelligence may be incorporated in the +system using the ML/AI based algorithms, to +increase the system computational capacity. +3) The 6G applications are largely based on the +AR/VR/MR/XR, having large power consumption +and real time simulation and therefore need proper +energy +efficiency +and +power +optimization +techniques for its effective implementation. It can +therefore prove to be an important future aspect +that needs to be considered. +4) Also the minimum infrastructure costs, with an +energy +efficient +performance +are +important +parameters that need to be considered while +implementing the proposed system. +Hence it can be concluded by saying that, all the above +enlisted parameters, will eventually pave a way towards the +establishment of intelligent and a configurable touch +enabled system, interfacing with the B5G/6G Wireless +Communication Network. We have further proposed an end +to end touch interfacing architecture to be implemented in +the real time B5G/IoT based wireless communication +network as described ahead in this section. +The IoT infrastructure includes physical objects integrated +in the WCN, which are designed to provide intelligent in- +house service to users. Thus the IoT system comprises of +five layers as the perception, connectivity, network, +transport and the application layers starting from the device +end. The network layer in turn includes the connectivity +with the edge/fog computing as well, as the data storage +functionality within itself as given in Fig 9, altogether +forming the middleware component of the system. +The proposed Touch based intelligent and IoT configured +system may therefore be realized as an end to end +architecture in the Fig.10. This end to end architecture is +therefore to be realized in three phases which include: +1) The TI enabled real time touch communication +network establishments at the transmitting end +complete with its bilateral feedback system. +2) Similar TI enabled real time touch reception system +at the receiver end that is operated by the remote +industrial machinery and robotic system. +3) Lastly it requires an intelligent interfacing algorithm +to simultaneously operate both the ends and a +suitable slicing mechanism to fulfill almost all the +URLLC and IoT enabled intelligent Touch based +tactile application. +VIII. +END TO END TOUCH SYSTEM +ARCHITECTURAL MODEL +The E2E system architectural model of the proposed +intelligent touch technology has been represented in Fig.10. +The real time applications at the user ends satisfying the +desired output at the destination may range from AR/VR +based transmission and communication system, real time +online gaming, remote medical service accessibility like +tele-surgery and remote health consultation and remote +industrial operation involving the major role of robots and +automatic machinery. The system model has been divided +into three components and therefore may be analyzed in +three phases. These being categorized as: +1) PHASE1: The bilateral communication at the +transmitter comprising of the tactile based user +devices like the robotic arm, AR/VR gear and +gesture based haptic devices, real time online +gaming console and many end, more, interacting +with the BS, the local server (LS), transmitting the +data to the middleware through the gateway. +2) PHASE2: It represents the bilateral communication +at the receiver end, accomplishing the intelligent, +touch based applications ranging from intelligent +healthcare facilities like remote surgery, remote +industrial operations, real time online classrooms +and many more. +3) PHASE3: It acts as the mediating controller, +connecting the above phases and is therefore +responsible for implementing intelligence in the +proposed system. The AI/ML algorithms are +therefore routed in this phase. The aforementioned +phases may be separately analyzed as the +transmitter end, the receiver end and the +processing end. +A. TOUCH BASED TRANSMITTER SYSTEM +The transmitter end of this end to end architecture of the +intelligent touch interfaced system may be represented as +phase 1, comprising of the tactile user devices like the +robotic arm, augmented or virtual reality gear, gesture +operated tactile devices, real time online gaming modules. +The Fig.11 presents the touch enabled flow process at the +transmitter end. The process may begin with the +abovementioned devices at the user end requesting for +diverse high data rate and IoT based URLLC applications + + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +31 + +FIGURE 10: Touch interfaced end to end architecture. +varying from the remote medical consultancy or tele- +surgical operation to the assembling and modeling of the +machinery pertaining to the remote industrial function. The +user may request for a particular service by the means of a +touch sensor or an AR/VR device, virtually furnishing the +data towards the server, to be routed further. +The further the convenience may be at the individual +viewpoint, making use of tactile enabled robotic system for +household function. In addition it may have a future +application as in virtual e-commerce or virtualized or +holographic shopping where the customer may virtually +access or try the product by tactile sensing and touch +enabled interface between the user end and the linking +server forming a bilateral communication feedback system +at the user/customer end. +The TI based applications place a request to the local +server in order to check for the channel availability. The +request is forwarded when the channel is available and if +the channel appears to be busy, the entire process starts all +over again from the first step, where the user must ask for a +service. Following the channel access granted to the user, +and prior to sending the data it tries to establish connection +with the server. This is followed by the handshaking +request-response procedure between the channel and server. +The server gives acknowledgement on the channel +availability and thus firmly establishes the connection +between the customer and the server, virtually accessing, +forwarding and routing the data further to the gateway. +Here ‘A’ represents the data to be forwarded to the +gateway. A similar kind of procedure is conceded at the +receiver end explained in the next section. +This middleware is to be positioned between the edge and +the cloud computing layers, successfully participating in the +network slicing and providing the requisite virtual platform +for supporting different tactile applications. The slices may +be controlled, scheduled and allocated using different +ML/DL based algorithms, successfully implementing the +proposed touch based network. +B. TOUCH BASED RECEIVER SYSTEM +The flow process at the receiver end of the touch +interfaced system is presented in Fig.12. The point ‘A’ +which represents the user end devices at the transmitter end, +further routes the information signal in the channel via BS, +router and gateway. The gateway connects the transmitter +with the middleware. The process therefore briefly +illustrates the role of the middleware infrastructure with its +connectivity at both the transmitter and receiver end +devices. + +IEEEAccessAINL Algorithm +GI +.- Feedback.- +INTELLIGENCE +U1 +(-Feedback +DELIVER +PROCESS +Remote Server +INPUT +Receiver +Transmitter +U2 +Middle ware +BS +Smartclassroom +Remote Industry +U3 +LS +U1-User1(Robotic Arm +U2-User2-ARVR +GI +U3-HapticGesturebasedCommunication +！ +Tele-surgery +Un-Online real ime Gaming +BS-Base Station +LS-Local Server +GW-Gateway +Intlligence-AINL AlgorithmThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +32 +CHECK CHANNEL +AVAILABILITY +START +Touch Technology Flowchart +REQUEST FOR +SERVICE +TELE- +SURGERY +VIRTUAL +SHOPPING +REMOTE INDUSTRIAL +ASSEMBLING/MODELING +ROBOTIC USE AT +DOMESTIC LEVEL +BUSY +REQUEST TO THE +SERVER +AVAILABLE +SEND DATA +HANDSHAKING +AT SERVER +AVAILABLE +NO +YES +ACKNOWLEDGEMENT +RECEIVED +CONNECTION +ESTABLISHED +A +SERVER + +FIGURE 11: Touch enabled flow process at the transmitter end. + +This middleware exists at the junction of edge and cloud +computing layer. Middleware performs the virtualization +process in conjunction with network slice abstraction and +allocation, fragmenting the existing physical network into +multiple virtual networks. The techniques and algorithms +required for the proficient implementation of the slices +bearing n number of virtual networks may perhaps be +applied in the middleware layer. The middleware further +connects to the LS on the receiver side and establishes +connection. +The process similar to that in the transmitter and is +followed with the middleware forwarding the information +data to the server after establishing connection with the +server via handshaking mode. The LS at the receiver end in +turn indicates whether or not the channel is available for +signal +transmission +by +means +of +request +and +acknowledgement process. The signal is processed as the +output after the channel availability is declared at the LS. +The receiving end of the proposed system may perhaps be +represented by point ‘B’. +This point ‘B’ may further endow the requested TI +enabled touch based applications at the destination end. +Hence the user/customer successfully gets the desired +output. It may be the successful tele-surgical operation or it +may the remote industrial operation. It may also consist of +the successful modeling and assembling of robotic and +machinery components. Additional applications vary from +the smart and virtual classroom to the virtual holographic + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +33 +A +USER END (Tx) +ROUTER +BS +GATEWAY +GATEWAY +MIDDLEWARE +REQUEST TO THE +SERVER +SEND DATA +SERVER +HANDSHAKING +AT SERVER +AVAILABLE +CONNECTION +ESTABLISHED +ACKNOWLEDGEMENT +RECEIVED +NO +B +TELE- +SURGERY +REMOTE INDUSTRIAL +ASSEMBLING/MODELING +VIRTUAL +SHOPPING +ROBOTIC USE AT +DOMESTIC LEVEL +DESTINATION END +GATEWAY + +FIGURE 12: Touch enabled flow process at the destination end. + +shopping and e-commerce, with successfully accessing the +desired product without actually having to lay a hand on it. +It is possible to virtually access and analyze the product +through holographic imagery and the transmission and +reception of the information may be established by +implementing the TI based touch configured system, which +will prove to be an impacting factor in 6G system. +C. TOUCH BASED MIDDLEWARE SYSTEM +The middleware system is an integral and the most +important component of the proposed intelligent touch +configured system. It therefore helps in integrating and +configuring both the transmitter and the receiving +user/application ends. Hence it is imperative that both its +connecting ends have the same configuration and +dimensionality for the system to act smoothly and without +any undesired latency. The LS at the user end routes the +requested information via BS, router and gateway to the +middleware infrastructure. At this juncture the information +requested is virtualized in numerous network functions +enclosed in several network slices. +This system too comprises of a layered infrastructure, and +we have considered a 4-layered centralized middleware +network in Fig.13. These layers may be categorized as: +input layer, processing layer, delivery layer and the output +service layer. The input layer forwards the input data from +the transmitter end; the processing layer helps analyze the +data from the transmitter end; the processing layer helps +analyze the data using the virtualization and network slicing + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +34 +A +B +TRANSMITTER +CENTRALIZED +NETWORK +RECEIVER +LAYER OPTIMIZER +PRIORITY +POWER +SECURITY +Layer Optimizing Algorithm +Priority Algorithm +Power Usage Algorithm +Security Algorithm +Input +Process +Deliver +Output Services +(application) +FIGURE 13: Middleware component system +process. The delivery layer categorizes the slices as per +their use cases and data rate and bandwidth requirements, +whereas the output service layer arranges the final +application slices to be delivered at the destination end. +Therefore the proposed requisite intelligence in the +system is to be established via this middleware +infrastructure. There is a bilateral feedback system at both +the transmitter and receiver end which helps in transmitting +the +feedback +and +incorporating +the +intelligent +techniques/algorithms at both the connecting ends. The +input to this system is all the way through the transmitter +user end represented by ‘A’. Consequently the input data +fed to the middleware may be arranged in a sequence +considered and governed by factors like optimization, +priority, power consumption and security. +The next subsection further elaborates the architecture +and functioning of the middleware to be used in the +proposed touch based system. Hence at the output end these +factors may be implemented as their corresponding +techniques and algorithms like the layer optimizing +algorithm, the priority based algorithm, the power usage +algorithm and the security based algorithm, so as to obtain +the desired output result. The AI/ML/DL algorithmic +implementation of the above listed factors, in wireless IoT +and B5G/6G systems has been a research interest of many +scientists and researchers. +Thus for the real time implementation of the intelligent +touch enabled system we call for an interfacing architecture +that integrates the existing B5G technology with the 6G +technological interface, enabling the TI enabled touch +connectivity +through +intelligent +network +slicing. +Consequently the subsequent sections describe the +interfacing architecture of the present scenario with the next +generation system along with its future scope and +applications towards the implementation of the intelligent +touch system. +IX. +TOUCH TECHNOLOGY IOT MIDDLEWARE +The IoT is a vast network of connected smart devices that +aim to make the surrounding environment intelligent and +autonomous. Thus most vendors do not care about +compatibility of their products with other competitive +brands, which is one of the major challenges that IoT +paradigms face in machine to machine communication. +This problem has been addressed in several ways, one of +which is the enforcement of universal standards, which is +extremely difficult to implement. Another approach has +been suggested, which is the implementation of middleware +software to facilitate communication between these devices. +Middleware can be defined as the software that offers +interoperability between incompatible applications and +devices; as well as shielding customers from the smart +object’s complexity. Hence, it serves as a software bridge +between applications and things, allowing IoT systems to +communicate and collaborate more effectively with one +another. There are a plethora of middleware solutions +available, whether proprietary or open source, that are +provided by various companies, with the majority of these +solutions being very similar to one another. +However, there are no benchmarks, performance +indicators, or performance measurements that allow us to +compare various systems. Following our examination of +several articles[152]–[154], we have come to the following +conclusions about some of the issues faced by IoT +middleware: +1) ABSTRACTION AND INTEROPERABILITY: IoT +middleware aids in allowing various smart devices +to interface easily with each other in order to +facilitate collaboration and data exchange among +heterogeneous devices. +2) DEVICE DISCOVERY AND MANAGEMENT: This +attribute allows IoT devices and services to be +located in their network domain where the IoT +environment infrastructure is primarily dynamic +because all newly joined devices must announce +their existence and services. +3) SCALABILITY: The IoT middleware must be +scalable and must provide APIs in order to list all +IoT devices, their capabilities, and their services, +among other things. +4) DEVICE CATEGORIZATION: APIs must also +allow users to categorize devices based on +capabilities, manage devices based on remaining +energy, report IoT device problems to users, and +perform problem load balancing procedures. +5) BIG DATA AND ANALYTICS: Because of the +fragile nature of wireless sensor networks, part of +the detected data may be incomplete, requiring the +middleware to take this into account and +extrapolate incomplete data using a suitable +machine learning method. +6) PRIVACY: Since this majority of data coming from +IoT applications and services is related to human +daily life, security and privacy issues must be +considered when transferring and processing it, +necessitating the development of middleware that +addresses these issues. +7) CLOUD SERVICES: Cloud computing is the vital +layer of any IoT system. The data captured +through sensors will be stored and analyzed in a + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +35 +centralized cloud, and, as a result, IoT middleware +should run well in various types of clouds as +shown in Fig. 14. +8) CONTEXT DETECTION: Ambient data collection +applications and real-time reactive applications are +the two types of IoT applications. In the first, +sensors collect data that will be processed offline +later to obtain reasonable information that will be +used for similar scenarios in the future, while in +the second, systems must make a real-time +decision based on the sensed data. + +FIGURE 14: Cloud and IoT enabling technologies. +A. ARCHITECTURE OF TOUCH BASED IOT +MIDDLEWARE +Following are types of IoT middleware architectures that +are currently available, which are classified based on the +services that they provide[155]: +1) SERVICE ORIENTED ARCHITECTURE (SOA) OR +SERVICE BASED SOLUTION: +Users and developers are given the ability to employ or +add different types of IoT devices that can be used as +services in the service oriented middleware architecture +(SOA). Three layers make up SOA architecture: the +physical layer, which contains actuators and sensors, the +virtualized layer, which contains cloud and infrastructure +servers that are responsible for performing various +computational operations, and the Applications layer, +which contains all services and utilities as shown in Fig.15. +Access +control, +storage +management, +and +event +management are just a few of the general intermediate layer +services accessible. + +FIGURE 15: Service oriented IoT Middleware architecture. +SOA is a powerful middleware that may be deployed on +nodes that communicate with cloud servers or on a +powerful gateway that sits between the cloud levels and the +IoT layer. As a result, this sort of middleware is +incompatible with resource-constrained devices and does +not allow for device-to-device communication. The most +widely used service-based middleware is discussed here +and summarized in the Table XII. +a) Link Smart (Hydra): A web service platform that +intends to reduce the heterogeneity of different +devices and entities in the IoT ecosystem, as well as +control all types of smart devices regardless of their +communication protocols, such as Zigbee, RFID, +Bluetooth, Wi-Fi, and so on. This middleware is +unique and it allows IoT devices to be used as +services by embedding the required services. Health +care, agriculture, and home automation are few of +the IoT applications that can be managed with Link +Smart [156]. +b) Kaa: It is an open-source platform that enables the +creation of IoT solutions and is administered by +Cyber vision Inc and Kaa IoT technologies. By +utilizing a web-based GUI built on the Apache +platform, it is possible to create data delivery +schemes, support multi-tenancy on servers, and +generate endpoint software development kits (SDK). +Kaa enables direct or indirect communication with +endpoint devices, while encrypting their data using +AES29 and RSA 30encryption methods[154]. + +29 AES: Advanced Encryption Standard +30 RSA: Rivest Shamir Adleman + +IEEEAccessService-Based +Middleware +Actor-Based +Cloud Based +Middleware +Mas +SECaas +Middleware +Cas +Event Based +DBaas +Midleware +Midleware +Naas +Cloud Services +Configurations +Paas +HTTP +Intemet/Cloud of Things +MQTT +laas +Enabling Technologies +Saas +IoT/CoT +AMQP +Protocols +XMPP +Arduino +REST +COAP +Computational +RPL +Hardware +Ciond Senices +Raspbery PI +Soas: SfwareamaSenice +Sensing&Communication +Pans:PlafomasaSenice +NFC +Technologies +faofastuctesSenice +Smart Phones +OS +Nas:NetorkasaSenice +CaaCotaiersaService +Mans:MoritringasaSenice +RFID Tags +SECans:SecwriysaSenice +TEEE802.15.4 +DBasS:Datrbase asaSenice +6LowPAN +ZigBeeApplication +个个个 +Application&Services(browserbasedapp,smart +health,smartcities) +88 +nn +1 +Virtualization +Accesscontrol +Queuing manager +Virtual sensor +Webservicesinterface +manager +品 +Eventprocessing +Storage +engine +Cloud Service +IoT Devices +Sensorsandembeddeddevices(wearables. +smartwatch,cameraetcThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +36 +TABLE XII +COMPARISON OF COMMONLY USED SERVICE-BASED MIDDLEWARE ATTRIBUTES. +Service Based +Middleware +Link smart (Hydra) +KAA +GSN +Thing speak +Aura +Deployment type +PaaS, SaaS +IaaS +PaaS +PaaS +IaaS, SaaS +Network +connectivity +HTTP, REST,MQTT +MQTT, CoAP +HTTP +MQTT, REST API +MQTT, HTML +Data format +supported +JSON +REST,JSON,API +JSON, Sen ML +XML, Thingspeak, API, +JSON +REST ful API,JSON +Programming +Language +C#, .NET, Java, Python, +PHP, JavaScript +C,C++, Java +C, Java, Ruby +MATLAB +Python, PHP, C++, +JavaScript +Session +persistence +Machine learning +algorithm +- +- +Using MQTT +Aura session +Stream processing +CEP queries, Esper EPL +- +SQL queries +MATLAB +Aura library +Applications + Device abstraction + Stream mining + Live data management + Data storage + Online ML + Analytics + ML + Event reporting + Visualization + Network structure +visualization + Data +stream +processing + Data plotting + Real-time analytics + Event reporting + Visualization + Real-time applications +connecting to a GUI + Online video services + Billing system + Consoles and mobile +devices and Smart TVs +Service discovery +REST API +MQTT with Kaa +protocol V1 +REST HTTP query, +Sbt 0.13+, Java JDK +1.7, Scala 2.11 +- +Environment +management +Security and +Privacy + Encryption + Authorization + Authentication + Encryption + Authentication + Access control mode + Encryption + Authentication + Authorization + +c) Global Sensor Network (GSN): Its objective is to +provide a standardized platform that enables +adaptable deployments, sharing, and integration of +heterogeneous Internet of Things (IoT) objects. This +platform is built to meet the requirements of +physical and virtual sensors and actuators, whether +they are connected via a wired connection or +wirelessly. GSN is a Java platform that can be +placed on IoT cloud or servers, and it allows a series +of wrappers to feed the system with collected line +data that is later processed using XML specification +files[157]. +d) Thing Speak IoT: It is a MATLAB-developed +analytic open-source platform service that allows +people and things to communicate. In order to +collect, visualize, and analyze real-time data from +devices and sensors, Thing Speak provides users +with tools that allow them to use the HTTP protocol +over the internet to store and retrieve data from +them[158]. +e) Aura: This middleware facilitates the development +of pervasive mobile IoT applications by abstracting +device +differences +and +allowing +them +to +communicate freely. Aura makes an effort to +optimize the screen backlight and CPU in order to +improve performance while also reducing power +consumption. In engaging with events, Aura uses +two concepts: proactive and reactive. In a proactive +concept, system layers respond immediately to the +higher layer, whereas in a reactive concept, all layers +adapt their resource and performance based on +demand[159]. +2) CLOUD BASED MIDDLEWARE: +User options are limited in the cloud-based middleware +framework due to the limited number and variety of smart +devices connected to IoT applications. In addition, because +different use cases can be programmed and then determined +in advance, the sensed data can be collected and interpreted +with relative ease and accuracy. The operational component +of this middleware is restricted by the resources available in +the cloud computing environment. +Although IoT functions have a general presence in the IT +architecture, like storage systems and computation engines, +these functions are represented and controlled by APIs +where IoT services are accessed by either cloud-based +RESTful APIs or by vendor-provided applications as shown +in Fig.16. The most widely used cloud-based IoT +middleware is listed ahead and summarized the Table XIII. + +FIGURE 16: Cloud based IoT Middleware architecture. + +IEEEAccessVendor provided Web App +www +Cloud System APIs +8 +REST +Vendor provided Mobile App +API management +Cloud Service +RESTful API +IoT Devices +Sensors and embedded devices(wearables. +smartwatch,camera etc)This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +37 +TABLE XIII: + COMPARISON OF COMMONLY USED CLOUD-BASED MIDDLEWARE ATTRIBUTES. +Cloud Based Middleware +AWS IoT +Azure IoT Hub +IBM Watson IoT +Google Cloud IoT +Oracle IoT +Deployment type +IaaS, PaaS +IaaS +IaaS, PaaS +IaaS, PaaS +PaaS +Interoperability +Web services +Azure IoT SDK +MQTT +REST APIs +Oracle Service +bus +Network connectivity +MQTT, HTTP, Web- +socket +HTTP, AMQP, MQTT +over Web-socket +MQTT, HTTP +MQTT, HTTP +MQTT, HTTPs +Applications + Real-time analytics + AI/ML + Event reporting + Visualization + Real-time analytics + ML + Event reporting + Visualization + Real-time analytics + ML + Event reporting + Visualization + Real-time analytics + ML + Event reporting + Visualization + Real-time +analytics + Event reporting + Visualization +Technologies for +application development + AWS Cloud Trial + Amazon Cloud Watch + Kenisis + Amazon ML + SQL database + Azure tables + Azure Cosmos DB + Cloudant + No SQL Database + Firebase Google’s Big +Data tool + Big Query + No SQL +Database +Service discovery + Discovery API + AWS Lambda + AWS App Mesh + Amazon Route 53 + Netflix eureka + Azure container services +with Kubernetes + Eureka + Consul + Discovery +knowledge graph + Watson discovery + Consul + Zookeeper + JAVA WSDP +Security and Privacy + Encryption + Authorization + Authentication + Auditing + Authorization + Authentication + Encryption + Authorization + Authentication + Authentication + Authentication + Authorization +a) AWS IoT: In order to manage cloud services, such +as allowing millions of connected devices to +communicate securely and easily with other devices +and cloud applications, Amazon developed this +platform. AWS IoT enables customers to build IoT +applications that collect, process, analyze, and sense +data in order to make appropriate decisions without +the need for infrastructure management by utilizing +AWS services such as Amazon Kinesis and Amazon +Cloud Watch. AWS IoT customers can also always +monitor all devices that communicate with their +applications[160]. +b) Azure IoT Hub: It is a central platform developed +by +Microsoft +for +managing +bidirectional +communication between IoT applications and the +devices to which they are linked. Because of Azure’s +extensive capabilities, it enables clients to develop +full-featured, scalable IoT solutions that provide +secure and reliable communication between the +hosted cloud and connected devices. Azure is a +Microsoft product. When it comes to controlling IoT +connected devices, Azure IoT Hub supports a +variety of messaging patterns, including request- +response, file upload from devices, and device to +cloud telemetry[161]. +c) IBM Watson IoT: This platform, which is built on +top of the IBM Cloud, allows users to connect and +control a variety of IoT appliances, sensors, +industries, and home appliances. Using IBM +Watson, its clients can create and manage their own +IoT applications and appliances. They can also +extract KPIs from their data and use them to control +their tools and applications, as well as process their +collected data using historical and real-time +analytics. IBM Watson also offers a block chain +service[162]. +d) Google Cloud IoT: Essentially, it is a fully +managed device that is composed of a set of tools +that provide a comprehensive solution for secure and +easily connecting and processing of data generated, +whether they are located in the cloud or at the +network edge. Google cloud IoT aspires to develop +models capable of optimizing client business, +anticipating problems, and increasing operational +efficiency[154]. +e) Oracle IoT: A cloud-based service platform that lets +users create a real-time IoT solution to be linked +with +enterprise +applications +while +leveraging +rigorous security cloud capabilities and cutting-edge +edge analytics. Furthermore, it integrates IoT data +quickly and easily into customer business. Oracle +IoT enables clients to connect their devices to the +cloud, which will aid them in making critical +strategies and decisions in their businesses[163]. +3) ACTOR BASED MIDDLEWARE FRAMEWORK: +In terms of functionality, it is a lightweight middleware +that can be implemented at the sensory, gateway, and cloud +computing +layers. +Unlike +other +middleware, +the +computational operations of this middleware are distributed +across multiple layers, including the sensory layer and +mobile access layer. The sensory swarm is the outermost +layer, made up of sensors and actuators, while the mobile +access layer, made up of gateways, smart phones, +Raspberry Pi, and laptops, is the intermediate layer, and the +cloud is the innermost layer. The middleware which is also +the actor host is designed to be lightweight and can be + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +38 +TABLE XIV: +COMPARISON OF COMMONLY USED ACTOR-BASED MIDDLEWARE ATTRIBUTES. +Actor Based +Middleware +Calvin +NODE-RED +Ptolemy Accessor Host +Akka +Deployment type +IaaS +PaaS, SaaS +- +- +Interoperability +Actor model (event driven) +Web services +Accessor +Aggregate programming +Network connectivity +MQTT +HTTP, MQTT +HTTP,HTML +HTTP,HTML +Data format supported +JSON +JSON +JSON, XML +JSON +Programming +Language +C, Python +JavaScript, Node.js +C, C++, JavaScript +Java, Scala +Session persistence +Distributed hash table +MQTT +Local file system +Akka persistence library. +Stream processing +Data flow processing +Node-red-contrib-cep +Discrete event director +Akka hop, Akka stream library +and Apache Flink +Applications + Distributed applications + Runtime applications + Connecting to IoT + Connecting and binding +databases + Collecting and storing IoT +data in event driven +applications. + Finite state machine +applications + Web applications + Real-time streaming + Real time applications + Building powerful and +concurrent Web applications +Service discovery + Calvin control APIs + Bonjour /Avahi + Discovery.js + Discovery function + Akka discovery method + Kubernetes API + AWS + Consul + Marathon API +Security and Privacy + Authorization + Authentication + Authentication + Encryption + Authentication + Encryption + Authentication + Authorization + Encryption + +embedded in any layer of the application stack as shown in +Fig.17. +A storage device, for example, may not be included in +the actor-based middleware used on a smart watch. If the +storage device is provided by an actor, it can be +downloaded from a cloud repository whenever it is +required. The most widely used actor-based middleware is +discussed here and summarized in the Table XIV. +a) Calvin: It is an open source IoT platform developed +by Ericsson to be used on energy-constrained smart +devices because it offers a portable and lightweight +unified programming architecture with input and +output ports that define the interfaces. Furthermore +Calvin can also be used at the edge of IoT +ecosystems to reduce long-distance connections, +lowering latency and reducing power consumption +of IoT devices. Their main advantage is its ability to +move from one environment to another[164]. +b) Node-RED: It is an open source IoT platform +developed by IBM and built on the node.js31 +programming language. Because of its small +footprint, this platform can be used at the edge of an +IoT network, while on the server side, a JavaScript +platform with an event-driven module and non- +blocking I/O is used. It allows users to build IoT +applications by dragging and dropping connected +blocks that represent IoT components. The platform +drawbacks include the fact that it does not support +service discovery and only allows for password +authentication for security[165]. + +31 Node.js: It is an open source, cross platform and run time environment for +executing JavaScript code on the server side. + +FIGURE 17: Actor based IoT Middleware architecture + +IEEEAccessLayer +ccess +Application &Services +browserbasedapp,smart health,smart cities +Mobile +888 +88 +smatcty +Ptolemy's Swarmlet/Node-Red/Calyin host +84 +Cloud Service +Actor Host +system +04 +Sensorsandembeddeddevices(wearables +smartwatch.camera etcThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +39 +TABLE XV: +COMPARISON OF COMMONLY USED EVENT-BASED MIDDLEWARE ATTRIBUTES. +Event Based Middleware +Hermes +Gryphon +Rebeca +Fiware +Deployment type + PaaS +SaaS +PaaS +PaaS +Interoperability +Active message abstraction, 5 +layered architecture by Fenix, +Pegasus +Information flow graph +between devices and +broker +HTTP,SNMP,RMI +IoT Agent Framework +Library +Network connectivity +ACL, HTML,XML +HTML, HTTP +HTTP, SNMP, Java RMI +MQTT, HTTP web socket +Data format supported +JSON, Hermes XML +JSON, NYSE, NASDAQ +XML +HTTP,JSON +Programming Language +C, Java, Python +Python, Java +.NET,C#, Java +C++, Java +Session persistence +Java persistence +Buffered stream, JMS +persistent events +Fault tolerance plug-ins, +sliding window scheme +Apache scheme, My SQL, +Postgre SQL. +Stream processing +Open RTSP +Relational Subscription +Scheme +- +Fiware Kurento, Web RTC +Applications + Internet based distributed +applications + Large scale ubiquitous +applications + Web service + Exchange connections + Ledger +accuracy +guarantee + Fault tolerance + Monitoring + ML + Quantitative analysis + Monitoring and +management + Fault tolerance + Publishing methods + Collecting and processing +data + Visualization +and +data +analysis + Data access control + Monetization + Publication + Communication + Publish/subscribe +Technologies +for +application development + Type base routing algorithm + Service agents + Java message service +(JMS) + BKS+99 + Information flow graph + Publisher-hosting broker + Java management +extension + Object oriented API + Fiware Content broker +Service discovery + Service agents + Yellow page service + Discovery component + Matchmaker service agent +- +Publish/subscribe +mechanism + Selection component Fiware +NGSI + REST ful API +Security and Privacy + Encryption + Authentication + Authentication + Auditing + Authorization + Authentication + Encryption + Authentication + Authorization + +c) Ptolemy Accessor Host: Professor Edward Lee +created this open source platform in 1996 to design, +simulate, and model embedded and real-time +devices. The underlying concept of this platform is +that an IoT system is constructed from software +components that interact and communicate with one +another +through +messages +sent +through +interconnected ports on a computer network[166]. +c) Akka: It is a collection of open source libraries and +a free actor-based platform that was created to allow +developers to create distributed and run-time +applications in either the Java or Scala programming +languages. It enables users to meet business +requirements without having to write large low-level +code, resulting in high performance, fault tolerance, +and reliability. Akka also allows multi-threading, +isolates communication between applications and +their devices, and provides a clustered architecture +with excellent availability[167]. +4) EVENT BASED MIDDLEWARE FRAMEWORK: +In order to improve the development of distributed +systems, middleware that supports the publish/subscribe +paradigm is being developed and implemented. According +to this definition, this paradigm is a communication +infrastructure that aims to provide clients with general- +purpose services by assisting them in dealing with the +heterogeneity and complexity of large-scale and distributed +environments as shown in Fig.18. The event-based +middleware hides some of the complexity of distributed +applications from the programmer, which will make it +easier to develop and program much different functionality +in the future. The most significant differences between +these architectures are their openness to supporting new IoT +device types, the types of middleware services or +computational units they support, and the locations where +the IoT middleware can be embedded or deployed. + +FIGURE 18: Event based IoT Middleware architecture + +IEEEAccessPublish +Publish— +Notify +←Subscribe-Subscriber +Publish +Publish→> +Notify→> +←Subscribe-Subscriber +Publish +Publish +Notify→> +←Subscribe-S +Subscriber +Event Service +Broker NetworkThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +40 +IoT middleware based on SOA is implemented on +servers and in the cloud. Because the middleware can be +implemented in all tiers and IoT devices can perform +computation where it is most advantageous, actor-based +delivers the best latency and scalability for large scale +linked IoT devices. While these architectures provide some +level of security and privacy, cloud-based architecture +requires users to place their trust in the cloud provider to +protect the privacy and integrity of their data. +Since a middleware cannot be implemented within the +physical device and the data exchanged between physical +devices and the middleware can be compromised, there is a +weak security link between the physical devices and the +middleware in both service and cloud-based architecture. +To ensure QoS, the middleware must have a service +discovery component that allows new services to be made +accessible on demand and failed services to be dynamically +replaced. The most widely used event-based middleware is +discussed below and summarized in the Table XV. +a) Hermes: An event-based, scalable middleware +designed to make distributed and large-scale +applications easier. To address big size and dynamic +situations, Hermes provides self-managed event +brokers based on a P2P routing layer. It features an +adaptive solution that takes into account failed +event-broker events and routing stacks, all while +maintaining the event-broker network. Hermes +middleware has two versions, both of which share +the majority of the codebase and are intended for use +in distributed and large-scale systems as well as in +communication and implementation among event +brokers[155]. +b) Gryphon: It is a highly scalable publish/subscribe +middleware designed to distribute large amounts of +real-time data over the network. Gryphon is a Java- +based interface that enables the development of web +applications and the creation of a robust, redundant, +publish/subscribe, and content-based multi-broker. +This middleware is completely secured and offers +simple yet efficient routing and event handling. It +also uses a messaging information flow paradigm +(BKS+99) to specify communication between +publisher and subscriber[168]. +c) Rebeca: +This +middleware +is +based +on +publish/subscribe technology and focuses on the +design of efficient routing algorithms and use of +professional software engineering methodologies to +implement large-scale business applications. To +avoid network flooding, Rebeca employs advanced +routing techniques. It incorporates interoperability +and subscription merging capabilities into its +services to facilitate location mobility and reduce the +size of the routing table. The event scope function +abstracts away the implementation details of a +service, such as transmission policies, security, data +transmission +methods, +external +and +internal +interfaces, and notification representation[169]. +d) Fi WARE: It enables distributed IoT devices and +applications to communicate in an efficient, flexible, +secure, and scalable manner. It was created to +facilitate the control and monitoring of a variety of +IoT applications, including logistics, retail, and +smart cities. This platform is comprised of numerous +components, including APIs, reusable modules, and +massive code, all of which enable a user to create an +IoT application. A collection of sensed data from +IoT sensors is captured via REST API and later sent +to a dedicated server called the broker. FiWare has +developed an API for querying and storing various +IoT contents, which enables any application +registered as a content consumer to retrieve the +necessary data from the broker. This platform has a +component called an adapter that is in charge of +transferring a certain type of material to subscriber +applications[170]. +X. +INTERFACING ARCHITECTURE WITH THE +EXISTING NETWORK +The architecture in the Fig.19 represents the real time +interfacing of the existing 5G/B5G network scenario with +the next generation 6G all the way through the network +slicing phenomenon. The slicing aspect comprises of the +intelligent cloud slicing, the RAN slicing and the +application slicing. The cloud slicing intends to facilitate +the cloud computing and storage access to the edge devices +and users via different slices catering to different users +simultaneously. The RAN slicing is beneficial in +connecting and routing the end users and devices to their +receiving ends, accordingly making use of techniques like +spectrum sharing, beam forming, cognitive antenna and +radio transmission, tactile support and transmission system +through different slices. +All the applications accessed at the user end owing to +the slicing phenomenon are furnished by the application +slicing technique. The applications accessed may range +from +robotics, +tactile, +haptic +and +touch +based +communications to the real time online gaming using the +AR/VR/XR with the UHD video streaming. Other +applications may range from the remote industrial +application (IIoT), tele-medicine and tele-surgery, smart +classroom with UHD video streaming. The autonomous +vehicular and UAV system may be operated by application +slice providing the IoT connectivity along with the URLLC +of 1ms. +The complete autonomy of the system and devices +connected to the internet with the minimum latency and +high data rate facilitate an intelligent and autonomous +functioning system. The smart cities comprising of the +smart homes, smart traffic monitoring systems and the rest +instill another level of intelligence in the existing wireless +network system. The subsequent section highlights some of +the future aspects concerning the application of intelligent +technologies in the wireless 6G communication system +followed by some of the collaborative and ongoing projects + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +41 + +FIGURE 19: Intelligent interfacing architecture of the existing communication system with the next generation technologies (B5G/6G). + + +IEEEAccessCLOUD SERVICES +INTELLIGENT +CLOUD SLICING +Smart airways +Smartraihw +SMARTHOME +WEARABLE +EDGEDEVICES +GREENCOMIUNICATION +Car securi +Smart parking +RAN SLICING +回 +PACKETDATAGATEWAY +TACTILE SUPPORI +NE +USSTOP +AVNETWORE +SmartParking +AUTONOMOUSVEHICLE +AR/VR/XF +BASESTATION +BO +IMA +D +ROBOTICS +APPLICATION SLICING +TELESURGERY +SMARTCITY +8 +(lloT +NAN( Neighborhood Area Network) +5" Generation Network +Next Generation Network +Technologies +ScenarioThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +42 + +FIGURE 20: Touch induced human to robotic interactions with emotional intelligence. + +implemented +in +the +B5G/IoT +and +6G +wireless +communication systems. +XI. +APPLICABLE USE CASES OF INTELLIGENT +TOUCH TECHNOLOGY +This section explores the touch technology framework +and future application elements, with a focus on tactile- +based haptic communication integrated with learning +approaches to enable efficient network transmission. The +touch-enabled tele-diagnosis, robotic interaction, and haptic +sensation-based shopping experience are just a few of the +potential research aspect areas to look into. +A. CASE 1- ROBOTIC INTERACTION +Another application of touch technology could be tactile- +based robotic interaction between humans and robots, with +the bot system being able to recognize and respond to +human emotions [171], [172]. Emotions are interfaced into +the machinery by creating a database in the system using +AI/ML algorithms. The following are some examples of +such applications: +1) ROBOTIC PETS: +It entails the integration of AI algorithms and +functionality into robotic systems, as well as the +introduction of emotional intelligence (EI) into +them. The robotic pet dogs [173] is a dog-shaped +robot capable of learning and detecting human +gestures, face and eye movement, and responding +appropriately. By introducing a touch-based +interface into the system, these robots are capable +of reacting to human touch feelings. Those who +are elderly or mentally ill and are looking for +companionship to relieve their loneliness will find +these useful, as they will aid in their emotional +development as a result of the interaction. +2) CLEANING/DOMESTIC ROBOTS: +These robotic systems are capable of following +user instructions and performing household chores. +These might be driven by popular AI-based home +automation systems like Alexa to provide +complete automation, allowing them to hear and +act on the user’s vocal commands rather than their +restricted instruction library. The introduction of +touch-based +sensation/actuation +into +the +framework enables them to automatically respond +and act in response to the sensory simulations +provided by the environment, thereby saving time +that would otherwise be spent on user instructions, +programming, and the interface itself. +In this way, the touch-enabled robotic-human interaction +system with emotional intelligence is illustrated in Fig.20. +B. CASE +2-AR/VR +BASED +ENTERTAINMENT/ +SHOPPING EXPERIENCE +AR and VR technology are becoming the two important +innovation factors that promote technological progress. +Here the AR glasses provide a realistic perspective for +viewing augmented reality content, while VR headsets +provide an immersive sensory experience. Therefore AR +and VR technology are becoming the two important +innovation factors that promote technological progress. The +virtual reality headsets provide users with an immersive +sensory experience by allowing them to view AR content +from a realistic perspective. At the same time, AR/VR +technology has opened up a slew of new advertising and +marketing possibilities. + +IEEEAccessTactile support +Neaning/DomesticRobots +engine +gNB +Tactilesupport +engine +RoboticPets +gNB +Serving gateway +Packet data gateway +TactileCommunication System +Emotional Intelligence Sensation +USER +Tactile/Touch +Human-RobotInteraction +FeedbackSystemThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +43 + +FIGURE 21: Touch technology enabled virtualized shopping experience. + +Around 75 percent of business owners anticipate adopting +AR/VR technology in the next two years, and global +AR/VR market spending will be more than double. +Moreover, AR/VR users are becoming more common in the +world. As consumers increasingly rely on e-commerce and +online shopping, there is a high demand for augmented +reality content. WIMI plays a significant role in the AR +shopping market and also uses ‘AI+AR’ to support other +industries such as advertising, entertainment, and e- +commerce[174]. +Immersive AR/VR solutions help bridge the gap between +online and offline purchasing for consumers, and more and +more people are seeing AR/VR as a valuable tool for +discovering products and getting brand services. AR/VR +can currently improve the shopping experience through +virtualization, with technology such as a virtual mirror +assisting in a virtual garment trial before purchase being +implemented[175]. Hence it acts as a future aspect that +complements +the +integration +of +tactile/haptic-based +intelligence with AR/VR technologies to further enrich our +virtual shopping experience using touch technology. +In this regard, the illustration below shows how the +proposed methodology can be integrated with existing +AR/VR technology to further enhance the real-time user +experience. The Fig.21 illustration above depicts a +graphical representation of the virtualized shopping +experience provided to users via intelligent touch +technology. +C. CASE 3-TACTILE AND HAPTIC SENSATION +BASED TELEDIAGNOSIS FOR CONTACT FREE +COVID-19 CASES EXAMINATION. +Tactile internet-enabled wireless communication systems +have been integrated into the conventional healthcare +system to create the smart healthcare system. Tactile- +enabled healthcare systems, such as telemedicine, tele- +surgery, and remote tele-diagnosis, could all benefit from +the suggested touch technology. In today’s technological +environment, let us consider a practical scenario in which a +medical surgeon is working from a smart surface or console +that is connected to a telecommunications network in +Chennai, India, while the patient is lying on an operation +table at Fortis Hospital in New Delhi thousands of miles +away[176]. +The medical surgeon can remotely control the movement +of a multi-armed surgical robot to perform gall bladder +surgery on the patient by utilizing the smart surface and +other communication technologies. Through the use of a +tactile +communication +network, +the +doctor +can +communicate with and instruct the robot, while at the +patient’s end, a multi-machinery robot performs operations +on the patient in accordance with the doctor’s instructions. +Introducing intelligence into the robotic system, which +allows the robot to learn on its own while performing +operations, can further enhance the technological benefits +of the proposed touch technology as illustrated through +Fig.22. +The procedure will be made easier if the machine is +capable of sensing and responding to the user’s touch +sensations and interactions. Thus, an attempt is made to +incorporate artificial intelligence (AI) into the system while +providing instructions through the tactile communication +network. Additionally, if a sense of touch or emotion is +introduced in the form of emotional intelligence (EI), the +robot will be able to comprehend and execute the user’s +instructions without delay. This method could be used for +both remote and local diagnosis. + +IEEEAccessPacket data gatenay +Seninggatewa +rtualizedClothestrialbyUser +Tactile +engine +DIG +RCADE +SHOPPING COMPLEX +N +AI +RTGLTNTELGENG +TACTILECOMMUNICATIONSYSTEM +INTELLIGENCE +Shopping Experience +USERDEVICES +Virtual MirrorandTrial Room +AR/VRinterface +with User Device +TACTILE/TOUCHFEEDBACKSYSTEMThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +44 + +FIGURE 22: Touch technology enabled contactless tele-diagnosis and surgery. + +In the current covid-19 pandemic, localized tele-diagnosis +employing robots could be used to allow on-duty clinicians +to undertake contactless diagnosis and testing on infected +individuals. That in turn will lessen their chances of +contracting the virus from the patients and thus lessen the +strain on them during such a pandemic crisis or emergency +situation. +XII. +RESEARCH CHALLENGES AND FUTURE +ASPECTS +This section comprehends some of the conclusive future +aspects concerning the incorporation of the reconfigurable +and intelligent technologies like AI/ML/DL in the B5G/6G +and IoT governed wireless communication system. In the +end, intelligent technology machine learning has become +one of the promising tools of artificial intelligence for the +intelligent integration of wireless communications in the +next generation. Their futuristic scope may be steered +towards their relevance in channel estimation and detection, +inferring user location and behavior, resource allocation, +iterative learning, computational intelligence like neural +networks and decision making process. Some of these have +been discussed below: +A. RESEARCH CHALLENGES: +1) INTELLIGENT +CHANNEL +ESTIMATION: +The +futuristic scope in the ML based channel estimation +technique lies in the fact that it may be put to use in +direct scenarios without any need for training. The +only way to be able to learn the channel features of +various user environments can be that a system is as +smart to understand its parametric needs or in other +words that such a generalized system requires a vast +amount of pre-collected communication data to be +used by ML/DL algorithms. +2) NON/SEMI-AUTONOMOUS DEVICE +DISCOVERY: +Human intervention in IoT components, such as +device discovery, renders these applications non- +scalable and prone to error. Because of this +limitation, interacting IoT devices like IoT +middleware are unsuitable for self-adaptive +applications +such +as +M2M +communication +systems. +3) HETEROGENOUS ENVIRONMENTS: Since most +smart IoT devices and middle wares only handle +one or two types of heterogeneous components, +this is considered a major issue that must be +addressed. Due to the fact that non-autonomous +and inflexible services and devices restrict the +support of IoT applications, it is critical that new +approaches address and resolve the heterogeneity +of IoT environments, particularly in large scale +networks. +4) SERVICE LEVEL AGREEMENT: To provide +customers with an agreed level of service, three +components should be considered: a model that +precisely defines all functional and nonfunctional +services required by consumers, automatic service +to ensure a high level of QoS and adaptation, and a +monitoring tool for SLA services. Human +intervention in current autonomous networks will +be phased out in favor of the development of +intelligent IoT devices. +5) QOS LEVELS: Since there is no mechanism in +place to guarantee a specific level of QoS for non- +functional +IoT +services, +researchers +should +develop procedures for optimizing and monitoring +QoS levels. + +IEEEAccessTactile support +engin +Packetjlatagatewa +Serving gatewa +Y +appor +Doctor(user)attheremotelocation(miles +awayfromthepatient) +Servinggateway +TactileCommunicationsystem +Feedbackreceivedbyoperatingdoctor +through surgical robot +Artificial Intelligence introduced in +robotic system +Touch Sensation (Feelings/EI)This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +45 +6) PRIVACY AND SECURITY: As a result of the +resource-constrained devices in IoT environments, +the majority of autonomous and semi-autonomous +application services restrict security, authorization +and authentication mechanisms, among other +things. Hence for the intelligent network to +communicate between the cloud, gateway, and +sensors securely and efficiently, security and +privacy must be both end to end and lightweight. +B. FUTURE RESEARCH ASPECTS: +1) ALGORITHMIC MODELING: Almost all relevant +and physical modeling/construction should be seen +as +an +integral +step +toward +algorithmic +implementation using applicable ML tools, and +DNN is a key technology component in this +regard. +2) ROBOTIC FEELINGS: The implementation of +tactile based haptic communication in conjunction +with touch enabled gesture recognition is another +application +that +could +benefit +from +this +technology. This could be used in the inculcation +of feelings and emotions in robotic systems. +3) INTELLIGENT HEALTH MEASURES DURING +PANDEMIC +LIKE +COVID-19: +Telemedicine, +remote surgery, and treatment using intelligence +and haptic sensations may also benefit from it, +requiring a suitable algorithm with appropriate ML +tools for efficient implementation and risk-free +operation. So it may also be useful in the +contactless administration to the patients during +pandemics such as the one that occurred in +COVID-19. In this way, the risk of doctors +becoming infected with the virus is reduced +because the diagnosis will be done by a robotic +system that will use haptic and touch sensations +that have been introduced. +4) EFFICIENT NETWORK MANAGEMENT: As a +result, a paradigm shift is required for the efficient +design of B5G/6G networks in order to leverage +AI/ML and take advantage of big data analytics to +improve the overall performance of future +networks. +5) PROACTIVE NETWORKING: Future networks +require a prediction mechanism to help predict and +anticipate the future while allocating network +resources proactively. As a result, it aids in the +prediction and analysis of traffic patterns while +determining off peak times on various spectrum +bands so that incoming traffic demands can be +properly allocated over a given window. +6) BEHAVIORAL +LEARNING: +Predicting +user +behaviors will result in better network resource +utilization and will allow us to optimally allocate +end-to-end network sources in an online fashion, +which would be impossible without the assistance +of AI and ML techniques. +XIII. +RECENT RESEARCH AND PROJECTS +CONCERNING 6G WIRELESS +COMMUNICATION SYSTEMS +Numerous 6G research and development activities have +already begun on a global scale and this section summarizes +the most significant 6G research activities underway at the +moment[68]. +A. 6G FLAGSHIP (MAY 2018-APRIL 2026) +The 6G Flagship[177] is an eight-year research initiative +that focuses on the wireless smart society and ecosystem +enabled by 6G technology. Being funded by the Academy +of Finland, this project intends to realize B5G networks +from +the +very +outset +towards +its +phase +of +commercialization, and to develop the new 6G standards +for the future digital societies. It aims to develop essential +technology components of 6G mobile networks in areas +such as wireless connectivity, distributed intelligent +computing, security, and privacy. In addition to human-to- +human +communication, +the +research +focuses +on +communication between devices, processes, and objects. +This, in turn, enables a highly automated, smart society that +will permeate all aspects of life. Ultimately, the 6G flagship +project is to conduct large-scale pilots with a test network +using industry and academic support. +B. HEXA-X: A FLAGSHIP FOR 6G VISION AND +INTELLIGENT FABRIC OF TECHNOLOGY ENABLERS +CONNECTING HUMANS, PHYSICAL AND DIGITAL +WORLDS (JAN 2021-JUNE 2023) +HEXA-X[178] is the first flagship project of the European +Commission for implementing 6G vision and establishing +an intelligent fabric of technology enablers for integrating +the human, physical, and digital worlds. HEXA-X is a +European industry-academic collaboration that intends to +prepare the way for the next generation of wireless +networks through exploratory research. Its objective is to +connect humans, with the physical and digital worlds +through a fabric of 6G essential enablers. In order to +achieve this goal and vision, the HEXA-X project is +concentrating on building critical technical enablers in the +following areas: +1) High-frequency and high-resolution long-range +access using New Radio access technologies. +2) Connected intelligence to future networks via AI- +powered air interface. +3) Network +disaggregation +and +dynamic +dependability are enabled by 6G architectural +drivers. +C. TERA FLOW: SECURED AUTONOMIC TRAFFIC +MANAGEMENT FOR A TERA OF SDN FLOWS (JAN +2021-JUNE 2023) +Tera flow[179] is working on developing a cloud native +SDN controller for B5G/6G networks. This novel SDN +controller is compatible with the contemporary NFV and +MEC frameworks. It will also provide new features for +traffic flow aggregation, service layer management, +infrastructure layer network equipment integration, AI/ML- +based security and forensic evidence for multi-tenancy +networks. + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +46 +D. DAEMON: NETWORK INTELLIGENCE FOR +ADAPTIVE AND SELF LEARNING MOBILE NETWORKS +(JAN 2021- JUNE 2023) +The major goal of the DAEMON project[180] is to enable +high-quality Network Intelligence (NI) for 6G systems, +which will completely automate network administration. +The +project +includes +an +end-to-end +B5G/6G +NI +architecture, which can be fully coordinated with the NI- +assisted features. DAEMON performs a systematic analysis +of each NI task that is solved using AI models and also +provides a solid set of guidelines for incorporating machine +learning into network functionalities. A major goal of the +DAEMON project is to focus on existing B5G network- +specific AI methods that go beyond the current trend of +integrating AI into network controllers and orchestrators. +E. 6G BRAINS: BRINGING REINFORCEMENT +LEARNING INTO RADIO LIGHTWEIGHT NETWORK FOR +MASSIVE CONNECTIONS (JAN 2021-JUNE 2023) +The +6G +BRAINS[181] +project +is +focused +on +implementing multi-agent DRL for 6G radio links using AI. +In order to improve massive connection over D2D assisted +highly dynamic cell free networks, a novel comprehensive +cross-layer DRL driven resource allocation solution will be +required to perform resource allocation for Sub 6GHz/mm- +wave/THz/optical +wireless +communication +(OWC) +medium. This +significantly improves the +capacity, +reliability, and latency of future industrial, intelligent +transportation, and e-health networks. +F. SOUTH KOREA MSIT 6G RESEARCH PROGRAM +The Ministry of Science and ICT (MSIT) [182] in South +Korea is working on a bold strategy to be the first country +to deploy 6G networks. 6G services are expected to be +commercially available in Korea between 2028 and 2030, +according to the South Korean government. The initial +deployment is expected in 2028, followed by a mass +commercial deployment in 2030, with a total investment of +$169 million in R&D for 6G technology. The preliminary +goal is to deploy a 6G pilot in five important areas, +including digital healthcare, immersive content, self-driving +cars, smart cities, and smart manufacturing, by 2026[183]. +The 6G research program’s objectives are as follows. +1) Attain a data rate of 1Tbps. +2) Latency reduction of 0.1ms for wireless networks +3) Increases the range of connectivity coverage to up +to 10 km from the ground. +4) AI must be integrated into the network to ensure +that everything is covered. +5) By implementing security by design, it is possible +to protect the network. +G. JAPAN 6G PROMOTION STRATEGY +Japan invests approximately 50 billion dollars in the 6G +development project. This initiative aims to strengthen +collaboration between the public and private sectors in the +field of 6G research and development. Furthermore, by +2025, this 6G promotion strategy seeks to establish and +exhibit the 6G system’s basic technologies, as well as put +new technologies into practice by 2030[184]. +H. 6TH GENERATION INNOVATION CENTER +With the continuation of the 5th Generation Innovation +Center (5GIC), the University of Surrey in the United +Kingdom launched the 6GIC[185] in 2020 to focus on 6G- +related research activities across two themes. +1) AMBIENT INFORMATION: There is a use of the +advanced wireless technologies, high resolution +sensing, and highly accurate geo-location methods +to improve the fusion of virtual and physical +environments. This will enable a new level of 6G +digital services by better connecting of human +senses with ambient and remote data. +2) UBIQUITOUS COVERAGE: It has a role in +emphasizing on increasing the quality and range of +6G +communication +network +coverage. +The +research will focus on expanding coverage +indoors, utilizing intelligent surfaces, and satellite +technology to enable 6G services to be available +globally. +I. INDUSTRIAL 6G PROGRAMS +The Table XVI summarizes several industrial research +programs focused on the development and implementation +of 6G. While Table XVII and Table XVIII give an overall +projection of the ongoing recent projects in B5G/IoT and +6G respectively in the Appendix I +XIV. +CONCLUSION +The internet, with the advances in technology, drastically +affects the human lifestyle, in profound ways, transforming +various facets of life via interactions between the +individuals at virtual level throughout most of the +applications. +The +wireless +technologies +have +thus +transformed many elements of life including the business, +living standards and the infrastructure. In a never-ending +quest for elegant solutions to various problems, the society +is always on the lookout for new avenues of progress. +Therefore the motivation behind seamless connectivity has +resulted in the evolution of wireless communication from +1G to 5G. +TABLE XVI + INDUSTRIAL AVENUES ON 6G DEVELOPMENTS +Reference +Industries +Research projects/collaborations +[186] +Sony, +NTT and +Intel +Cooperative research on 6G technologies to +be commercialized by 2030 +[187] +Huawei +6G wireless technology research at the +Canadian Research Center and 13 other +universities. +[188] +SK +Telecom +A 6G-based commercial networking project +with Nokia and Ericsson. +[189] +Samsung +Commercialization is anticipated by 2028, +along with 6G services that incorporate XR, +a high-fidelity mobile hologram, and a +digital replica. +[190] +LG and +KAIST32 +The opening of a new Research Center +aimed at developing the 6G network +standard. +[191] +NTT +Demonstrating a 100Gbps communication +solution using OAM at 28GHz with MIMO. +[192] +Tektronix +Wireless fiber, a 100Gbps communication +solution. + +32 KAIST: Korea Advanced Institute of Science and Technology + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +47 +Despite the recent development, work in this field is still +underway all over the world with the aim of deploying the +6G communication network by 2030, thus making it one of +the most in demand research fields, with the potential to +revolutionize personal life of individual and society, +business and communication systems. The real time +applications from the individual user’s point of view are +perceptible in both professional as well as domestic fields. +It may vary from e-health, smart appliances to the smart +classroom based enhanced learning. +From the industrial point of view, the evident outcomes +are discernible in departments like automation and +industrial manufacturing, logistics, business and process +management along with the intelligent transportation of +people and goods. Therefore the current existing +technologies make the IoT concept viable but do not +necessarily fits well with the expected scalability and +proficiency criteria of the 6G system. Therefore many +problems do exist in the technical process and therefore, a +societal reform is vital in the technology, universally +conceptualizing the IoT connectivity. +A complete interoperability of the integrated devices is +required, provided with a high percentage of smartness, +maintaining the trust, safety and protection in the system +for creation of technologies that emphasize technical +requirements. However, the most intriguing aspect of the +next generation 6G standard is the incorporation of an +intelligent interface linking the existing communication +standards with the recently researched ones, so as to +competently fulfill the high data rate and the stringent +latency requirements. +In other words, all we lack is an intelligent interface that +is capable enough to integrate and enable a tactile based +haptic (touch based) communication in the existing B5G +network right from its source to its destination, complete +with its feedback mechanism, altogether incorporating +intelligence in the system so as to satisfy the stringent +latency requirement of 1ms. +This paper therefore provides the preliminary insight to +and answers the above mentioned challenges by providing a +comprehensive survey of the touch based intelligent +communication system using network slicing and TI +coupled the intelligence like AI and ML, in 6G. +APPENDIX I: +A. COLLABORATIVE AND ONGOING PROJECTS IN +B5G/IOT +Table XVII illustrates the recent collaborative projects in +B5G and IoT based wireless communication networks. +B. GLOBAL LEVEL ONGOING PROJECTS FOR 6G +DEPLOYMENT +Table XVIII summarizes all the ongoing research works +and projects pertaining to 6G wireless communication +system along with their application. + +TABLE XVII +COLLABORATIVE AND ONGOING PROJECTS IN B5G AND IOT WIRELESS COMMUNICATION SYSTEMS. +Ref +Research Project/ +Group +Institution +Research area +Source +[74] +5G Evolution and +6G +NTT- +DOCOMO +Performance +enhancement +of +mobile +communication through exploring high frequency +bands and improvement in wireless technologies. +https://www.nttdocomo.co.jp/english/binary/ +pdf/corporate/technology/whitepaper_6g/DO +COMO_6G_White_PaperEN_v3.0.pdf +[193] +5G exchange +(5GEx) +European +Commission +Multiparty +collaboration +and +multi +domain +orchestration for multiple carriers in 5G/B5G +network infrastructure. +https://ieeexplore.ieee.org/document/790148 +1;doi:10.1109/MCOM.2017.1600197 +[194] +MATILDA +European +commission +A holistic 5G E2E operational framework with +smart and unified orchestration and management +supporting edge/cloud computing. +http://www.matilda-5g.eu/ +[195] +Strategic Research +and Innovation +Agenda (SRIA- +2021-27) +European +Technology +Platform Net +world 2020 +Integration and management of AI and ML for +supporting network and applications like slicing, +URLLC networks, satellite communication with +seamless fog/edge and cloud orchestration. +https://bscw.5g- +ppp.eu/pub/bscw.cgi/d367342/Networld2020 +%20SRIA%202020%20Final%20Version% +202.2%20.pdf +[196] +ML for future +networks including +5G (ML5G) +ITU +Identification of gaps, issues and standardization +concerning the ML for future networks for +enhancing the network interfacing algorithms, +protocols and architecture. +https://www.itu.int/en/ITU- +T/focusgroups/ml5g/pages/default.aspx +[197] +AI and applied ML +TIP +Application of AI and ML for network planning, +operation and optimization while leveraging the +customer behavior driven optimization experience. +https://telecominfraproject.com/artificial- +intelligence-and-applied-machine-learning/ +[198] +Network data +analytics function +(NWDAF) +3GPP +ML function facilitating the monitoring of the +network slice status concerning the purpose of the +third party involvement. +http://www.tech-invite.com/3m29/tinv-3gpp- +29-520.html +[199] +ITU-T Study group +13 +ITU +Evaluation of intelligence in the future networks +including the IMT-2020. +http://handle.itu.int/11.1002/1000/14133 +[200] +6Genesis Flagship +Program +Academy of +Finland +Implementation and testing of the key enabling +technologies for 6G using the existing state of art +5G test bed. +http://jultika.oulu.fi/files/nbnfi- +fe2019081624413.pdf +[201] +Zero touch +provisioning +CISCO +Device +configuration, +management +and +orchestration to the local Ethernet in remote +locations via dynamic host configuration protocol +(DHCP)-IP. +https://www.cisco.com/c/en/us/td/docs/switc +hes/lan/catalyst3850/software/release/16- +5/configuration_guide/prog/b_165_prog_385 +0_cg/zero_touch_provisioning.pdf +[79] +What should 6G be? +KAUST. +Human centric perspective of 6G vision as research +channel in post 5G era. +http://hdl.handle.net/10754/661147 + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +48 +TABLE XVIII + GLOBAL-LEVEL ONGOING PROJECTS ON 6G ALONGSIDE THEIR ASSOCIATED APPLICATIONS +S.No +Ongoing +Projects +Institute/ +Organization +Enabling Technologies +Applications +Source +1. +6G Flagship +Academy of Finland + +To enable THz +communication. + +AI/ML/FL implementation + +Blockchain/DLT + +ZSM + +NTN/3D networking + +VLC + +Quantum computing + +Extended reality + +Autonomous driving + +Intelligent healthcare + +Personalized Body area +networks + +Industry 4.0/5.0 +[177] +2. +HEXA-X +European Commission + +THz communication + +AI/ML/FL + +Compressive sensing + +Swarm networking + +UAV based networking + +Internet of everything (IoE) + +Industry 4.0/5.0 + +Collaborative robots +[178] +3. +Tera flow +5GPPP + +AI/FL + +Blockchain/DLT + +ZSM + +Autonomous driving + +IoE +[179] +4. +DAEMON +European Commission + +AI/FL + +ZSM + +Compressive sensing + +UAV connectivity + +Autonomous driving + +IoE + +Intelligent Heathcare +[180] +5. +6G BRAINS +5GPPP + +THz communication + +AI/FL + +ZSM + +Swarm networking + +Compressive sensing + +VLC + +UAV navigation and +connectivity + +Collaborative autonomous +driving + +IoE +[181] +6. +MSIT +South Korea + +THz communication + +AI/FL + +Smart surfaces + +VLC + +XR + +Collaborative robots and +autonomous driving + +IoE + +Smart grid 2.0 + +Industry 4.0/5.0 + +Intelligent Healthcare +[182] +7. +JAPAN +JAPAN + +THz communication + +AI/FL + +Smart surfaces + +Swarm networking + +VLC + +Quantum computing + +UAV mobility + +XR + +Autonomous driving + +IoE + +Smart grid 2.0 + +Industry 4.0/5.0 + +Intelligent Healthcare +[184] +8. +6GIC +University of Surrey, +UK + +THz communication + +AI/FL + +Compressive sensing + +NTN/3D networking + +IoE + +Industry 4.0/5.0 + +Hyper Intelligent IoT +[185] +ACKNOWLEDGMENT +The authors gratefully acknowledge the support provided +by 5G and IoT Lab, DoECE, Shri Mata Vaishno Devi +University, Katra, Jammu +REFERENCES +[1] +A. 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Accessed: Jun. 14, +2021.[Online].Available:https://www.cisco.com/c/en/us/td/docs/sw +itches/lan/catalyst3850/software/release/16- +5/configuration_guide/prog/b_165_prog_3850_cg/zero_touch_pro +visioning.pdf + Mantisha Gupta (S’ 21) received +the B.E degree in Electronics and +Communication Engineering from +Jammu University, Jammu and +Kashmir, India in 2017 and the +M.Tech Degree in Electronics and +Communication Engineering from +Shri Mata Vaishno Devi University, +Katra, Jammu and Kashmir, India +in 2019, where she is pursuing the Ph,D degree in +Electronics and Communication Engineering. Her research +interest includes the emerging technologies involving the +B5G/6G and IoT enabled wireless communication and +security network and currently she is doing her research on +IoT +configured +networks +in +B5G/6G +wireless +communication systems. She is a student member of +Institute of Electrical and Electronics Engineers (IEEE). +. Rakesh K Jha (S’10, M’13) is an +Associate +Professor +in +the +Department of Electronics and +Communication +Engineering, +Indian Institute of Information +Technology, +Design +and +Manufacturing, Jabalpur (IIITDM +Jabalpur). He is carrying out his +research +in +wireless +communication, power optimizations, wireless security +issues, and optical fiber communication. He has done B. +Tech +(Hon's) +in +Electronics +and +Communication +Engineering and M.Tech from NIT Jalandhar (Hon's), India +in 2008. Received his Ph.D. degree from NIT Surat, India +in 2013. He has completed his 10th exam from govt. High +school and Class 12th from Science College. He has +published more than 101 Journal Papers out of which more +than 61 SCI Journal papers including IEEE Transactions, +IEEE Journal, Elsevier, Springer, Taylor & Francis, +Hindawi, etc. He has published more than 25 Interference +including ITU-T, IEEE ANTS, INDICON, IEEE ANTS, +and APAN. Dr. Jha’s one concept related to the router of +Wireless Communication has been accepted by ITU +(International Telecommunication Union) in 2010.He has +received the young scientist author award by ITU in Dec +2010. He has received an APAN fellowship in 2011, 2012- +Srilanka, 2016, and in 2017-China, 2018-Singapore,2018- +New Zealand, 2019-South Korea, and a student travel grant +from COMSNET 2012. He is a Senior Member of IEEE, +GISFI, and SIAM, International Association of Engineers +(IAENG), +ACCS +(Advanced +Computing +and +Communication Society), CSI, etc. He has filed 8 Patents +out of which 4 are published. Dr. Jha had 10 years of rich +academic, Industrial, and research experience in various +institutes/University including NIT-Surat, Capgemini India +Pvt. Ltd and SMVD University. He has also served as an +organizing member and TPC member for several national +and international conferences. He has organized many +workshops and has also been invited as a resource person in + +IEEEAccessThis work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ +This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI +10.1109/ACCESS.2022.3148473, IEEE Access + M. Gupta et al.: Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey +54 +many workshops organized by prestigious research +institutes. He has guided 05 Ph.D. students, 01 Submitted +the thesis, 03 Defended Pre-Ph.D. Synopsis and 03 students +are presently pursuing. He has guided more than 15 M.Tech +and more than 41 B.Tech students for various projects. +More than 4001 citations in his credit in the area of wireless +communication. +PROF. +SANJEEV +JAIN, +born at Vidisha in Madhya +Pradesh in 1967, obtained his +Post +Graduate +Degree +in +Computer +Science +and +Engineering +from +Indian +Institute of Technology, Delhi, +in 1992. He later received his +Doctorate Degree in Computer +Science & Engineering and +has over 24 years’ experience in teaching and research. He +has served as Director, Madhav Institute of Technology and +Science (MITS), Gwalior. He has worked as a vice +chancellor at Shri Mata Vaishno Devi University, Katra. +Presently he is a Professor in the Computer Science +Department, Central University Jammu, Jammu and +Kashmir. Besides teaching at Post Graduate level Professor +Jain has the credit of making significant contribution to R +& D in the area of Image Processing and Mobile Adhoc +Network. He has guided Ph.D. Scholars and has undertaken +a number of major R & D projects sponsored by the +Government and Private Agencies. His work on Digital +Watermarking for Image Authentication is highly valued in +the research field. He is also a member of Association for +Computing Machinery (ACM). + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +IEEEAccess \ No newline at end of file diff --git a/gNE3T4oBgHgl3EQfHwn2/content/tmp_files/load_file.txt b/gNE3T4oBgHgl3EQfHwn2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..32563c0a2895b63f1b78d14de81d1d0a65b625a5 --- /dev/null +++ b/gNE3T4oBgHgl3EQfHwn2/content/tmp_files/load_file.txt @@ -0,0 +1,4574 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf,len=4573 +page_content='This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': 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+page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Doi Number Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey Mantisha Gupta1, Student Member, IEEE, Rakesh Kumar Jha2, Senior Member, IEEE, Sanjeev Jain3, Member IEEE 1School of Electronics and Communication Engineering (SoECE), Shri Mata Vaishno Devi University, Katra, Jammu, J&K, India, 182320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2Associate Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=', Department of Electronics and Communication Engineering, Indian Institute of Information Technology, Design and Manufacturing, Jabalpur (IIITDM Jabalpur).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' (e-mail: jharakesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='45@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Computer Science Department, Central University Jammu, J&K, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' (e-mail: dr_sanjeevjain@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='com ) Corresponding author: (e-mail: jharakesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='45@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This work was supported by the 5G and IoT Lab, SoECE, TBIC, TEQIP-III at Shri Mata Vaishno Devi University, Katra, Jammu ABSTRACT Touch enabled sensation and actuation is expected to be one of the most promising, straightforward and important uses of the next generation communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In light of the next generation (B5G/6G) system’s need for low latency, the infrastructure should be reconfigurable and intelligent in order to be able to work in real time and interoperable with the existing wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It has a drastic impact on the society due to its high precision, accuracy, reliability and efficiency as well as the ability to connect a user from far away or remote areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Such a touch-enabled interaction is primarily concerned with the real time transmission of the tactile based haptic information over the internet, in addition to the usual audio, visual and data traffic, thus enabling a paradigm shift towards establishing a real time control and steering communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Due to the existing system’s latency and overhead, it creates delays and limits the usability of the future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In light of the aforementioned concerns, this study proposes an intelligent touch-enabled system for B5G/6G and IoT based wireless communication network that incorporates the AR/VR technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The tactile internet and network slicing serve as the backbone of the touch technology which incorporates intelligence from techniques such as artificial intelligence and machine/deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The survey also introduces a layered and interfacing architecture complete with its E2E solution for the intelligent touch based wireless communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is anticipated for the next generation system to provide numerous opportunities for various sectors utilizing AR/VR technology in robotics and healthcare facilities, all with the intension of helping in addressing severe problems faced by the society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Conclusively the article presents a few use cases concerning the deployment of touch infrastructure in automation and robotics as well as in intelligent healthcare systems, assisting in the diagnosis and treatment of the prevailing covid-19 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The paper concludes with some considerable future research aspects of the proposed system with few of the ongoing projects pertaining to the development in the incorporation of the next generation (6G) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' INDEX TERMS 6G, AI, AR, intelligence, IoT, ML, network slicing, tactile internet, VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' INTRODUCTION Mobile and wireless communications have been playing a decisive role in the current economy with the technologies like 2G,3G,4G,5G,GPRS,EDGE that successfully satisfy the user end with a significant role in the business, education, logistics and other primary industrial applications, effectively connecting the majority of the world’s population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These, in the present day are proficient enough to connect to the devices and people for an unprecedented exchange of multimedia and data content, enjoying its fastest growth in the history due to its enabling technologies, encouraging its widespread deployment, further intensifying the communication and the industrial sector[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As per the Cisco Visual Networking Index (VNI), there is an effective forecasting of the impact of the visual networking applications on global networks, incrementing from about 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='5 exabytes in 2017 to an expected surge of 77 exabytes towards 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The compound annual growth rate (CAGR) is expected to be about 74 percent of the present mobile traffic, 66 percent of the cellular (Wi-Fi) traffic, accompanied by the smart phones, dominating more than 90 percent of the mobile data traffic in the coming few years[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccesSThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='BASE STATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='PARK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='NAN( Neighborhood Area Network) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart home ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='HOTSPOT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='HOTSPOT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='HOTSPOT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart Parking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart grid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='D2D Pair ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='BUS STOP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart classroom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart City ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='E Health ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Education ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart Home ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Car security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart parking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Surveillance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='and security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart railways ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart airways ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='V2V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='FIGURE 1: General architecture of existing wireless and IoT based scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The Internet of Things (IoT) has therefore been a novel paradigm that is swiftly accelerating in the modern wireless communication scenario, connecting billions of devices with a seamless access of internet and applications, altogether forming an internet of everything (IoE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Conclusively the main strength of the IoT lies on the significant impact it has on variety of facets of everyday life and user behavior [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Since then, there has been a worldwide increase in the development of cellular network over the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Table I hereby provides a list of all the commonly used acronyms throughout the paper for better understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Many technical challenges have instantiated the designing of a robust wireless network, capable of delivering the necessary performance to support the emerging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The previous generations have seen a paradigm shift in the cellular technology and unlike these, the B5G/6G is said to be an integrative structure of the present 5G air interface spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This in combination with the LTE and Wi-Fi provides a seamless user experience, accompanied by the universally high coverage rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The existing 5G core network inculcates an unprecedented flexibility and intelligence in the upcoming 6G system with an improved spectrum regulation, along with the energy and cost efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Along the lines, the system also introduces extreme base stations with high device densities and an unparalleled number of antennas, as well as high carrier frequencies with large bandwidths [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' From the onset of 2020 onwards and till this date, the existing wireless communication networks have been standardized and deployed globally, with eMBB, MMTC and (URLLC) being the key 5G/B5G communication scenarios [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='The widely researched, IoT-enabled wireless network architecture, compatible with the B5G/6G system has been effectively illustrated in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The figure represents more or less every possible IoT configured services and applications, requiring high data rate and low latency, interacting with the surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' All these together constitute a smart system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This smart system entails all the possible smart devices, gadgets and sensors, all in all actuating a smart and automatized infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This infrastructure therefore highlights the utilization of the D2D communication network[9]–[11], the massive MIMO[12], small cell access points (SCA)[13], the IoT[14] with the network cloud [15], [16], all in all forming a part of the said 5G/B5G cellular framework[17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' All these along with the number of other sensor based interactions, make the existing system automatic enough to ease the human effort and save time, all the while considering the exponential surge in the data traffic, operating the millions of devices that are connected to the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The key technologies in this framework incorporates the spectrum sharing [19] with the cognitive radio [20], the interference management [21], the ultra dense network (UDN) [22], the mm-wave [23], 5G/B5G cloud and RAN [24], [25] and SDNs [26]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The existing 5G network therefore rides on the coattails of an explicit New Radio (NR) interface accompanied by a considerable number of virtualization technologies like the IEEEAccesSThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 3 TABLE I LIST OF COMMONLY USED ACRONYMS Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Definition Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Definition Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Definition Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Definition Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Definition 1G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2G,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The market driven allocation and reallocation of bandwidth are the few efficacious parameters in 5G/B5G system [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The smart applications constitute the entire automated smart city, comprising of the smart infrastructures in our day to day lives, ranging from a smart traffic monitoring system, with an efficient V2V/V2X interactions, providing a competent sensor based collision/accident detection system to an equally competent smart parking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The other applications may vary from the smart health and education facilities like remote health counseling, online smart classroom with the teacher-student interaction, the smart grid system and the smart homes, collectively forming the smart neighborhood area network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The virtualization facilitates an advanced computation of the network resources and their allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These, reasoned with their indispensable applications, facilitate a profitable proposition like network slicing [35]–[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The IEEEAccesSThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 4 virtualization in the core network (CN) [38]–[40] with the billions of miscellaneous IoT devices [14], [41], improves the integration of the past and present cellular and Wi-Fi standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore provides a ubiquitous high rate, low latency, giving a smooth experience to all the users in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The prime objective of the existing communication standard has always been to fulfill the demands of increase in capacity, the improvement in the data rate, the reduction in the latency, to provide a better quality of service and experience (QoS and QoE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore to meet these requirements, drastic improvements have been made and are still ongoing, in order to update the existing cellular architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence the next generation network is said to be agile enough to revert back the intensified network complexity, regardless of handling diverse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The existing 5G/B5G networks thus have to rely on the self organization and virtualization approaches, to deal with the disproportionate heterogeneousness and complexity of the network, associated with the massive amount of devices [42]–[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='In the forthcoming years, intelligence in the system is required to yield with such massive connectivity of IoT devices in the existing network with a minimum latency and network complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The B5G infrastructure thus has to focus on the considerable scaling and enhancement of the mobile network by incorporating open interfaces to support vertical segments in the network [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Such vertical segments are most often the third parties that do not own a particular network infrastructure but require the networking services with their specific requirements, along with their latest business solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Automotive manufacturing has been one of the most notable vertical segments in the existing communication system, requiring competent networking capabilities combined with the IoT and edge-cloud services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This in turn helps in the progress of a number of applications like autonomous driving, bird eye view, real-time assessment of road conditions, to name a few[46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The mobile internet in B5G/6G will thus make provisions for human to human (H2H) interactions with the primary goal of connecting the machines and gadgets to construct an IoT interface which is often built on D2D and H2H interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore, to solve the drawbacks such as latency, poor data rate and compatibility, high complexity, privacy and security, the next generation reconfigurable IoT allows for a real time control labeled as the ‘Tactile Internet’ (TI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' According to the ITU1, it is a network that combines extremely low latency with a high degree of reliability, scalability, and security[47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TI here provides an improved and virtualized environment which is most likely feasible for the commercial applications like tele-operation with haptic communication like remote surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It has therefore been an onset towards revolutionizing every fragment of the society right from the education and healthcare with their 1 ITU: International Telecommunication Union prospected future applications varying from inculcating sensations and sentiments using the intelligent robotic applications to the smart health facilities like remote surgery to the virtual shopping experience at the user end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The applications evolving the modernized WSN like the smart and automated homes and appliances, vehicles, factories, remote sensing and monitoring, augmented and virtual reality(AR/VR) and quantum computing based applications have an IoT as a common backbone, altogether forming an Internet of Everything (IoE)[48], [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The touch enabled sensation and actuation is expected to be one of the most fundamental applications of the B5G/6G communication technology due to its potential, simplicity and convenience, taking into consideration the real time scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For this reason, the ultra-responsive internet thus helps enable a real-time control of the physical tactile-based haptic devices, bringing in a paradigm shift toward an intelligent and touch-enabled technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ‘Why do we need a touch technology?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' is the most anticipated question here, taking into consideration all the previously-known aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The TI although being the most researched domain of the B5G/6G framework, its exploration towards the onset of an intelligent touch based technology is still in the infancy and has thus been emphasized upon in this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SCOPE OF THIS SURVEY The opening gambit for such technological advancement takes into account the subsequent technical update of the wireless communication network with the contemporarily researched 6G system [50], [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore encourages a real time interaction of humans with their environment, with few instances like the actuation of sensors causing the tactile sensation and the real time control/interface in our body system resulting from touching such surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore defines a new human-machine(H2M) interaction framework enabling a physiological latency of human beings to build a real-time interactive system, with their applications ranging from robotics to healthcare to the autonomous driving including the use of virtual/augmented reality (AR/VR/XR/MR)[52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For this reason the ‘Tactile Internet’ is regarded as an impetus and a cornerstone for the deployment of the touch technology and is expected to influence the development, innovation as well as the revolution of the healthcare, education, entertainment, manufacture, automation and smart grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This survey therefore paves a concrete path towards the initiation of the intelligent and touch enabled technology in the B5G/6G and IoT based wireless communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' MOTIVATION AND CONTRIBUTION OF THIS SURVEY The main purpose of this paper is to present a comprehensive and the state of art proposal motivated towards deploying an intelligent and touch enabled technology interface in the B5G/6G and IoT based wireless IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 5 TABLE II COMPARATIVE ANALYSIS OF PROPOSED SURVEY WITH THE EXISTING SURVEYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Ref no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Year Key contribution Technology used Communication system used Intelligence technique used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Touch technology Issues addressed [68] 2021 6G applications like holographic telepresence, e health and in body networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6G B5G/6G based mobile communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AI,ML and edge intelligence No An in-depth look at the latest 6G innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [67] 2021 Space-air-ground-sea integrated communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6G Network Slicing and Tactile Internet in 6G AI No Addressing the coverage requirements in terrestrial and NTN2 like satellite and UAV with high data rates and network security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [66] 2020 A novel architecture that employs computing resources in a cross-layered infrastructure to enable network computing SDN,NFV,TI Tactile internet, transport and cross-layered protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' No Leverages transport and network layers to increase network effectiveness in regards to congestion control and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [65] 2020 To provide virtual and logically independent slices for obtaining and deliver slice services from the infrastructure provider to the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ML,DL, 5G network slicing Network slicing with intelligence and virtualization ML,DL, SMDP3, N3AC4 No Performance optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [64] 2020 Design and operation of B5G wireless network using AI/ML technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AI,ML,B5G Network slicing and intelligence AI,ML No Overview of ML/AI algorithms with channel modeling and estimation with network management and optimization in B5G wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [63] 2019 ML application in the 5G/B5G WCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ML,DL ML,DL,DNN5 No MAC layer based resource management, networking and mobility management, and localization in the application layer using ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [62] 2019 Integration of robotics, human-computer interactions and virtual control environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AI, Edge Computing Network slicing with Tactile internet and AI AI No An insight on the Role of Tactile internet in industrial systems as well as an enabling factor of the Industrial Revolution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0) with RAS6 and Virtual control networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [58] 2018 Incorporation of deep reinforcement learning (RL) to handle cognitive smart cities services and improves their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ML, DNN Deep RL No Incorporate ML with a high level intelligence in the smart city services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [56] 2018 Role of the Network slice Orchestration and Management, Network slice Broker in 5G/B5G network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5G/B5G, NFV, SDN, Cloud and Edge Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Network slicing with virtualized fog/edge computing No 5G network slicing use cases with the E2E slice orchestration and management in eMBB, MIoT, eV2X, URLLC networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [60] 2017 Two types of feedback: Kinesthetic (based on force, torque, velocity, position) and Tactile (based on texture, friction, touch) AI and predictive analysis AI based predictive system AI No Enabling touch transmission and actuation in real time using TI, having a control on the real and virtual objects i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=', H2H and MTC interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [47] 2016 The E2E Tactile architecture in real time has 3 domains: Master, Network and Control domain, in TI and 5G network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' HSI, SDN, NFV Tactile Internet based haptic communication AI No Touch transmission in real time using robotics, haptic equipment, by means of a communication network combining the TI and the 5G network with their applications in industry, automation, healthcare, VR/AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Our paper 2021 2021 To propose an intelligent touch based system in B5G/IoT system incorporating AR/VR B5G /6G network slicing, TI, IoT, ML TI and intelligence based Touch communication system in B5G/6G AI,ML,DL, Hybrid model Yes Layered and interfacing architecture for intelligent touch system for E2E solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2 NTN: Non Terrestrial Network 3 SMDP: Semi Markov Decision Process 4 N3AC: Network slicing Neural Network Admission Control 5 DNN: Dense Neural Network 6 RAS: Robotic Autonomous System IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 6 network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' We first throw light on the next generation 6G WCN describing its key requirements and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The survey then subsequently emphasizes on the role of network slicing in B5G/6G network to sustain a high connectivity of massive number of devices without any latency and buffering at the user end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The network may be sliced into Cloud and RAN services and applications at the customer end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore enables an easy access, storage facility and virtualization, with each slice serving different applications on the same physical network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result the network slicing and the tactile internet in the B5G/6G are therefore the backbone for incorporating a touch enabled infrastructure with an induced intelligence by the AI/ML/DL implementation in the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Through this paper we propose here an intelligent touch configured system that would be applicable in the B5G/6G WCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' We hence summarize our contributions as follows: 1) We describe the well researched B5G/6G communication system while throwing light on some of its key parameters and prominent use cases with their applicability in the next generation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) We provide a discussion on the Tactile Internet along with its applications in the B5G/6G networks as an enabler of intelligent touch based system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) We introduce an intelligence in the B5G/6G network with a descriptive mention of few of the intelligence learning techniques in the wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) We further provide a backdrop of the intelligent touch configured technology involving the orchestration of the network slicing with the tactile internet allied with the intelligence and IoT connectivity in the B5G/6G wireless domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) We propose a layered and an end to end (E2E) interfacing architecture enabling the touch technology interface in the B5G/6G domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6) We further discuss the research challenges and further explore the research areas of the next generation (6G) deployment and ultra low latency tactile based applications in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' COMPARATIVE ANALYSIS WITH THE EXISTING SURVEYS The Table II indicates a complete summary of the existing survey papers on the network slicing and the tactile internet with some of the intelligent technologies like ML/DL implemented in the wireless cellular communication network in 5G/B5G scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In contrast to the other surveys, our survey provides a comprehensive overview of the proposed intelligent touch enabled technology in the B5G/6G and IoT configured WCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Farris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [6], describes the essentiality of the mobile edge computing (MEC) for supporting a wide range of user centric applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These have an important role in the smart city scenarios presented in Taleb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [53] where a smart MEC based architecture is significant for reducing the core network traffic while guaranteeing an ultra short latency for the existing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus MEC here acts as a key factor in enhancing QoS and attaining the 1ms requisite latency for the 5G/B5G mobile systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is accompanied by a considerable number of virtualization technologies like NFV, SDN and SD-RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The virtualization in turn promotes an advanced computation and allocation of the network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Myriad existing studies have provided a comprehensive overview of the technical challenges and applications associated with the network slicing, TI and the emerging intelligence in B5G/IoT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The works [39], [45], [54] provide comprehensive reviews of literature on network slicing in B5G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Richart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [39] discusses resource slicing and their allocation in virtual networks powered by SDN and NFV, as well as how these can be distributed appropriately to network slices without impairing the efficiency of other slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' While Afolabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [45]describes the state of art network slice life cycle architecture operating across the multiple domains thereby enabling an effective network programmability and flexibility with the creation, management and orchestration of the network slices, utilizing the massive IoT and multimedia broadband connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Foukas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [54] reviews the concept of network slicing and proposes a generalized layered architecture consisting of an infrastructure layer, a network function layer, and a service layer, along with their associated benefits and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The existing progress in the B5G network slicing with its key trends along with their corresponding potential challenges is presented in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The authors in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [56] throw light on the various concepts of network slicing and softwarization7 in the B5G technology with their applicability across the RAN and core network, altogether establishing an E2E slicing infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The intelligent tools like AI/ML/DL play significant role in outlining automation, deployment and disposition of different applications the existing as well as the next generation networks (B5G/6G) [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The work of Mohammadi and Fuqaha [58] provide an intensive facet of the most prevalent deep reinforcement learning in structuring of the cognitive smart cities and its applicability concerning the energy consumption with garnering of water and agricultural utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The Kafle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [59] successfully highlights the various ways of applying AI/ML techniques for the automation of network functions in different configurations, ranging from development organizations to industrial forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' One such intelligent application of the B5G/6G requiring the 1ms latency is Tactile Internet (TI) which is quite efficient in establishing a bilateral communication between the humans 7Network softwarization is the notion of designing, architecting, deploying and administrating network components, largely based on software programmability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It enables flexibility, adaptability, and even total reconfiguration of a network on the spot[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 7 FIGURE 2: Structural organization of paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' and machines, forming a HSI, which ultimately enables a haptic communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The term ‘Tactile Internet’ coined by Gerhard Fettweis in [46], has been a key enabler in fulfilling the need for a higher data space, essentially resulting in a continuous increase in the storage and computation among various cellular devices connected to the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore the TI is centered on H2M interactions with the devices using the B5G and IoT connectivity, enabling haptic and tactile sensations at both the transmitting and receiving end forming a bilateral communication and feedback network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The works of Maier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [52] highlight the commonalities and various subtle differences between the TI, IoT and the B5G vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The authors in Simsek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [47] highlight the critical requirements and architectural approaches for TI, as well as the technical issues and challenges associated with the resource management, core networking and edge cloud/ AI capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The Aijaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [60] helps to examine a number of the stringent design challenges to revolutionize the tactile internet, providing an enhanced haptic perception with a 1ms round trip delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Authors in Antonakoglou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [61] explain the evaluation of methodology and technology assessments for the necessary haptic communication infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The examination of the advancements in tele-operation over long distances has an effect on haptic communication, while using the Tactile Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore as per [62], the Tactile internet is a key enabler for realizing industrial revolution(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0) by users and devices in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The work by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [63] explains how machine learning may help with resource control at the MAC layer, network and mobility management in the network layer, and application layer localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' While C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='X Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [64] gives an overview of ML/AI technologies while addressing issues like channel modeling, estimation, network management and optimization in B5G wireless communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Following which D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Bega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [65] encourages exercising ML approach towards the market optimization while maximizing infrastructure provider monetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence to thoroughly optimize the availability of the computing resources, authors in [66] present a novel tactile based flexible next generation internet architecture (FlexNGIA) that capitalizes on the coexistence of transport and network layers to provide better congestion control and reliability services via cross- layered network computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [67] gives an insight of the next generation 6G communication highlighting its parameters specifically emphasizing its coverage requirements for the functionality in terrestrial as well as non terrestrial environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The paradigm aspect of this work is the integration of the space- air-ground and sea based communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='De.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Alwis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [68] discusses several 6G use cases, including holographic telepresence, e-health, and in-body networks that require extremely high data rates, ultra-low latency and high reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, the continuous IEEEAccessSec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='l-A: Scope ofthis survey Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='I-B: Motivation and contribution of this survey Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='l-C: Comparative analysis with the existing surveys Se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='I-A: Transition from 5G to B5G/6G Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='I-B: An Overview of the B5G/6G Communication System Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='II-C:6G enabled service cases metrics and driving trends Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='II-A: Slicing Process in B5G/6G Network Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='IV-A: Introduction Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='IV-B: Tactile Internet Architecture Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='V-A: ML in B5G/6G Wireless Communication System Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='V-B: Generalized Work Flow proces for Machine Learning Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='V-C: MLtechniques in B5G/6G communication system Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VI-A: Intelligent touch based system Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VI-A: Appiation Layer Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VII-B: Data Accumulation/Storage Layer Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VI-C: Edge/Fog Computing Layer Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VI-D: Connectivity Layer Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VII-E: Perception Layer Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VIII-A: Touch Based Transmitter System Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VII-B: Touch Based Receiver System Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VII-C: Touch Based Middleware System SeX-A:Architectureoftouch based IoT Middleware Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='IX-B: Existing IoT Middleware Platfoms Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='XI-A: Robotic Interaction Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='XI-B: AR/VR based Enterainment/Shopping Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='XI-C:Tactile and Haptic Sensation Based Tele-diagnosis forcontact feeCovid-19cases examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 8 FIGURE 3: Timeline of wireless communication technology evolution from 1G to 6G penetration of mobile platforms, robotics, human-computer interaction, and autonomous agents in virtual environments will distinguish future communication and industrial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The remainder of the paper follows the structure depicted in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In Section II and III we discuss the next- generation 6G system in detail with its main parameters, service and applications along with network slicing and virtualization concepts in 6G system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Section IV delves into the architecture and state-of-art uses of the tactile internet, which is a crucial component of the touch interface system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Section V introduces the intelligence that is to be incorporated into the next-generation wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Section VI explains the proposed intelligent touch technology, including its layered and E2E architecture in Section VII and VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Section IX discusses some basic major components of the proposed system, while Section X presents an interfacing architecture with the existing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Section XI contributes some potential and applicable use-cases of the proposed system, while Section XII highlights some of the research challenges and upcoming future aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Section XIII summarizes the recent research and few of the ongoing projects on 6G before concluding in Section XIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' INTRODUCTION TO NEXT GENERATION TECHNOLOGY (6G) The 6G communication era anticipates how humans will engage with digital virtual worlds beyond 2030, as the projected digital transition with the existing B5G networks have already begun and will continue to evolve over the next decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Although 5G/B5G is recognized for network cloudification via micro service architecture, the next generation 6G network is strongly linked to the intelligent network orchestration and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' New digital virtual worlds with the connected intelligence must have novel technologies that support these communication and networking challenges beyond 2030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The 6th generation wireless communication network is anticipated to consolidate the terrestrial, aerial, and maritime communication into a robust network that would be more reliable, faster, and capable of supporting a large number of devices with ultra-low latency requirements while remaining cost-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Due to the exponential growth in the number of IoT devices, the next generation systems must achieve high spectral and energy efficiency (SEE), low latency, and massive connectivity to provide for services like smart traffic monitoring, VR navigation, telemedicine at the user end, along with digital sensing using a full HD video transmission in connected autonomous devices like drones and robots[69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The timeline in Fig 3 shows the evolution of the wireless communication network from 1G to the recent 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The next section explains why the shift from 5G to 6G is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TRANSITION FROM 5G/B5G TO 6G As 5G/B5G networks are consistently deployed, the inherent limitations of this system are being exposed, in comparison to its original assertion as a platform for IoE applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is gradually more difficult for the existing multiple access techniques, to cope up with the exponentially growing IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, the 5G communication systems, already being implemented in the world today are incapable of supporting these many IoT IEEEAccess1980 1G AdvancedMobile Phone Service (AMPS) 3G/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='5G/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='75G TotalAccessCommunicationSystem(TACS) : FDMA, Analog voice CDMA-2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='UMTS WCDMA, HSPA Digitalvoice+Data Packet switching,Broadband 2G/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='5G/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='75G 2000 1990 GSM, GPRS,EDGE : TDMA/CDMA 2010 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='95G/4G Digital Voice LTE, LTE-A, Wi-Max OFDMA/SC-FDMA 4G/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='5G Packet switching LTE-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Wi-Max 2016 MIMO OFDMA/MC-CDMA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Wirelessbroadband MIMO, m-MIMO Mobile Internet D2D,Hetnet HDVideo streaming All IP (IPV4/IPV6) with unified LANWAN/PANandWLAN 5G/B5G 2020 B5G/6G NOMA,Hybrid(OMA+NOMA) SM-MIMO, THz comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' mm-Wave,Beamforming Cloudization,Softwarization IMT-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='5G-NR Virtualization,Slicing mMTC,eMBB, URLLC 2030 Intelligence(Al/ML/DL) Cloudization (Cloud/Fog/Edge) Quantum Computing, Blockchain(DLT) Softwarization,Virtualization, TactileInternet,Fullyautomated Slicing (SDN/NFV) vehicles, Holographic verticles, loT/loE,AR/VR, V2X, UHD and Digital sensing, and underwater 360°vide0s,Smartcities, communication Tele-medicine and wearable devicesThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 9 TABLE III: COMPARISON OF PERFORMANCE ATTRIBUTES BETWEEN 5G, B5G AND 6G COMMUNICATION SYSTEMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Performance attributes 5G B5G 6G Application types eMBB URLLC mMTC Reliable eMBB (ReMBB) URLLC mMTC Hybrid( URLLC + eMBB) MBRLLC mURLLC HCS MPS Architecture Dense small base stations operating at sub-6GHz in conjunction with umbrella macro base stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Mm-wave small cells with a range of approximately 100m for fixed access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Denser small cells operating at sub-6 GHz with umbrella macro base stations Mm-wave cells with a diameter of less than 100m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For mobile and fixed access, cell- free smart surfaces with high frequency are supported by mm- wave small cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Drone-carried base stations and tethered balloons provide temporary hotspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Frequency bands Sub 6 GHz Mm-Wave for fixed network accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Sub 6 GHz Mm-Wave for fixed network accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Sub 6GHz Mm-wave for mobile network accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' High frequency and THz bands above 300GHz are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Non RF technologies like VLC, Optical fiber communication etc Spectral and Energy Efficiency (SEE) 10x in bps/Hz/m2/Joules 100x in bps/Hz/m2/Joules 1000x in bps/Hz/m2/Joules Data rate 1Gb/s 100Gb/s 1Tb/s E2E delay 5ms 1ms <1ms Radio-only delay 100ns 100ns 10ns Processing delay 100ns 50ns 10ns E2E reliability requirement 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='999% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='9999% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='99999% Interoperable devices Smart phones Sensors Drones Smart phones Sensors Drones XR equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Sensors and DLT devices CRAS Smart implant system XR and BCI equipment devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The requirement for faster data rates has fueled the evolution of wireless networks, which has necessitated a continuous 1000-fold increase in network capacity[70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As the demand for wireless capacity continues to surge, the emerging IoT system, which connects millions of people to billions of machines, has resulted in a radical paradigm shift from the rate centric eMBB services from the previous eras towards URLLC and intensified mMTC services, as per the 3GPP, which is working on the implementation of 5G/B5G standard[71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Although it can be asserted that the evolutionary aspects of the existing 5G supporting the data hungry eMBB services have gained a significant momentum, while the promised revolutionary disposition systems, operating exclusively at high mm- wave frequencies have yet to materialize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Despite the fact that today’s linked 5G systems are easily capable of supporting the most fundamental IoE and URLLC services (such as factory automation), it is still debatable whether or not they will be able to deliver the smart city services based IoE applications in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Conversely, the initial and the existing B5G deployments are most likely to rely on the low frequencies (sub 6GHz) to support mobile data transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' While on the other hand, an enormous influx of new IoT services such as XR (including AR/VR/MR), flying vehicles, and connected autonomous systems would most likely derail 5G’s original purpose of supporting small packet and sensing-based URLLC applications[72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus, in order to successfully operate these IoE services, a wireless system must simultaneously provide a high level of reliability, low latency, and a high data rate for a wide range of heterogeneous devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These new services necessitate the resolution of novel and distinct challenges, unprecedented in terms of their complexity including the tradeoff between latency, throughput and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Not only do these services help entail new approaches for an effective regulation and handling of performance and challenges but also aid in exploration of frequencies beyond 6GHz range in order to create a self sustaining and intelligent wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This aforementioned network is capable of provisioning and orchestrating communication, computing, control, IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 10 TABLE IV COMPARISON OF VARIOUS 6G SERVICE CASES WITH PARAMETERS AND APPLICATION AREAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='No 6G service cases Performance Attributes Application Areas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' FeMBB (Further enhanced Mobile Broadband) Enhanced broadband in densely populated areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Operates in THz communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Enhanced multimedia applications like 4D video gaming, mobile TV, connected wearables and sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Public transportation High speed trains Smart cities 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' umMTC ( Ultra massive Machine Type Communication) Reliable connectivity with massive scale (trillions of devices) of connected devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Improves connection density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Enables IoE with ultra dense cellular IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SigFox and LoRa as potential technologies for enhanced connectivity and network coverage in 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Internet of Industrial Smart Things (IIsoT) Smart buildings Internet enabled supply chains, logistics, fleet management and water quality monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Natural/wildlife sensing Forest monitoring 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ERLLC (Enhanced Reliable and Low Latency Communication) End to End fast turnout time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Intelligent framing and coding with efficient resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Intelligent UL/DL communication Remote robotic surgery using smart surfaces and intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Telemedicine Internet of Healthcare (IoH) Remote Robotic Surgery XR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ELPC (Extremely Low Power Communication) Uses Intelligent Reflecting Surfaces (IRS) known as Reconfigurable Intelligent Surfaces (RIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Reduces hardware dependency and Tx-Rx complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Reduced energy consumption with passive array transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Smart homes Smart cars UAVs 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' LDHMC (Long Distance and High Mobility Communication) High mobility and seamless communication for long distances (>1000km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Accurate channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Use of FBMC and UFMC as alternative to OFDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Deep sea tourism High speed transportation Space sightseeing 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' MBBLL (Mobile Broad Bandwidth and Low latency) MEC to attain end to end low latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Low complexity mechanism for VR experience by user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Mobile AR,VR 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' MLLMT (Massive Low Latency Machine Type) Data availability, ultra scalability and low latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Time critical applications where decision making takes fraction of seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Automation, controlling and monitoring of industrial 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Home and building automation UAVs IoT enabled Healthcare 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' MBBMT (Massive Broadband Machine Type) Touch based experience with high data rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Massive IoT connectivity in densely populated areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Tactile sensations captured by sensors/devices converts to digital data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Tactile Internet localization and sensing of the scenarios that are best suited for IoT needs[73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' So to address these issues, a game- changing 6G wireless system is required, with a design that is organically tuned to the performance requirements of IoE applications and associated technical advancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AN OVERVIEW OF THE B5G/ 6G COMMUNICATION SYSTEM The 6G wireless communication network, which is currently being researched, is expected to integrate terrestrial, aerial, and maritime communication systems into a robust network that is more reliable, fast, and capable of supporting a large number of devices with ultra-reliable and low-latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The AI, ML/FL, quantum communication, blockchain/DLT, beyond 6GHz and towards Terahertz communication, TI, swarm UAVs, Zero touch network and service management (ZSM), large intelligent surfaces (LIS), NTN and 3D networking, VLC, compressive sensing with an efficient energy transfer and harvesting are just few of the currently proposed by the ongoing researches worldwide[74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Owing to a massive growth in the number of IoT devices, realization of the advanced services like smart traffic monitoring, VR based navigation, smart medical facilities like tele-medicine and HD video transmission in drones and robots is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence the B5G/6G communication systems aim to achieve high SEE, low latency, and massive connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The ever-increasing number of IoT devices makes it difficult for the existing multiple access strategies to handle such a huge number of devices, therefore requiring a more extensive network in order to make use of IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 11 FIGURE 4: Network slice functionality in B5G/6G network the massive bandwidth capabilities offered by B5G/6G communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' According to the work by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [75], a speculated vision and functionality of 6G networking scenario provides a technological framework and requirements for industries in the future generation communication system with cell less architecture, decentralized networking, and resource allocation with 3D radio interoperability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The next generation wireless network comprises of a large number of linked devices with numerous base stations (BSs) and access points (APs), each of which will serve multiple devices at the same time, forming a coordinated multipoint (CoMP)[76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence more data would be transmitted via future wireless communication networks, since most of the value-added apps and services rely significantly on the data exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore large devices will generate a massive quantity of data, which will need high-performance processing units and backhauling connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The following subsections go through the new 6G enabled driving trends, metrics, and use service cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6G ENABLED SERVICE CASES, METRICS AND DRIVING TRENDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' With the new performance metrics, new technological trends step up to redefine the prevalent B5G applications by morphing the classical URLLC, eMBB, and mMTC into something entirely new and innovative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, Table III gives a comparison of some of the key performance attributes of the 5G, B5G, and 6G wireless communication systems[73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Various countries have initiated projects aiming at the research and deployment of B5G/6G communication networks, as discussed in Section XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The research on 6G is accelerating, and has been documented in recent works like [77]–[79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' According to the recent research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' a variety of possible 6G applications have been classified as mobile broad bandwidth and low latency (MBBLL)[80],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' massive broad bandwidth machine type (mBBMT)[78],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' massive low latency machine type (mLLMT)[81],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' further-enhanced mobile broadband (FeMBB)[82],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' extremely reliable and low-latency communications (ERLLC)[83],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ultra-massive machine-type communications (umMTC)[44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' long- distance and high-mobility communications (LDHMC)[84] and extremely low power communications (ELPC)[85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Detailed descriptions of each of these are provided in the preceding Table IV, along with their parameters and application areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' NETWORK SLICING IN B5G/6G BASED IOT NETWORKS The forthcoming B5G/6G networks aim to serve a wide range of applications, thus recognizing the 5G epoch as the century of mobile telecom networks, all the while promoting dedicated use-cases and endowing unequivocal services, to meet diverse user requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence the technologies like UHD, multimedia, AR/VR/XR therefore needs a faster speed and a relatively higher capacity and connectivity, compared to the mission-critical applications like IoT/MIoT and autonomous systems require an ultra- low latency and ultra-reliable operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessSERVICE LIFECYCLE E2E SERVICE OPERATION ANDMANAGEMENT MANAGEMENTLOOP SERVICELAYER V2X COMMUNICATION AMF LSMF SYSTEM BASE STATION UPF ASSURANCE NETWORK LAYER FULFILLMENT SMART CITY AND UPF AMF SMF CONNECTIVITY ORCHESTRATION BASE STATION DATA ACQUISITION,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' PROCESSING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ABSTRACTION AND DISTRIBUTION () (g) RESOURCES AND FUNCTIONS MANAGEMENT ANDORCHESTRATION (MANO) RAN CORE = ORCHESTRATION ORCHESTRATION () ( FOGNODE ROUTER GATEWAY TRANSPORT NFV&MEC EDGECLOUD WIRELESSAND CORE/CENTRAL ORCHESTRATION ORCHESTRATION FIXEDACCESSNETWORKS CLOUD RESOURCES AND FUNCTION LAYER INFRASTRUCTUREORCHESTRATIONThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 12 FIGURE 5: Network slice orchestration and management The 6G cellular framework formulation anticipates to be accomplished on the existing and researched 5G/B5G technology, thus supporting a surplus of network services with miscellaneous performance requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The advancement of cellular networks and resulting generation- wide improvements is motivated primarily by the desire to enable better data-based services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A variety of aspects render 5G important, including the mm wave spectrum distribution and reallocation of bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Virtualization with a billion individual networks in the CN and IoT facilitates the convergence between previous and present cellular and Wi-Fi requirements[83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This in turn offers a pervasive high rate and low latency experience for network customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The most significant part of the B5G/6G infrastructure comprises of the network service and its development platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is highly capable of improving the network scalability while fulfilling the user requirements by utilizing existing services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The network slicing functionality in B5G/6G domain has been depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The virtualized infrastructure here has provisions for slice instances, and collectively functions with the infrastructure resources from another slice instance[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A set of homogenous APIs are made available for creating an abstraction layer to facilitate with the slice management while controlling its virtual resources during its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These slices can therefore be accessed by different tenants or third parties using these APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Here SLA act as the slice blueprints, using which the tenant specifies its requisite slice characteristics ranging from topology, management, control and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The slice lifecycle is regulated by the service lifecycle management loop, openly accessed by all the functioning slices[45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The management and orchestration (MANO) offers an integrated and a holistic approach towards the regulation of network slicing and the NFV management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It offers a standardized level of data abstraction followed by the adapt specification of its network infrastructure together with its service management and implementation process[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This section discusses the concept of network slicing in B5G/6G communication networks with its functionality, management and orchestration in RAN and CN and finally its application in the 6G with the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SLICING PROCESS IN THE B5G/6G NETWORK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The existing B5G is most likely to consider a variety of business and service quality requirements like the enhanced capacity coupled with the intelligent traffic and offloading techniques accompanied with a highly complex and heterogeneous network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' All of these fulfill the required performance criterion together with an autonomous network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A high data rate guarantees a high level of end user service quality with an unlimited mobile broadband connectivity in the jam-packed areas like stadiums, concerts and shopping centers, by means of the terminals having the AI capabilities [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The reduced latency with high data rates are capable of supporting the UHD streaming from the cloud technology and improvised VR devices and other wearable computing gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore provides a faster web downloading while enabling a premium user experience in services like YouTube streaming, Netflix and so on with a high video resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The network slicing is a fundamental key for the B5G/6G technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus the network slices here are an end to end concept of the next generation technology where the slice operator supports a massive amount of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Here each of them in the long run requires a multiple end to IEEEAccess- B SMART HOME EMBB AR/VR/XR wiE [ UHD VIDEO STREAMING Service Layer Network Orchestrator SMARI CITY MMTC NetworkSlices + Resource Sensor IoT Application Service Allocation Orchestrator Infrastructure Layer X AUTONOMOUS VEHICLES (IIoT) REMOTE SURGERY SMART CLASSROOM Application/Business LayerThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 13 FIGURE 6: Network slicing in next generation networks (B5G/6G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' end individual and logical networks referred as network slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' They are categorized into three components, each of them governing the RAN, core and transport domain[56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The RAN and core slices consist of the application context and personalities respectively with the transport slices being connectivity between the RAN and core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Each individual domain has a controller, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' the RAN controller, core controller as well as the transport controller, all of these supported by an end to end orchestrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' To realize the B5G/6G networks, 5G network slicing plays a crucial role in guiding about slice utilization for automation, assurance and optimization of transport slices involving various low latency and high reliability applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These applications may range from automated vehicles, tactile applications, smart devices and so forth[87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='5 diagrammatically illustrates the network slice orchestration process applicable in B5G/6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' At the back end, the resource allocation takes place in the infrastructure layer where resources are provided to the individual slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The slicing applications are managed by the network orchestrator in the service layer, so as to enable an effective application slice management and orchestration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It can be undoubtedly claimed that the most defining feature and in other words the ‘secret sauce’, for the 5G/B5G success is the E2E network slicing, which will be applicable in 6G networks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence the network slicing is responsible for the optimal resource efficiency and flexibility in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore enables the implementation of new business models as NaaS, supporting various mission critical use cases including the industrial automation(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0) and availment of remote health facilities[50], [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The network slicing architecture pertaining to the B5G/6G system has been illustrated in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The slicing process is therefore described as the three sub processes, interlinked with each other: the intelligent cloud slicing, the RAN slicing and the application slicing, all of which are functional in the B5G/IoT enabled networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Commencing from the lowermost stratum we have various applications that furnished at the consumer end ranging from the real time online gaming with a UHD streaming to the live online classroom teaching sessions, efficiently making use of the AR/VR technology to deliver the required information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The other applications may also involve the use of robots in real time actuations and tactile and touch based haptic communications, which may be put to a practical use in industrial operations, automation in the robotics and machinery, vehicles and UAV fleet to accomplish the services requested by the users at the customer end in the form of application slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The succeeding layer is that of RAN slicing, taking place in between the CN and RAN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=', at the backend of the network, routing the clients with the final applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is capable of enabling an effective resource and spectrum allocation with power and energy efficient cognitive radio network system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' All of this is in the form of RAN slices and successfully connects the edge devices with the cloud network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The cloud computing and storage enables intelligent cloud slicing technique, where the cloud enabled applications are accessed in the form of cloud slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These support different edge devices at the user end while facilitating the requested services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The following section describes the tactile mode of communication in B5G/6G system that is to be incorporated with the aforementioned slicing techniques, so as to deploy and configure the proposed touch interfacing wireless system with incorporated intelligence which is discussed in detail in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TACTILE INTERNET IN B5G/6G SYSTEM The internet initially was designed and indented to be a reliable and interoperable means of communication across the globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' With time, not only has it evolved to convey a large amount of content, but it also helps enhance the real IEEEAccessCS1 CLOUDSERVICES CS2 INTELLIGENT CLOUD SLICING FOGNODES CSN SMART HOME W WEARABLE AIML DEVICES EDGEDEVICES RS!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' RS2 GREENCOMMUNICATION COGNTTIVERADIOTRANSMISSION RAN SLICING SERVINGGATEWAY PACKET DATA GATEWAY RSN TACTILE SUPPORT UAWNETWORK AUTONOMOUSVEHICLES VIRTUALCLASSROOM AS2 AR/VR/XR AS3 Tube HAPTICCOMMUNICATION fa ROBOTICS UHD VIDEOSTREAMING TELESURGERY APPLICATION SLICING Y ASN SMARTCITY (rom)This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 14 FIGURE 7: Tactile internet E2E architecture with bilateral feedback system world experience of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The tactile internet thus improves remote real-time physical engagement with existing and virtual items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Whereas integrating TI with touch interface will provide a two way interactive experience that blurs the boundaries between the real and the virtual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The works in [89]–[92] examine the technical requirements of TI and its ability to support future smart city applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore provisions for a real- time control with the physical and haptic sensations to be experienced by the users remotely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Unlike the traditional internet applications, the new tactile internet (TI) intends to serve as a medium for remote, real-time physical interactions with actual, physical objects, including humans, machines, and processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The tactile internet enables enhanced virtualized remote classroom instruction with the participants in collaboration with the remote environment collectively amid the presence of remote and virtual resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus, the data rates of wireless communications have been increasing, which is primarily due to innovation in electronics and the latest communications technology, including text messaging, video streaming, emails, and file sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The TI is centered on the H2M interactions with the haptic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' To facilitate haptic communication, the transmission of data via tactile internet creates a network that is both extremely reliable and extremely responsive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Haptic interaction is a type of interaction that includes the use of remote touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It refers to the kinesthetic perception of information conveyed by the muscles and joints of the body via force, torque, position, and velocity, as well as the tactile perception of information conveyed by the mechanoreceptors of the human skin via surface texture and friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Sharing information through kinesthetic mode facilitates a global control loop with stringent latency requirements, while containing a feedback of generally audio/video type[93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Now with the enabling of haptic data, the TI enables a networked control system (NCS) supporting the connected sensors and actuators while controlling highly dynamic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore it helps in digitally transmitting the sense of touch from one place to another, facilitating the URLLC network in the B5G/6G system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In other words, the tactile internet’s purpose is to provide a remote and dynamic way for people to experience physical haptic or the touch based control, while exchanging closed-loop information between the virtual and physical worlds[90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Wireless communications can thus be a medium for controlling and directing real and virtual objects using such a platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This revolutionary technology continues to transform healthcare, transportation, education, logistics, smart grid systems and many more, hence covering a major portion of the economy sector in the society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It thus provides sub-millisecond connectivity for the healthcare applications like remote surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This section draws attention to one of the most popular applications of the B5G/6G communication system: ‘Tactile Internet’ and helps review its parameters, applications and the basic architecture, while focusing on its application in the intelligent 6G and IoT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' INTRODUCTION The ‘Tactile Internet’, as coined by Gerhard P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Fettweis, has been a catalyst for the economic development and creativity and in bringing a new stage of maturity in adapting technologies for a changing global environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Given that cellular communications connects a vast majority of people worldwide, it is therefore imperative to connect the technology as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' According to IEEE P1918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1 working group, the Tactile Internet may be defined as a IEEEAccessCommand signals Serving gateway Packet data gateway ngin gNB Tactile suppor engine Human operator and HSI Teleoperator (Slave robot) Slave Domain Master Domain gNB Serving gateway Packet data gateway Haptic feedback Bilateral ControlThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 15 network of networks that can be accessed, perceived, and manipulated by people or machines on a remote, real or virtual basis in real-time perceptions[94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The tactile internet application offers the standard required latency needed to guide and control real and virtual objects without causing cyber sickness, revolutionizing education, accessibility and traffic, healthcare, athletics, culture, games, and the smart grids, thereby profoundly shaping our community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The i-phone for instance has an astounding haptic interface, provided by the gyroscope and the modern touch screen technology, which has been a welcome step that could drastically transform the way we connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Additionally, we have an instinctive (or innate) sense of our surroundings, which has a tactile understanding of the real time connection with our world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The tactile feedback and a phone give our hands the sensations, which in turn enable our whole system to be modulated with its proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Whereas inside the vehicle it modulates several sensors and controls elucidating a real time human-machine communication interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus all TI needs is a highly responsive, smart, and reliable connectivity in order to provide a medium for intelligent, real-time touch, control, sensing and operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' With high availability of TI, accompanied with its very fast reaction time and reliability, the human interaction with the machines enables a new dimension by creating an interactive, real-time system, which revolutionizes almost every segment of the society[94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Taking into account the industrial dimension, TI is an interconnected system of specialized components and applications used in industrial environments to monitor and operate the physical equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence the works in [62], [89], [92] contribute to the role of TI in industrial systems by examining its potential in emerging and future industrial sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Against this backdrop, the goal of this study is to identify and address the cutting-edge challenges to implement an intelligent touch enabled system via tactile internet in the perspectives of B5G/6G based wireless and IoT communications networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus, tactile internet serves as a key to realizing the vision of “Touch Technology”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The TI has an expected potential and future scope through the increasing penetration of mobile and cell platforms with the robotic-human and computer interactions in the virtual control environments, for interconnecting people, machines, appliances and processes in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In order to accomplish this, we present our discussion with an overview of haptic communication via tactile internet architecture, with an emphasis on its potential development in the touch-enabled framework in 6G and IoT mobile networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TACTILE INTERNET ARCHITECTURE The haptic or the touch sensation helps ascertain a connection between the humans and peripheral environment in a way analogous to the auditory and visual senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore occurs bilaterally as a touch, sensed by imposing a motion or a movement in an environment by some reacting force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The haptic communication thus provides an additional dimension and advantage over the traditional audio visual communication for a real time control and accessibility in the distant and remote environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The tactile internet architecture consists of a radio access network (RAN) and a core network (CN), both of which are expected to meet critical requirements for TI functionality realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Each of the three domains in the tactile E2E architecture can be separated into three sections: the master domain, the network domain, and the controlled (slave) domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The master framework comprises of an operator, either human or machine, and an operating system interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Using various coding techniques, this interface acts as a master robot or as a controlling device, converting the operator’s input into tactile input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' If the controlling device is haptic in nature, it lets human interact with objects in virtual or real environments through physical means such as touching, feeling, manipulating, or controlling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is primarily responsible for the controlled domain’s operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In case of a network controlled system, the master domain includes a controller that issues command signals to the sensor or actuator system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A domain, where both robotic machines and other objects at distant locations in a controlled environment, and directly accessed by the master domain via command signals is referred to as a controlled domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' When remote operations are carried out via haptic feedback, energy is transferred between the master and the controlled domains and the global control loop is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As apparent from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='7, the components in the E2E tactile internet architecture are explained as below: 1) MASTER DOMAIN: Generally, an HSI/HMI8 is a robot where the user may touch, feel, and move virtual and real-world items while directing the actions in the slave domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) SLAVE DOMAIN: The slave domain is controlled by a tele-operator, through different command signals from the master domain, interacting with its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) THE NETWORK OR CONTROLLER: The network domain kinesthetically integrates the person with its surroundings and distant environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) TACTILE SUPPORT ENGINE: Being on the edge network, it effectively offers AI capabilities that are crucial in system stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) HAPTIC DEVICE: Enables the tactile communication, which means a user may touch, feel, and engage with virtual or real-world things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, the most common design for a tactile haptic device is a linkage-based system that consists of a robotic arm connected to a stylus and capable of applying force on its tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus, the growing degree of freedom (DOF) is an essential phenomenon for the envisaged applications that 8 HSI/HMI: Human System Interface or Human Machine Interface IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 16 integrate the network interface in a direct and indirect connection with a cellular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Since they both include sensations rather than conventional multimedia, it is essential to differentiate between the tactile internet and the haptic communication, which are comparable to traditional multimedia like speech, data, and video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' To summarize, haptic communication networks include communications across the wired and mobile internet, as well as applications that operate on the tactile internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This means that the haptic communications and the tactile internet have a service and medium connection with each other, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' NEED FOR INTELLIGENCE The existing and the future B5G/6G wireless network are expected to endow the users with an improved coverage and high data rates with a better cost efficiency, resource utilization, scalability, adaptability and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence the B5G/6G wireless communication system is anticipated to be a backbone of the digital revolution in the next generation network providing a ubiquitous reliable and practically an instantaneous connectivity for the humans and machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The Artificial Intelligence (AI) may be regarded as the ‘processing and simulation of the human intelligence by machines’ and therefore has a great potential in working out several intractable and unstructured problems containing large amount of data[84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In other words AI may be defined as a science of constructing computers that are capable of performing tasks requiring human-like cognitive intellect[95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' All the while AI has therefore been a widened approach for the machines to be able to smartly carry out assigned tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ML on the other hand is presently the widely accepted application of AI, empowering the machines to train and learn from large datasets and perform tasks without any need for explicit programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Next in order is deep learning (DL), a subset of ML that analyses the artificial neural networks (ANNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These have more number of hidden layers to replicate the human brain, making it one of the most widely used ML techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' DL therefore has a lucrative application in fields like computer vision, bioinformatics and speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Such induced intelligence in the wireless communication network not only reduces the manual effort in network deployment, configuring and management but also helps in an improved system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It also asserts the adaptability and reliability of the communication network by taking robust decisions in real time according to the prediction and behavior of the users and network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence due to the recent advances and research in the intelligence (AI/ML), a wide range of novel technologies like self driving cars, voice assistants, holographic telepresence, e-health and wellness applications, pervasive connectivity in smart environments, industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 applications, massive robotics with the unmanned mobility in 3D, AR/VR have become possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6G wireless networks thus offer a broadband network, which is fast, instantaneous and safe, in order to enable mass data exchange at various frequencies using a wide range of technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In addition, these technologies are moving towards intelligent devices in IoT that will demand a more reliable, stable, efficient, and a secure connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence the complex connected devices therefore require a dynamic communication network to address their inherent complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The future wireless networks will eventually need a self-organizing and configuring capability alongside their cooperation and coordination between the different nodes and communication layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It enables us to effectively meet challenges like coverage, spectrum, and energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The continuous acceleration of the machine type communication (MTC) devices adds to the existing ultra dense network’s complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore many such applications supported by B5G network must attain a short transit time and low latency with high reliability, availability and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The majority of these are resource constrained and unable to rely on their bounded resources and thus call for an uninterrupted and safe operation as its main concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Consequently these applications owing to their delay and bandwidth constraints cannot be moved to the cloud or network controller[96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Furthermore these devices generate diverse range of datasets with a large scale of erroneous or missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Many wearables with VR/AR, intelligent products and support systems, and other data hungry use cases have a built-in end infrastructure to afford and deliver content based services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus in order to incorporate an intelligent system, an intelligent and content aware approach must be implemented for the planning, design, analysis and optimization of such network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This necessitates integration of the network systems with their data sources, decision making and cyber physical infrastructures, as well as sensing and communication networks [97]–[99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Conclusively the favorable conditions for the implementation of the intelligent learning techniques in 6G wireless networks range from: 1) Network interoperability with the distribution, affordability and accessibility of computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) The predictive nature of the network characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) Accessibility of a considerable amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore a densely integrated wireless network may be engineered by adopting artificial intelligence (AI) principles combined with the incorporation of machine learning (ML) techniques with reasoning and decision making mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Accordingly the development of an intelligent touch based system calls for a promising development in facilitating the efficient resource distribution in the cloud, fog and edge nodes, aiming to put together system intelligence and data processing abilities in close proximity with the original data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' MACHINE LEARNING IN B5G/6G WCN ML as a member of the quintessential AI technology has nowadays become a key component of 6G wireless communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ML is said to be of a plausible advantage in the communication system owing to fact that the dynamic nature of the wireless communication channels complicates the channel interference models in the B5G scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore ML techniques are capable of extracting information from the unknown channel by learning from the communication data while taking into account previous learning experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Furthermore, the rapidly growing number of wireless access points necessitates a global optimization of communication resources as well as a fine-tuning of the system design[73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' However the existing approaches together with the massive amount of resources complicate the tasks concerning the optimization and correlation of the system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In contrast, advanced ML techniques like as deep learning and probabilistic learning can represent highly nonlinear relationships and aid in the determination of optimal system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Sequentially, ML aids in the realization and instillation of learning-based adaptive network configurations by identifying and evaluating their behavioral patterns in advance rather than reacting to unanticipated outcomes[63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, the current cellular networks that were built and managed based on the preceding premise may be unable to keep up with the growing complexity of data produced and therefore fail to provide the necessary capacity, dependability, and flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Now as response, the network may be unable to respond fast enough to expected occurrences, thereby compromising real-time communication services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Because the majority of AI/ML algorithms are not purpose-built for the wireless communication networks, it is difficult to apply them directly to the B5G/6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' All the above arguments call for an intelligent communication interface in the real time, facilitating a stable and efficient connectivity within the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The 6G wireless communication system guarantees a wide range of frequency bands, including sub-6GHz, mm-wave, THz, and optical bands, while also increasing the computational complexity in the channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a consequence, comprehending new channel characteristics for modeling new channel scenarios is a lengthy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Owing to the significantly high computational channel complexity in many situations, traditional techniques may aid in certain approximations and assumptions to help simplify the channel modeling and processing methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This ensures the balance between accuracy and complexity tradeoffs of both channel modeling and processing methods is beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' GENERALIZED WORK FLOW PROCESS FOR MACHINE LEARNING For most prevailing and functional machine learning algorithms, the generalized work flow process in basic steps is described and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 8 [100]: 1) PROBLEM FORMULATION: Since the ML training process is time consuming, it is critical that the problem be appropriately formulated at the start of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Moreover there should be a strong correlation between the problem and the information gathered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Classification, clustering, and decision making are three important types of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The proposed model should also be considered within these three categories to aid in the identification of learning model as well as data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Improper problem formulation results in an unsatisfactory learning model and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) DATA COLLECTION: There are two types of data collection: offline and online data collection, where data collected in real- time is used as model feedback in online data collection and is also used for re-training of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In contrast, offline data may be retrieved from the source without an Internet connection [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' By utilizing monitoring and measurement tools, online and offline data can be collected efficiently, securely and stored for model adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Data collection marks the beginning of training and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Validation and testing are set in motion after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) DATA ANALYSIS: It is divided into two types: preprocessing and feature extraction where preprocessing is used to reduce noise from the gathered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The data’s features are then extracted, which is a prerequisite for learning and training [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The types of characteristics that can be extracted from the network include packet level features and flow level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The packet size, mean, root, and variance are extracted at the packet level and mean flow length and mean number of packet flow features extracted at the flow level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) MODEL CONSTRUCTION: While iterative process selection, training, and tuning are all necessary aspects of the model selection process, they are applied differently and a suitable learning model must be chosen depending on the dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The training stage entails training the model with the dataset that will be collected at the start of the stage, whereas the tuning stage have the model learn for itself by comparing it to the trained data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) MODEL VALIDATION: Cross validation of the testing process is used to check the model’s accuracy, which aids in optimizing the model and preserving the overall efficiency of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6) DEPLOYMENT AND INFERENCE: Throughout the deployment and inference stages, the model’s trade-offs and stability are monitored to ensure accuracy and to determine the optimal sequence of steps to be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 18 TABLE V MACHINE LEARNING TECHNIQUES AND THEIR APPLICATIONS IN B5G/6G SYSTEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ML Techniques Definition Type Principle Applications in 5G /B5G communication system m- MIMO Small Cells D2D Hetnets Small cell,D2D, Hetnet clustering Spectrum sensing and allocation Resource allocation Anomaly/ fault detection QoS requirement Outage mgmt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SINR improvement Channel estimation/ detection CSI Behavioral learning Cognitive radio Energy harvesting Smart grid SVM9 Data point separation using a hyper plane or kernel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SL Classifier function: Linear /non linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' \uf0fc \uf0fb \uf0fb \uf0fb \uf0fb \uf0fc \uf0fb \uf0fb \uf0fc \uf0fb \uf0fb \uf0fc \uf0fc \uf0fc \uf0fc \uf0fb \uf0fb KNN10 Test point decision by voting of the k nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SL Non parametric lazy learning algorithm for classification and regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' \uf0fc \uf0fb \uf0fb \uf0fb \uf0fb \uf0fc \uf0fb \uf0fb \uf0fc \uf0fb \uf0fb \uf0fc \uf0fc \uf0fc \uf0fc \uf0fc \uf0fb K-Means Clustering Segregation of n data points into K clusters, each associating to cluster with nearest mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' UL Iterative refinement with cluster allocation to the data point with least ED11 from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' \uf0fb \uf0fc \uf0fc \uf0fc \uf0fc \uf0fc \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb Bayesian Learning Data points trained by GM12, EM13, HMM 14 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SL A posteriori probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' \uf0fc \uf0fb \uf0fb \uf0fb \uf0fb \uf0fc \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fc \uf0fb \uf0fb \uf0fc \uf0fb \uf0fb PCA15 Relevant information extraction from large data sets using orthogonal transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' UL Data sets reduction into principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' \uf0fb \uf0fb \uf0fc \uf0fc \uf0fb \uf0fc \uf0fb \uf0fc \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fb \uf0fc \uf0fc Q Learning A model free RL where an agent has an access to a set of possible states and environment, with no concern of rewards or transition between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' RL Off policy RL to get best action for the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' \uf0fb \uf0fc \uf0fc \uf0fc \uf0fb \uf0fc \uf0fc \uf0fb \uf0fc \uf0fc \uf0fc \uf0fb \uf0fb \uf0fc \uf0fb \uf0fc \uf0fc MAB16 Multiple agents, sequentially taking actions receive random reward to achieve a steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' RL Trade off between the best action and information to achieve a larger reward in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' \uf0fb \uf0fc \uf0fc \uf0fc \uf0fb \uf0fc \uf0fc \uf0fb \uf0fb \uf0fb \uf0fb \uf0fc \uf0fb \uf0fc \uf0fb \uf0fc \uf0fc MDP17 A discrete time stochastic control state transitioning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' RL A single agent with partly random and partly controlled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' \uf0fb \uf0fc \uf0fc \uf0fc \uf0fb \uf0fc \uf0fc \uf0fb \uf0fb \uf0fb \uf0fb \uf0fc \uf0fc \uf0fc \uf0fc \uf0fc \uf0fc 9 SVM: Support Vector Machine 10 KNN: K Nearest Neighbor 11 ED: Euclidean Distance 12 GM: Gaussian Mixture 13 EM: Expectation Maximization 14 HMM: Hidden Markov Model 15 PCA: Principal Component Analysis 16 MAB: Multi Armed Bandits 17 MDP: Markov Decision Process IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 19 FIGURE 8:Machine learning basic workflow process[100] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' MACHINE LEARNING TECHNIQUES IN B5G/6G WIRELESS COMMUNICATION SYSTEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A few existing works containing a comprehensive overview of the existing intelligent techniques with their application in the B5G/IoT wireless communication systems are discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Authors in [99], [101] talk about different learning techniques in IoT applicable scenarios while taking into consideration their input information and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' While the works in [103], [104] provide fundamental concepts about the state of art AI based technologies applied in the existing network, making the system competent enough to accomplish self configuration, self optimization and self healing of the concerned system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Additionally authors in [58] discuss framework and application of the popular deep reinforcement learning (RL) technique that is vital in the engineering of the cognitive smart cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Furthermore the articles in [105], [106] further provide a comprehensive survey of the existing RL algorithms and their stability and behavioral adaptability with the other learning agents in the implemented network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Speech recognition, bioinformatics, and computer vision are just a few of the applications for machine learning that make efficient use of the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Machine learning is primarily used for prediction and classification, but it also plays a role in performance prediction and intrusion detection in networking systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore to make decisions directly, machine learning constructs models that can learn from data without adhering to a set of rules[107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ML hence allows a model to enter a self-learning mode without having to be explicitly trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' To learn system characteristics that cannot be represented by an explicit mathematical model, ML models therefore are used as computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These models are employed in tasks including categorization, regression, and intelligent agent- environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Using basic arithmetic calculations, the model can efficiently complete the task once the system characteristics are learned[108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Models may be trained by providing them with data sets, and when they are exposed to fresh data, they are able to learn, forecast, and develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' There are three types of machine learning algorithms: supervised learning, unsupervised learning, and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In supervised learning, a model is trained on labeled datasets and then learns on its own by comparing the training dataset to the predicted output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This method is commonly used for classification and regression issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Unsupervised learning is a type of machine learning that uses an unlabeled dataset to detect patterns and relationships in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is mostly used to solve clustering and association problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' During reinforcement learning, an agent interacts with a system set and learns how to map all information about action, without any training data[109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This section provides a brief overview of some of the most commonly used artificial intelligence and machine learning techniques, with Table V illustratively defining and highlighting various machine learning techniques and their functionality in B5G and 6G wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The works in [110]–[113] encompass few of the researched investigations concerning the ML application on the B5G/6G wireless communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' However most of the existing works exercise intelligence on a limited part of the existing wireless communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Some of the wireless channel characteristics influenced by intelligent learning techniques are discussed below: 1) CHANNEL MODELING USING ML: The ML helps deal with the channel modeling problem by implementing the model based approach, extracting the wireless channel features from the existing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ML has an efficacy in predicting channel feature, estimating channel parameter, CIR18 modeling, MPC19 and classification of scenarios pertaining to the concerning channel environment, derived out from the above cited works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) CHANNEL MEASUREMENT: A channel model based on a feed forward neural network(FNN20) and RBFNN21 is shown in [114] which is functional in predicting channel properties like received power, RMS delay and angle spread(DS/AS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Therefore in addition to the transmitter and the receiver coordinates, their input 18 CIR: Channel Impulse Response 19 MPC: Multipath Component Clustering 20 FNN: Feed Forward Neural Network 21 RBFNN: Radial Basis Function Neural Network IEEEAccessStep:1 Problem Formulation (Prediction, Regression, Clustering,Decision making) Step:2 Data Collection No (Realtimedatacollection formodeltraining) Step:3 Step:5 Requirement Model Validation Data Analysis satisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' (Cross Validation, ErrorAnalysis) (Preprocessing, FeatureExtraction Yes 个 Step:4 Step:6 Model Construction Deployment and Inference (Offline training and Tuning) (Tradeoffonspeed,memory,stabilityandinference accuracy)This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 20 parameters also influence the distance and frequency of the Tx-Rx link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) NOISE : ML in [115] makes use of ANN to remove noise from CIR, while [116] makes use of CNN to identify the relevant wireless channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In CNN and wireless communication channels, MPC characteristics like as amplitude, latency, and Doppler frequency serve as input and output parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) CHANNEL ESTIMATION: In order to obtain accurate channel estimation, the work in [117] uses a 2D non linear complex support vector regression (SVR) in a rapidly fading and time varying multipath channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' On the other hand, the work in [118] refers to a deep learning-based channel estimation method for beam space mm-wave massive MIMO systems that can learn the channel structure from a huge number of training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) MASSIVE RADIO INTERFACE: ML algorithms help in analyzing the enormous amount of data produced by the massive MIMO arrays, where the conventional channel estimation and detection algorithms are rendered incapable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Deep learning methods and techniques, particularly image processing and video analytics, provide the most exciting algorithmic approaches[119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6) WIRELESS NETWORK LOCALIZATION: The continuous development and updating of wireless channel locations has been made feasible via automated learning from crowd-sourced data employing a large number of mobile devices, yielding precise localization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In exchange, it enables consumers to benefit from improved location-based services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 7) NETWORK MANAGEMENT: Machine learning and artificial intelligence has the ability to optimize a variety of tasks such as fault detection and user tracking over a wirelessly linked network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 8) RESOURCE MANAGEMENT: The resource management mechanism is only able to function once the system has memorized the states and conditions of the network users and their real time wireless environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This therefore helps improve the system performance with time and in turn helps the system incorporate an intelligent and dynamic decision making phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Following consideration of all relevant factors, we intend to incorporate intelligence into the tactile internet infrastructure in order to achieve a complete automation of the systems that surround us, taking into account the feasibility, interoperability, and functionality of the 6G and IoT-based wireless communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' An intelligent touch-infused technological framework is proposed in the next section, which incorporates intelligence from existing smart IoT infrastructure and interfaces it with next- generation systems by creating a tactile/haptic feeling as we interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' PROPOSED TOUCH TECHNOLOGY IN B5G/6G AND IOT NETWORK The proposed and futuristic anticipated system infrastructure is expected to encompass an intelligent and a reconfigurable touch enabled system that is pertinent in an IoT interfaced B5G/6G systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The intelligent Touch- based IoT paradigm can be made up of a variety of functional elements that help smart objects perform various functions such as sensing, actuation, identification, management, and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The touch based IoT system’s functional elements[3] can be summarized as follows: 1) SMART DEVICES: The primary components of the IoT based Touch System, performing sensing, actuation, and control functions are capable of sharing data with other applications and servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' To connect to other smart devices, each IoT device has to be prepared with numerous interfaces including the internet access, I/O interfaces for sensors, audio and video, storage and memory interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) FUNCTIONALITY: The device functionality ranges from smart- watches, wearable sensors, automatic cars, industrial machines, LED lights and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' From office automation and household appliances to manufacturing lines and commodities tracking, intelligent IoT techniques are used in a wide range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, IoT services must be used to improve IoT application development and accelerate installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) SERVICES: These are dedicated to identity and device modeling and are commonly grouped under the umbrella term identity services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Additional subcategories include information aggregation, discovery, control, collaborative awareness, ubiquitous services, and data analytics and publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) REMOTE ACCESS: As opposed to devices that use mechanical switches or buttons to remotely manage, IoT devices have either no human involvement or can be remotely managed without the need for human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) SECURITY: Taking into consideration the security aspect, as far as the data on wireless networks is concerned, especially with regards to denial of service (DoS), spoofing, and eavesdropping, the information is vulnerable to an array of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus, IoT systems use many security features, such as privacy, authorization, authentication, data IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 21 security, content integrity, and message integrity, to attempt to thwart these attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6) IOT APPLICATION: In order to provide IoT users with interfaces, the application layer supplies IoT users with various interfaces that enable them to monitor and control the various aspects of IoT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus the projected system is to be most likely based on the IoT functionality, and is to be expected to be implemented in the B5G/6G networks, where the system intelligence is of an utmost need so as to be compatible with the accelerating high data rate and in turn satisfy the low latency requirements of the next generation system[120], [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' INTELLIGENT TOUCH BASED SYSTEM The problem statement here describes the need for the intelligence in B5G system, and therefore can be elaborated as per the works in [122]–[125] stating that the ever since the evolution of the wireless network from 1G to the existing 5G/B5G and subsequently the 6G networks, have led to a tremendous ascend in the billions of connected devices bound together by IoT, forming an integrated IoE network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The increased traffic, due to growing number of devices, therefore requires high energy efficiency and lower latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The growing new use cases in the evolving B5G network incorporates the AR/VR/XR based smart systems including the smart road monitoring, the smart cities, consisting of the smart homes and IoT governed smart appliances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These systems are externally controlled and therefore greatly lack in intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence for the efficient functioning of these devices in the next generation 6G network, an intelligent system is required to govern the existing AR/VR based 5G/B5G network effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore requires an interfacing mechanism with the existing network infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' To implement this in real time, the B5G/6G enabled tactile internet proves to be a promising technology, which in turn can provide a new dimension to human to machine interaction by enabling haptic sensations and therefore a touch enabled communication interface, in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The touch-governed IoT system is expected to permeate many facets of the contemporary daily living, including the ability to sense, process, analyze, and infer environmental parameters from natural resources and ecologies to human environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The main aspect of this touch-based IoT network is its ability to intelligently connect to the other existing and future networks, to all the resources that are utilized by these networks, and to help accomplish that through the advancement of the prior networks and communication protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For this vision to succeed, we will need to advance beyond conventional mobile computing technologies and design an IoT system in which everything we can touch is connected and capable of acting as a smart and intelligent extension of ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' An in-depth knowledge of IoT issues span from an awareness and a better understanding of the IoT concerns and the complexities involving the size and depth of the pervasive communication network, software architectures, and data transfer and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This knowledge is used to create autonomous and intelligent devices in IoT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The primary goal is to set up a global network of connected smart objects and devices, all of which can connect to each other without human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Each object that has been embedded with a smart interface and connected to the user platforms and digital environments is assigned a virtual identity and interfacing, allowing it to connect and communicate with other embedded objects[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As we build our IoT network, our physical and virtual entities turn into virtual things in a cyber world, each with specific abilities that all IoT entities can use to realize personalization, specialization, and autonomy based on the communications protocols used to make the smart entities unique and provide them with virtual personalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The combination of B5G/6G network slicing technology and the TI application will thus prove to be a driving factor in the realization of this suggested reconfigurable and intelligent touch enabled framework, whose research is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Consequently, it can be concluded that the main goal of the IoT-based Touch technology system is to improve the lives of people, where all objects around us have the ability to figure out what we want, what we require, and what we like, as well as serve it accordingly without us having to explicitly command them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='9 presents the layered architecture of the proposed intelligent touch technology in B5G/IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence it requires an AI/ML technology, coupled with the B5G/6G based Network Slicing and Tactile Internet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' to implement and interface the intelligent touch based system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence the network slicing and tactile internet explained in the previous sections proves to be the cornerstone in the incorporation of a reconfigurable intelligent touch system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' LAYERED ARCHITECTURE FOR TOUCH TECHNOLOGY The Internet of Things connects millions of smart objects, increasing data traffic and necessitating the use of large processors and storage systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This emergence of IoT system together with the rising demand for the wireless capacity ultimately paves way for the futuristic intelligent Touch Technology system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Many factors, including as interoperability, scalability, QoS, and reliability, must be considered while designing such an IoT based intelligent infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, the required intelligent IoT architecture based on touch technology should have the following characteristics: 1) DISTRIBUTIVENESS: The IoT model for the proposed system should enable data collection from various sources and their processing by various smart entities in a distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) INTEROPERABILITY: IoT devices from different vendors must communicate to achieve common goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 22 FIGURE 9: Proposed layered architecture for intelligent touch technology22 22The architecture presents the IoT Layered Infrastructure comprising of : Perception,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Connectivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Edge computing(network),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Cloud Storage(transport) and Application layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' providing tactile-based communication and device interaction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' as well as efficient network slicing between the connectivity and application levels in order to transfer and completely support requested services and user requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessApplication Slicing Controller Domain Touch based sensors and Home Automation Touch based Healthfacilities like Tele-surgery, Remote medical consultancy Tactile Haptic communication system Touch/Tactile based robotics and automated vehicles 灭！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Automated vehicles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Tactile based robotics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart classroom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Remote industries(HloT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Haptic communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='lot sensors and devices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Ultra low latency of Ims ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Data Accumulation (storage) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Edge (Fog)Computing layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='@actile suppor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Tactile support ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Foy node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='engine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='I/MLAlgorithmfor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='touchimplementatior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Touch based applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Cognitive (smart antenna) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Basestation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Point ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Power optimization in Receiving antennas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Spectrum sharing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Rout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Router ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Gateway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Comnectivity layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='RAN Slicing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Command ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Human controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Program control system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='lio/video ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='withHST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='feedback system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Perception layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Master Domain with HSI (edge devices)This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 23 Additionally, protocols and systems must be designed in such a way that smart devices from a variety of manufacturers can exchange sensed data in an interoperable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) SCALABILITY: Any such IoT network is expected to have billions of objects connected at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Because these platforms run many applications and systems, hence such applications should be able to handle and process a huge amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) LIMITED RESOURCES: Both computing units and energy are considered highly rare units in regards with the limited resource availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) SECURITY: The feelings of helplessness and vulnerability that users have when they are under the control and dominance of an unknown external device could be detrimental to the given system deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This section presents the proposed layered architecture for the intelligent and touch enabled technology which is to be functionally implemented in the B5G/6G and IoT configured wireless communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This architecture, which is defined by the tactile internet and network slicing as its cornerstone components, can thus be broadly classified into master, controller, and network domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The master domain is comprised of user-facing edge devices such as a robotic arm, gesture-based glove, or a gaming console for real-time online gaming, which collectively form the HSI network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus, this layer contains various end devices and sensors, which aid in the touch- based sensory data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' All of this is located on the perception layer as per the layered architecture presented before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The perception layer is responsible for transmitting the generated sensory data to its required destination, over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The reliable and timely transmission of data from the perception layer devices to the edge computing layer is ensured by the efficient connectivity between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Connectivity layer is therefore responsible for processing and communication between the master devices accompanied with their effective routing and switching, enabling a reliable delivery of information across the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This layer also ensures the security of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The third layer is the edge computing or the Fog layer, where the data evaluation is done and is processed at higher levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The information is then processed and transferred by the tactile support from the end user to the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus the fog and edge network therefore facilitates a distributed computing, storage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' control and communication of the network functions with reduction in the latency, system response and cloud workload of the transmitted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is further connected to the RAN layer which is responsible for introducing intelligence in the system through radio network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The B5G/6G network slicing, virtualization and fragmentation takes place in this layer, the details of which are explained in the earlier sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence to further avail the tactile facilities, the RAN slicing is accomplished, fragmenting it into the isolated frameworks, each of which is further meant to furnish various high data rate applications simultaneously, while maintaining the latency of the requisite ≤1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The routed network endows with a provision for network slicing and storage of data, enabling access to the various cloud services, for effective data storage and acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The data acquisition process, before further furnishing the requisite applications, follows the data accumulation and abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Data accumulation involves the data capture and storage, to be used by the applications later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It also deals with the query based data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Data abstraction further virtualizes and consolidates the data at a place before its slicing and cloud storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The setup of the signaling procedure and protocol stack along with the physical layer optimization allows the accessible devices to direct and control the power of the sensors, edge, fog, and cloud-based platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The application layer further furnishes the applications due to application slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Here the applications required are entirely based on the TI and are intelligent and touch enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These may vary from the haptic communication by the haptic and IoT sensors, the automated remote industrial operation and monitoring, which is an example of the industrial revolution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It has a great potential in the evolution of healthcare, education, entertainment as well as edutainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Since the application layer is in charge of monitoring, controlling, and analyzing the data, it must be at the heart of all systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The business layer in the end provides these services at the consumer end i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' at the controller domain at the receiver at the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=" For an easy understanding, let's go over each layer of the architecture with their associated protocols and functionality." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Following the bottom up approach, the five layered IoT configured architecture of the proposed touch technology comprises of the following layers: 1) Perception layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) Connectivity (Data Link) layer 3) Fog/Edge computing (Network) layer 4) Data Accumulation (Transport) layer 5) Application layer As a result, we will go over each of these layers one by one, explaining their functionality and interoperability as we go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This section describes each layer one by one with its functionality and interoperability beginning with the application layer upto the perception layer following a top down approach [126] in Fig 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' APPLICATION LAYER As the front end of the IoT architecture, the application layer is where the majority of the IoT potential will be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This is because the application layer provides IoT developers with the user interfaces, platforms, and tools that are required to implement IoT applications such as IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 24 smart homes, intelligent transportation, smart health, and smart cities, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Furthermore, it is in charge of receiving the data that has been processed from the network layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The IoT application layer provides appropriate protocols and services needed to transmit messages at application level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' When choosing a communication protocol for a certain application, many elements should be considered, including power consumption, necessary BW, transfer and connection time, delivery guarantee, data security, and packet size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A description of a few protocols that are commonly used at the application layer is provided below and summarized in the Table VI: 1) MQTT: It uses middleware and apps to enable communication between IoT devices and the network in a variety of ways, including M2M, server to server, and machine to server, and it runs on the TCP/IP protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Additionally, it supports communication over low-bandwidth and unreliable links and is used for publishing and subscribing to lightweight message exchanges[127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) XMPP: It allows for low latency communication and minimal message delivery, making it ideal for video calls, instant messaging, and chats, as well as publish/subscribe systems, gaming, and IoT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Because of its simplicity and adaptability, it makes it possible to communicate between heterogeneous systems [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) REPRESENTATIONAL STATE TRANSFER (RESTFUL): The REST protocol is a collection of best practices, rules, and constraints for developing web services that enable data exchange and communication between various devices, as well as for developing distributed hypermedia systems with desirable properties such as modifiability and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' RESTful is a request-response and client-server architecture that allows clients to access server resources in IoT contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is based on the HTTP protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Because they are lightweight and straightforward protocols, RESTful APIs are considered to be a good choice for a variety of IoT applications[128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) CONSTRAINED APPLICATION PROTOCOL (COAP): In IoT applications, it enables resource- constrained and unsynchronized devices to communicate while providing flow control, reliable delivery, and simple congestion control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It uses multicast and unicast requests to enable publish-subscribe communication strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' CoAP uses UDP due to its small message size, low code footprint, and lack of TCP handshake overhead before transmission[129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) AMQP: It is widely used in commercial and business domains because it offers reliable and secure communication between heterogeneous devices and supports publish subscribe architecture based on an efficient and reliable messaging queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In addition, it uses the TCP protocol for increased reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The message queue and exchange queue models are used to transfer data over AMQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In the message queue model, messages are retained until they are transmitted to the receiver, whereas in the exchange queue model, messages are routed in a proper order[129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TABLE VI: SOME COMMONLY USED APPLICATION LAYER PROTOCOLS Protocol MQTT XMPP REST ful CoAP AMQP Standard OASIS23, Eclipse Foundations IETF IETF IETF, Eclipse Foundation OASIS, ISO/IEC24 TCP/UDP TCP TCP TCP UDP TCP Architecture Publish/Subscribe Publish/Subscribe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Request/Response Request/Response Publish/Subscribe Publish/Subscribe QoS: \uf0fc \uf0fb \uf0fc Confirmable, Non-confirmable, Acknowledge, Reset \uf0fc Semantics Connect, Disconnect, Publish, Subscribe, Unsubscribe, Close Get, Post, Put, Set, Result Post, Put, Delete, Cut Post, Put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Delete, Get-CON (Confirmable), Non (non- confirmable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ACK (acknowledgement),RST(Reset) Consume, Deliver, Publish, Get, Select, Ack, Delete, Nack, Recover, Reject, Open, Close Security TLS/SSL25 TLS/SASL TLS/SSL DTLS/IPSec TLS/SSL, IPSec, SASL Features For high latency and low BW networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For resource constrained devices Decentralization by server as there is no central master server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Flexibility Open standards High network overhead Scalability Easy implementation and interaction Flexibility Unsuitable for distributed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Reliability Lost packets retransmission Multicast support Resource monitoring Low overhead Scalability Reliability Performance security Heavy protocol as it requires memory and power resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 23 OASIS: Organization for Advancement of Structured Information Standards 24 ISO/IEC: International Organization for Standardization/ International Electrochemical Commission 25 SSL: Secure Sockets Layer IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='TABLE VII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SOME COMMONLY USED DATA ACCUMULATION LAYER PROTOCOLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Protocol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='TCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='UDP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='DCCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='STCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='TLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='DTLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='RSVP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Packet size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='20-40 bytes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='header ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='8 bytes header ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='12-16 bytes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='header ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='12 bytes header ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='5 byte header ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='224-1 bytes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='(handshake ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='message) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='16 bytes header ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Packet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='transport ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Segment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Datagram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Datagram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Datagram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Segment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Datagram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Datagram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Flow ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Congestion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Reliability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Performance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' robustness and network capacity improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Orderly data delivery between hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Transmission not possible if sequenced packet is not acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' No guarantee on packet delivery High packet loss Packets may arrive out of order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Retransmission needed when data is corrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Eliminates delay in out of order waiting data packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Supports strict, partial and unordered delivery modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Multi-homing support Congestion control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Flexibility for VoIP applications Multi-homing method Minimizes DoS attacks Dynamic IP addressing Prevent tampering by intruders Prevent passive listening by attackers Adds more latency Security and reliability for handshake message transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Unordered message queuing Retransmission timer to reduce packet loss probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Slower than STCP Data integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Error reporting Multicast comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' among heterogeneous devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' QoS routing Scalability issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' DATA ACCUMULATION/STORAGE LAYER: It interacts with the application layer to send and receive data without mistakes in the typical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The transmitting side here is in charge of breaking down messages received from the application layer into segments and sending them to the network layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Following that, the segmented messages received will be reassembled into messages that will be passed directly into the application layer by the receiving side of the communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This layer is responsible for ensuring the integrity and reliability of transmitted data by providing features such as packet delivery order, congestion avoidance, multiplexing, byte orientation, and data integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence known as the routing layer because it is in charge of routing data packets through the network area, where its protocols are in charge of packet ordering, error detection, and correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This section provides an overview of a few protocols that are commonly used at the data accumulation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The protocols are summarized in the Table VII: 1) TRANSPORT CONTROL PROTOCOL (TCP): It is a heavyweight, connection-oriented protocol, where the connection is not established until all the required data has been sent between each end device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TCP is suited for reliable communication since it requires acknowledgement messages to ensure each sending and receiving procedure, as well as support for retransmission of lost or corrupted packets and a flow control mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence, this protocol’s packet overhead is extremely high, resulting in increased power consumption from devices and thus making it impossible to operate on power-constrained devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TCP divides the data packet into multiple packets, each with an ordering number and source and destination IPs[130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) USER DATAGRAM PROTOCOL (UDP): It is a connectionless protocol that attempts to give protocols and applications that run over IP, an unreliable, minimum message queuing, message passing, and best effort transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' There is no requirement for end-to-end connectivity between communicating entities, which enables efficient communication for some applications that require real-time performance with low latency, such as video and voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' UDP does not provide a port with any attribute for addressing the source and destination functions and provides a data integrity check [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) DATAGRAM CONGESTION CONTROL PROTOCOL (DCCP): It establishes unicast bidirectional connections for datagrams with unreliable dynamic congestion control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These characteristics make DCCP ideal for applications that transmit large amounts of data and require a trade-off between reliability and timeliness, such as VoIP and media streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Due to its unreliability and absence of a receiving window, the flow rate of DCCP can be progressively increased[130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) STREAM CONTROL TRANSMISSION PROTOCOL (SCTP): It is a connectionless, message-oriented, IP transport layer protocol IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 26 similar to UDP that enables SCTP-based peer-to- peer (P2P) communication and reliable transmission for applications communicating over an IP network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, it inherits the majority of TCP’s functionality, such as packet recovery and congestion management[130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) TRANSPORT LAYER SECURITY (TLS): It was created to provide security channels among communicating peers and to give authentication, data secrecy, data integrity, and encryption to applications by preventing eavesdropping, message forging, and tampering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It runs on top of several transport layer protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is composed of two components: the handshaking protocol, which is responsible for authenticating communication ends, agreeing on shared keys, and negotiating cryptographic parameters and modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' and the record protocol, which divides the traffic into multiple records and protects them using the traffic keys[131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6) DATAGRAM TRANSPORT LAYER SECURITY (DTLS): It was created to secure datagram applications that do not require or provide dependable data delivery, such as datagram online gaming, internet telephony, and media streaming, which are deemed delay sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' DTLS is an enhancement of the TLS protocol that prevents message forgery, tampering, and eavesdropping when transmitting data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore, it should be able to cope with and resolve a variety of datagram difficulties, such as packet loss, packet reordering, and delay[132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 7) RESOURCE RESERVATION PROTOCOL (RSVP): It is a multicast and unicast control transmission protocol that was created to enable data stream transmission with a flexible, robust, scalable, and heterogeneous resource reservation setup at each router.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' There is also support for resource reservations in each node along the data path, multipoint to multipoint communication paradigm, cache management routers, and receiver initiated reservation[133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' EDGE/FOG COMPUTING LAYER: It is responsible for providing data with routing paths so that the packets can be transmitted across the network area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This layer creates logical connections, sends out error messages, and maintains the data transmission routing path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Furthermore, this layer contains all network devices such as switches, firewalls, bridges, and routers that are required to work with appropriate communication protocols such as 3G-4G-5G, Wi-Fi, infrared, ZigBee, and Fibre to the X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This layer is in charge of forming, addressing, and routing data packets, as it receives datagram packets from the transport layer and converts them to data packets before transmitting them to the destination side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This section gives an overview of some protocols commonly used in the edge/fog computing layer with the protocols summarized in Table VIII: 1) ROUTING PROTOCOL FOR LOW POWER AND LOSSY NETWORK (RPL): It is a tree-based, IPv6 proactive distance vector routing protocol developed by the routing-over-low-power and lossy networks working group to run commercial appliance networks with insecure connectivity, poor data rates, and substantial losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It has a storing and non-storing mode to reduce memory requirements and eliminate loops in low-resource applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is prone to high packet loss owing to congestion, has a long delay, and is vulnerable to assaults since it lacks end-to-end encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Overhead packets for control are flooded into the networks as a result[134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) COGNITIVE ROUTING PROTOCOL FOR LOW POWER NETWORK (CORPL): It is an extension of the RPL protocol, which was designed to be compatible with cognitive networks in order to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This feature ensures high packet delivery ratio, and keeps the network from colliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The CORPL routing mechanism takes advantage of an opportunity to select the most effective forwarding next hop from a pool of eligible neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' There are minimum collisions and duplication of data packets[135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) CHANNEL AWARE ROUTING PROTOCOL (CARP): It is a distributed protocol with a lightweight data package that was created for underwater and IoT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Network initialization and data forwarding are two steps in the routing operation performed by CARP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The receiving node updates its distance to the sink node, broadcasting the welcome messages with their ID and hop count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In data transmission, the sender sends a ping message to its neighbors, determining the best relaying node based on the link quality and information received from their ping messages, and then forwards data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' When selecting a relaying node, residual energy, network quality, and buffer space are all taken into account[136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) COLLECTION TREE PROTOCOL (CTP): A tree- based routing system that was created to give the greatest effort for any cast communication in networks with low energy demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' An early form of networking is where a source node (sink node) announces itself as the root node, where minimal cost is paid to deliver data to the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Other nodes connect to the root tree via lightning ads and then send their collected data into the sink node with the minimum amount available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' CTP, on the other hand, does not permit reverse routing from the sink to the sensors[137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) LIGHTWEIGHT ON-DEMAND AD HOC DISTANCE VECTOR ROUTING PROTOCOL- NEXT GENERATION (LOAD-NG): When compared to the on demand distance vector (AoDV) protocol, it is a more lightweight distance IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 27 vector and reactive protocol that is designed to provide a secure, scalable, and efficient routing in lossy and low power networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a reactive protocol, LOAD-ng generates on-demand route requests to discover a path to the target node and when data has to be delivered, the receiving unicast replies hop by hop from the destination node back to the sender node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' If a route is found to be broken, attempts to fix it are made, or an error message is sent to the requested node[138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6) AD-HOC ON DEMAND MULTIPATH VECTOR FOR IOT (AOMDV-IOT): It seeks to discover and establish connections between nodes and the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For each node, AoMDV-IoT generates two routing tables: an internet connecting table (ICT) and a routing table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In addition, it converts IP addresses into internet linking addresses (ILA) and when a node wants to be connected to the internet, the IP associated with the desired internet is connected to ILA so that the search function can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' If there is no internet node in ICT, the source node will broadcast the requested packet to update both tables until an optimal route to an internet node is discovered[139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' CONNECTIVITY LAYER: The IoT connectivity layer in the touch system architecture is comprised of a variety of communication protocols that are primarily responsible for providing services to the network layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence, it is in charge of connecting and transmitting signals from end devices to higher layers via routers and gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The IoT connectivity layer consists of a variety of communication protocols that, depending on the transmission range and coverage area, provide services to the network layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Some of the most commonly used protocols are reviewed further down this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 1) NFC PROTOCOL: A short-range protocol that allows mobile objects to communicate with one another over a few cm of distance and allows data to be transferred in seconds between the connected NFC devices that are in close proximity to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is RFID-based and thus uses an alternate magnetic field to connect devices that are either active or passive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In active mode, all of the TABLE VIII: SOME COMMONLY USED EDGE/FOG COMPUTING LAYER PROTOCOLS Protocol RPL CORPL CARP CTP LOAD-ng AOMDV-IoT Network Topology Mesh, Hierarchical Cognitive M2M networks, Mesh Tree-based topology, Mesh Grid Dynamic IoT network Routing metrics Connectivity, Link quality, BW, Reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Reliability, Collision risk End to end packet latency, energy consumption per bit, buffer spaces, packet delivery ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ETX26 of neighbors Hop-count Lifetime Hop count Network metrics MP2P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='P2MP27 communication P2P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' MP2P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' P2MP MP2P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' P2MP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='P2P MP2P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='P2MP P2P P2P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='P2MP Data rates Low data rates Low data rates Low data rates Low traffic rates Latency High Latency Delay of sensitive applications Low Latency High Latency High Latency Low Latency Algorithm Distance vector Distance vector Link state Distance vector Distance vector Distance vector Scalability \uf0fc \uf0fc \uf0fc \uf0fc \uf0fc \uf0fc Security \uf0fb \uf0fb \uf0fb \uf0fb Uses integrity check values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' \uf0fb Network mobility \uf0fb \uf0fb \uf0fc \uf0fc \uf0fc \uf0fc Applications \uf0b7 Home automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Industrial automation Building automation Smart grid and Smart cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Smart Grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Underwater WSNs application Commercial products Industrial WSNs Teaching Research Home applications Industrial application Mobile IoT applications 26 ETX: Expected Transmitted Count 27 MP2P: Multipoint to Point Communication ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='P2MP: Point to Multipoint Communication IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 28 TABLE IX: SOME COMMONLY USED SHORT RANGED CONNECTIVITY LAYER PROTOCOLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Protocol NFC 6LowPAN BLE Zigbee Z-Wave Network type P2P Star, Mesh Star Star, Tree, Mesh, Cluster, Hybrid Mesh Frequency band 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='56MHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='4GHz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='402-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='481) GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='4GHz, 915MHz, 868MHz 868(Europe), 908(USA), 900ISM Transmission range 10cm (10-100)m 100m (10-100)m 30m Data rate 106Kbps-424Kbps (20,40,60 )kbps 125kbps (12 Mbps) 250Kbps (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='6, 40, 200)kbps No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' of nodes 2 65000 65535 65000 232 Power consumption 15mA 15mA 30mA 5mW Routing protocols NFC possesses routing features RPL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AoDV RPL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6LoWPAN Zigbee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' RPL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AoDV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Zigbee network routing (ZBR) AoDV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' DSR Applications P2P data transfer Payment and ticketing applications Smart home Smart agriculture Industrial IoT Healthcare applications Mobile phones Smart homes Wearables and PC Security and privacy Healthcare Sports and fitness etc Smart home Medical monitoring AI sensors Consumer electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Home automation Smart lighting connecting devices generate a magnetic field, whereas in passive mode, one device generates a magnetic field while the others use load modulation to transmit data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Passive mode is energy saving and is widely used in today’s smart phones[140], [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) 6LOWPAN: It allows smart devices to connect to the internet via the IPv6 protocol while also taking into account the nature of wireless IoT networks by creating a very small header message format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It also removes obstacles to using IPv6 addressing protocol in IoT devices with limited processing power, data rate, and power[142]–[144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) BLUETOOTH LOW ENERGY (BLE) PROTOCOL: BLE is a low-power alternative to short-range wireless communication developed by the Bluetooth Special Interest Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Additionally, it allows for fast data packets to be transmitted at speeds up to 2Mbps in the ISM band[145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) ZIGBEE: The objective is to develop a low-cost, scalable and power-sipping wireless connectivity that is suitable for a wide range of controlling and monitoring purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' With intelligent routing and setup procedures, this protocol builds on IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='4’s features to enable high failure resilience and easy installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It also works well with other wireless communication technologies due to its strict security and listening techniques[146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) Z-WAVE: Smart light controllers and other sensors in home devices use this low-power wireless communication technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' With low latency transmissions and data rates of up to 200kbps, this technology operates over 900 MHz ISM bands[147], [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6) LOW POWER WIDE AREA NETWORKS (LPWAN) PROTOCOLS: LPWAN protocols are low-power, low-bandwidth, and low-cost protocols that are particularly useful for long-distance communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Furthermore, the devices that implement these protocols have transmission ranges ranging from 1m to 50 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The general characteristics of LPWAN protocols are as follows, followed by a brief discussion of each protocol: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These LPWAN protocols are implemented by low-power devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These protocols are limited to the transmission of small smart packets, typically 100 bytes or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Devices that implement LPWAN protocols are composed of extremely low-cost components, typically costing less than a few dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Within and outside their domains, these devices are designed to provide good coverage and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 7) LONG RANGE WIDE AREA NETWORK (LORAWAN): It is a physical layer communication protocol that uses very less power and has a battery life of up to ten years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' LoRaWAN specializes in M2M, industrial and smart city applications, which require long-range communication, ranging from 2 to 5 km in urban areas and up to 15km in suburban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The process of communication through large networks, which contain billions of intelligent devices, also promotes the data rates of this protocol in the complete duplex wireless medium, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3 to 50Kbps[149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 29 TABLE X: COMPARISON OF LPWAN PROTOCOLS Protocols LoRaWAN SigFox NB-IoT Topology Star of stars, Mesh Star Star Modulation CSS28 BPSK QPSK Frequency Unlicensed ISM bands: 868 MHz (Europe), 915 MHz (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' America), 433 MHz (Asia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Unlicensed ISM bands: 868 MHz (Europe), 915MHz (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' America), 433 MHz (Asia) Licensed LTE frequency bands Transmission range 5 km (urban), 20 km (rural) 10 km (urban), 40 km (rural) 1 km (urban),10 km (rural) Data rate 250bps-50kbps 100bps 200kbps No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' of nodes 1000 100 5500 Power/ Current consumption 50mW (3-50)µA 500mW-4W/ (19-49)mA Handover No end devices will be associated with a single base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' No end devices will be associated with a single base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' End devices are associated with a single base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Applications Smart city Smart logistical and transportation Industrial application Real time monitoring Video surveillance Smart farming Status monitoring Smart building Asset tracking and logistics Electric metering Manufacturing Automation Smart city 8) NB-IOT: It is a narrowband radio technology that was developed and standardized by the 3GPP to support IoT applications with low data rates and high complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It proposes a new radio access method based on LTE standards but with less capabilities in order to lower the power consumption of IoT devices with limited resources[150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 9) SIGFOX: It is a technology of narrowband and ultra narrowband for connecting a large number of power-controlled devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In order to operate, the protocol must operate on a frequency band of 868MHz, where the spectrum is split into 400 channels of 100Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Rural areas can receive signals from IoT devices that can transmit up to 140 packets per day at a data rate of up to 100bps, and 28 CSS: Chirp Spread Spectrum urban areas can receive signals that can reach distances of (30-50)km in rural areas and (3-10)km in urban areas[149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The Table X compares the characteristics of the LPWAN protocols discussed in this section[149]: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' PERCEPTION LAYER The major goal of this layer is to feel the physical characteristics of the entities that surround us and within the dominating IoT network, which relies on sensing technologies like RFID, WSN, and GPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It’s also in charge of translating sensory data into digital signals that may be transmitted via a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Indeed, embedded intelligence and nanotechnology play an important role in this layer, as they improve the processing capabilities of any object by inserting small chips or microcontrollers into everyday smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This layer consists of all user end devices (smart devices, wearables, sensors, actuators etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=') that are connected to the IoT and cellular network and capable of accessing and transmitting tactile sensations over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Additionally, there are some fundamental attributes that are an integral part of the end devices, as detailed in the Table XI [151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For the system to enable an intelligent and reconfigurable touch based system, both the master domain and the controller domain, need to have a bilateral connectivity with the intelligent RAN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is further governed by the AI/ML algorithms, responsible for intelligent resource allocation and data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence motivation for the proposed model lies entirely in the fact that to incorporate an entirely intelligent system in the 6G and IoT framework requires the AI/ML technology implemented on the B5G network slicing and its application, TI as the building blocks of the touch system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TABLE XI: FUNDAMENTAL ATTRIBUTES OF CONNECTED USER END DEVICES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='No Attributes Description 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Identification (ID) Each of the connected objects is assigned an ID based on conventional parameters such as universal product code, Media Access Control (MAC-ID), and IPv6 ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Meta information It contains device model, ID, revision, hardware, serial number, and manufacture data for each IoT device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=" Security controls Allows the device owner to employ security settings to limit the types of devices that can connect to the user's device." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Relationship management It enables each IoT device to establish, update, and terminate relationships with other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Service composition This component allows smart objects to interact, with the goal of providing the optimum integrated services to users and is also in charge of processing the data obtained from different objects to provide user with best solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 30 One of the most promising aspects of this proposal is that it is based on the assumption that 6G applications will have a high data rate and latency requirement of less than 1ms in the majority of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The application of the TI based intelligent systems are still in the nascent stage and therefore require a well defined technology to deploy the touch enabled technology in the next generation networks, combining the network slicing with the TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus an IoT enabled interface acts as a backbone of the proposed reconfigurable and intelligent tactile based touch communication wireless system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Summarizing the important requisites for the proposed touch based technology can therefore be listed below as: 1) An interfacing architecture with the existing B5G/IoT framework is required that can link the present B5G network with the next generation 6G network through IoT based intelligent and sensory devices and sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) The network slicing and TI prove to be a strong backbone in implementing the given framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Also the intelligence may be incorporated in the system using the ML/AI based algorithms, to increase the system computational capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) The 6G applications are largely based on the AR/VR/MR/XR, having large power consumption and real time simulation and therefore need proper energy efficiency and power optimization techniques for its effective implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It can therefore prove to be an important future aspect that needs to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) Also the minimum infrastructure costs, with an energy efficient performance are important parameters that need to be considered while implementing the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence it can be concluded by saying that, all the above enlisted parameters, will eventually pave a way towards the establishment of intelligent and a configurable touch enabled system, interfacing with the B5G/6G Wireless Communication Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' We have further proposed an end to end touch interfacing architecture to be implemented in the real time B5G/IoT based wireless communication network as described ahead in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The IoT infrastructure includes physical objects integrated in the WCN, which are designed to provide intelligent in- house service to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus the IoT system comprises of five layers as the perception, connectivity, network, transport and the application layers starting from the device end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The network layer in turn includes the connectivity with the edge/fog computing as well, as the data storage functionality within itself as given in Fig 9, altogether forming the middleware component of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The proposed Touch based intelligent and IoT configured system may therefore be realized as an end to end architecture in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This end to end architecture is therefore to be realized in three phases which include: 1) The TI enabled real time touch communication network establishments at the transmitting end complete with its bilateral feedback system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) Similar TI enabled real time touch reception system at the receiver end that is operated by the remote industrial machinery and robotic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) Lastly it requires an intelligent interfacing algorithm to simultaneously operate both the ends and a suitable slicing mechanism to fulfill almost all the URLLC and IoT enabled intelligent Touch based tactile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' END TO END TOUCH SYSTEM ARCHITECTURAL MODEL The E2E system architectural model of the proposed intelligent touch technology has been represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The real time applications at the user ends satisfying the desired output at the destination may range from AR/VR based transmission and communication system, real time online gaming, remote medical service accessibility like tele-surgery and remote health consultation and remote industrial operation involving the major role of robots and automatic machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The system model has been divided into three components and therefore may be analyzed in three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These being categorized as: 1) PHASE1: The bilateral communication at the transmitter comprising of the tactile based user devices like the robotic arm, AR/VR gear and gesture based haptic devices, real time online gaming console and many end, more, interacting with the BS, the local server (LS), transmitting the data to the middleware through the gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) PHASE2: It represents the bilateral communication at the receiver end, accomplishing the intelligent, touch based applications ranging from intelligent healthcare facilities like remote surgery, remote industrial operations, real time online classrooms and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) PHASE3: It acts as the mediating controller, connecting the above phases and is therefore responsible for implementing intelligence in the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The AI/ML algorithms are therefore routed in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The aforementioned phases may be separately analyzed as the transmitter end, the receiver end and the processing end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TOUCH BASED TRANSMITTER SYSTEM The transmitter end of this end to end architecture of the intelligent touch interfaced system may be represented as phase 1, comprising of the tactile user devices like the robotic arm, augmented or virtual reality gear, gesture operated tactile devices, real time online gaming modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='11 presents the touch enabled flow process at the transmitter end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The process may begin with the abovementioned devices at the user end requesting for diverse high data rate and IoT based URLLC applications IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 31 FIGURE 10: Touch interfaced end to end architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' varying from the remote medical consultancy or tele- surgical operation to the assembling and modeling of the machinery pertaining to the remote industrial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The user may request for a particular service by the means of a touch sensor or an AR/VR device, virtually furnishing the data towards the server, to be routed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The further the convenience may be at the individual viewpoint, making use of tactile enabled robotic system for household function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In addition it may have a future application as in virtual e-commerce or virtualized or holographic shopping where the customer may virtually access or try the product by tactile sensing and touch enabled interface between the user end and the linking server forming a bilateral communication feedback system at the user/customer end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The TI based applications place a request to the local server in order to check for the channel availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The request is forwarded when the channel is available and if the channel appears to be busy, the entire process starts all over again from the first step, where the user must ask for a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Following the channel access granted to the user, and prior to sending the data it tries to establish connection with the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This is followed by the handshaking request-response procedure between the channel and server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The server gives acknowledgement on the channel availability and thus firmly establishes the connection between the customer and the server, virtually accessing, forwarding and routing the data further to the gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Here ‘A’ represents the data to be forwarded to the gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A similar kind of procedure is conceded at the receiver end explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This middleware is to be positioned between the edge and the cloud computing layers, successfully participating in the network slicing and providing the requisite virtual platform for supporting different tactile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The slices may be controlled, scheduled and allocated using different ML/DL based algorithms, successfully implementing the proposed touch based network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TOUCH BASED RECEIVER SYSTEM The flow process at the receiver end of the touch interfaced system is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The point ‘A’ which represents the user end devices at the transmitter end, further routes the information signal in the channel via BS, router and gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The gateway connects the transmitter with the middleware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The process therefore briefly illustrates the role of the middleware infrastructure with its connectivity at both the transmitter and receiver end devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessAINL Algorithm GI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='- Feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='- INTELLIGENCE U1 (-Feedback DELIVER PROCESS Remote Server INPUT Receiver Transmitter U2 Middle ware BS Smartclassroom Remote Industry U3 LS U1-User1(Robotic Arm U2-User2-ARVR GI U3-HapticGesturebasedCommunication ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Tele-surgery Un-Online real ime Gaming BS-Base Station LS-Local Server GW-Gateway Intlligence-AINL AlgorithmThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='CHECK CHANNEL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='AVAILABILITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='START ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Touch Technology Flowchart ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='REQUEST FOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SERVICE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='TELE- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SURGERY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='VIRTUAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SHOPPING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='REMOTE INDUSTRIAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='ASSEMBLING/MODELING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='ROBOTIC USE AT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='DOMESTIC LEVEL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='BUSY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='REQUEST TO THE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SERVER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='AVAILABLE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SEND DATA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='HANDSHAKING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='AT SERVER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='AVAILABLE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='NO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='YES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='ACKNOWLEDGEMENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='RECEIVED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='CONNECTION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='ESTABLISHED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SERVER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='FIGURE 11: Touch enabled flow process at the transmitter end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This middleware exists at the junction of edge and cloud computing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Middleware performs the virtualization process in conjunction with network slice abstraction and allocation, fragmenting the existing physical network into multiple virtual networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The techniques and algorithms required for the proficient implementation of the slices bearing n number of virtual networks may perhaps be applied in the middleware layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The middleware further connects to the LS on the receiver side and establishes connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The process similar to that in the transmitter and is followed with the middleware forwarding the information data to the server after establishing connection with the server via handshaking mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The LS at the receiver end in turn indicates whether or not the channel is available for signal transmission by means of request and acknowledgement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The signal is processed as the output after the channel availability is declared at the LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The receiving end of the proposed system may perhaps be represented by point ‘B’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This point ‘B’ may further endow the requested TI enabled touch based applications at the destination end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence the user/customer successfully gets the desired output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It may be the successful tele-surgical operation or it may the remote industrial operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It may also consist of the successful modeling and assembling of robotic and machinery components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Additional applications vary from the smart and virtual classroom to the virtual holographic IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 33 A USER END (Tx) ROUTER BS GATEWAY GATEWAY MIDDLEWARE REQUEST TO THE SERVER SEND DATA SERVER HANDSHAKING AT SERVER AVAILABLE CONNECTION ESTABLISHED ACKNOWLEDGEMENT RECEIVED NO B TELE- SURGERY REMOTE INDUSTRIAL ASSEMBLING/MODELING VIRTUAL SHOPPING ROBOTIC USE AT DOMESTIC LEVEL DESTINATION END GATEWAY FIGURE 12: Touch enabled flow process at the destination end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' shopping and e-commerce, with successfully accessing the desired product without actually having to lay a hand on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It is possible to virtually access and analyze the product through holographic imagery and the transmission and reception of the information may be established by implementing the TI based touch configured system, which will prove to be an impacting factor in 6G system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TOUCH BASED MIDDLEWARE SYSTEM The middleware system is an integral and the most important component of the proposed intelligent touch configured system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It therefore helps in integrating and configuring both the transmitter and the receiving user/application ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence it is imperative that both its connecting ends have the same configuration and dimensionality for the system to act smoothly and without any undesired latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The LS at the user end routes the requested information via BS, router and gateway to the middleware infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' At this juncture the information requested is virtualized in numerous network functions enclosed in several network slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This system too comprises of a layered infrastructure, and we have considered a 4-layered centralized middleware network in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These layers may be categorized as: input layer, processing layer, delivery layer and the output service layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The input layer forwards the input data from the transmitter end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' the processing layer helps analyze the data from the transmitter end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' the processing layer helps analyze the data using the virtualization and network slicing IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 34 A B TRANSMITTER CENTRALIZED NETWORK RECEIVER LAYER OPTIMIZER PRIORITY POWER SECURITY Layer Optimizing Algorithm Priority Algorithm Power Usage Algorithm Security Algorithm Input Process Deliver Output Services (application) FIGURE 13: Middleware component system process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The delivery layer categorizes the slices as per their use cases and data rate and bandwidth requirements, whereas the output service layer arranges the final application slices to be delivered at the destination end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore the proposed requisite intelligence in the system is to be established via this middleware infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' There is a bilateral feedback system at both the transmitter and receiver end which helps in transmitting the feedback and incorporating the intelligent techniques/algorithms at both the connecting ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The input to this system is all the way through the transmitter user end represented by ‘A’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Consequently the input data fed to the middleware may be arranged in a sequence considered and governed by factors like optimization, priority, power consumption and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The next subsection further elaborates the architecture and functioning of the middleware to be used in the proposed touch based system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence at the output end these factors may be implemented as their corresponding techniques and algorithms like the layer optimizing algorithm, the priority based algorithm, the power usage algorithm and the security based algorithm, so as to obtain the desired output result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The AI/ML/DL algorithmic implementation of the above listed factors, in wireless IoT and B5G/6G systems has been a research interest of many scientists and researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus for the real time implementation of the intelligent touch enabled system we call for an interfacing architecture that integrates the existing B5G technology with the 6G technological interface, enabling the TI enabled touch connectivity through intelligent network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Consequently the subsequent sections describe the interfacing architecture of the present scenario with the next generation system along with its future scope and applications towards the implementation of the intelligent touch system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TOUCH TECHNOLOGY IOT MIDDLEWARE The IoT is a vast network of connected smart devices that aim to make the surrounding environment intelligent and autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus most vendors do not care about compatibility of their products with other competitive brands, which is one of the major challenges that IoT paradigms face in machine to machine communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This problem has been addressed in several ways, one of which is the enforcement of universal standards, which is extremely difficult to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Another approach has been suggested, which is the implementation of middleware software to facilitate communication between these devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Middleware can be defined as the software that offers interoperability between incompatible applications and devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' as well as shielding customers from the smart object’s complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence, it serves as a software bridge between applications and things, allowing IoT systems to communicate and collaborate more effectively with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' There are a plethora of middleware solutions available, whether proprietary or open source, that are provided by various companies, with the majority of these solutions being very similar to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' However, there are no benchmarks, performance indicators, or performance measurements that allow us to compare various systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Following our examination of several articles[152]–[154], we have come to the following conclusions about some of the issues faced by IoT middleware: 1) ABSTRACTION AND INTEROPERABILITY: IoT middleware aids in allowing various smart devices to interface easily with each other in order to facilitate collaboration and data exchange among heterogeneous devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) DEVICE DISCOVERY AND MANAGEMENT: This attribute allows IoT devices and services to be located in their network domain where the IoT environment infrastructure is primarily dynamic because all newly joined devices must announce their existence and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) SCALABILITY: The IoT middleware must be scalable and must provide APIs in order to list all IoT devices, their capabilities, and their services, among other things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) DEVICE CATEGORIZATION: APIs must also allow users to categorize devices based on capabilities, manage devices based on remaining energy, report IoT device problems to users, and perform problem load balancing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) BIG DATA AND ANALYTICS: Because of the fragile nature of wireless sensor networks, part of the detected data may be incomplete, requiring the middleware to take this into account and extrapolate incomplete data using a suitable machine learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6) PRIVACY: Since this majority of data coming from IoT applications and services is related to human daily life, security and privacy issues must be considered when transferring and processing it, necessitating the development of middleware that addresses these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 7) CLOUD SERVICES: Cloud computing is the vital layer of any IoT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The data captured through sensors will be stored and analyzed in a IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 35 centralized cloud, and, as a result, IoT middleware should run well in various types of clouds as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 8) CONTEXT DETECTION: Ambient data collection applications and real-time reactive applications are the two types of IoT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In the first, sensors collect data that will be processed offline later to obtain reasonable information that will be used for similar scenarios in the future, while in the second, systems must make a real-time decision based on the sensed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' FIGURE 14: Cloud and IoT enabling technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ARCHITECTURE OF TOUCH BASED IOT MIDDLEWARE Following are types of IoT middleware architectures that are currently available, which are classified based on the services that they provide[155]: 1) SERVICE ORIENTED ARCHITECTURE (SOA) OR SERVICE BASED SOLUTION: Users and developers are given the ability to employ or add different types of IoT devices that can be used as services in the service oriented middleware architecture (SOA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Three layers make up SOA architecture: the physical layer, which contains actuators and sensors, the virtualized layer, which contains cloud and infrastructure servers that are responsible for performing various computational operations, and the Applications layer, which contains all services and utilities as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Access control, storage management, and event management are just a few of the general intermediate layer services accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' FIGURE 15: Service oriented IoT Middleware architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SOA is a powerful middleware that may be deployed on nodes that communicate with cloud servers or on a powerful gateway that sits between the cloud levels and the IoT layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, this sort of middleware is incompatible with resource-constrained devices and does not allow for device-to-device communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The most widely used service-based middleware is discussed here and summarized in the Table XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' a) Link Smart (Hydra): A web service platform that intends to reduce the heterogeneity of different devices and entities in the IoT ecosystem, as well as control all types of smart devices regardless of their communication protocols, such as Zigbee, RFID, Bluetooth, Wi-Fi, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This middleware is unique and it allows IoT devices to be used as services by embedding the required services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Health care, agriculture, and home automation are few of the IoT applications that can be managed with Link Smart [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' b) Kaa: It is an open-source platform that enables the creation of IoT solutions and is administered by Cyber vision Inc and Kaa IoT technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' By utilizing a web-based GUI built on the Apache platform, it is possible to create data delivery schemes, support multi-tenancy on servers, and generate endpoint software development kits (SDK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Kaa enables direct or indirect communication with endpoint devices, while encrypting their data using AES29 and RSA 30encryption methods[154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='29 AES: Advanced Encryption Standard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='30 RSA: Rivest Shamir Adleman ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='IEEEAccessService-Based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Middleware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Actor-Based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Cloud Based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Middleware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Mas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SECaas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Middleware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Cas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Event Based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='DBaas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Midleware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Midleware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Naas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Cloud Services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Configurations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Paas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='HTTP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Intemet/Cloud of Things ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='MQTT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='laas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Enabling Technologies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Saas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='IoT/CoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='AMQP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Protocols ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='XMPP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Arduino ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='REST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='COAP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Computational ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='RPL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Hardware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Ciond Senices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Raspbery PI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Soas: SfwareamaSenice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Sensing&Communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Pans:PlafomasaSenice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='NFC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Technologies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='faofastuctesSenice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart Phones ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='OS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Nas:NetorkasaSenice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='CaaCotaiersaService ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Mans:MoritringasaSenice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='RFID Tags ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SECans:SecwriysaSenice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='TEEE802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='4 DBasS:Datrbase asaSenice 6LowPAN ZigBeeApplication 个个个 Application&Services(browserbasedapp,smart health,smartcities) 88 nn 1 Virtualization Accesscontrol Queuing manager Virtual sensor Webservicesinterface manager 品 Eventprocessing Storage engine Cloud Service IoT Devices Sensorsandembeddeddevices(wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' smartwatch,cameraetcThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 36 TABLE XII COMPARISON OF COMMONLY USED SERVICE-BASED MIDDLEWARE ATTRIBUTES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Service Based Middleware Link smart (Hydra) KAA GSN Thing speak Aura Deployment type PaaS, SaaS IaaS PaaS PaaS IaaS, SaaS Network connectivity HTTP, REST,MQTT MQTT, CoAP HTTP MQTT, REST API MQTT, HTML Data format supported JSON REST,JSON,API 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Esper EPL SQL queries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='MATLAB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Aura library ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Device abstraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Stream mining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Consoles and mobile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='devices and Smart TVs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Service discovery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='REST API ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='MQTT with Kaa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='protocol V1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='REST HTTP query,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Sbt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='13+, Java JDK 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='7, Scala 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='11 Environment management Security and Privacy Encryption Authorization Authentication Encryption Authentication Access control mode Encryption Authentication Authorization c) Global Sensor Network (GSN): Its objective is to provide a standardized platform that enables adaptable deployments, sharing, and integration of heterogeneous Internet of Things (IoT) objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This platform is built to meet the requirements of physical and virtual sensors and actuators, whether they are connected via a wired connection or wirelessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' GSN is a Java platform that can be placed on IoT cloud or servers, and it allows a series of wrappers to feed the system with collected line data that is later processed using XML specification files[157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' d) Thing Speak IoT: It is a MATLAB-developed analytic open-source platform service that allows people and things to communicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In order to collect, visualize, and analyze real-time data from devices and sensors, Thing Speak provides users with tools that allow them to use the HTTP protocol over the internet to store and retrieve data from them[158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' e) Aura: This middleware facilitates the development of pervasive mobile IoT applications by abstracting device differences and allowing them to communicate freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Aura makes an effort to optimize the screen backlight and CPU in order to improve performance while also reducing power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In engaging with events, Aura uses two concepts: proactive and reactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In a proactive concept, system layers respond immediately to the higher layer, whereas in a reactive concept, all layers adapt their resource and performance based on demand[159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) CLOUD BASED MIDDLEWARE: User options are limited in the cloud-based middleware framework due to the limited number and variety of smart devices connected to IoT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In addition, because different use cases can be programmed and then determined in advance, the sensed data can be collected and interpreted with relative ease and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The operational component of this middleware is restricted by the resources available in the cloud computing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Although IoT functions have a general presence in the IT architecture, like storage systems and computation engines, these functions are represented and controlled by APIs where IoT services are accessed by either cloud-based RESTful APIs or by vendor-provided applications as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The most widely used cloud-based IoT middleware is listed ahead and summarized the Table XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' FIGURE 16: Cloud based IoT Middleware architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessVendor provided Web App www Cloud System APIs 8 REST Vendor provided Mobile App API management Cloud Service RESTful API IoT Devices Sensors and embedded devices(wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' smartwatch,camera etc)This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 37 TABLE XIII: COMPARISON OF COMMONLY USED CLOUD-BASED MIDDLEWARE ATTRIBUTES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Cloud Based Middleware AWS IoT Azure IoT Hub IBM Watson IoT Google Cloud IoT Oracle IoT Deployment type IaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' PaaS IaaS IaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' PaaS IaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' PaaS PaaS Interoperability Web services Azure IoT SDK MQTT REST APIs Oracle Service bus Network connectivity MQTT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' HTTP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Web- socket HTTP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AMQP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' MQTT over Web-socket MQTT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' HTTP MQTT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' HTTP MQTT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' HTTPs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Real-time analytics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='AI/ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Event reporting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Visualization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Real-time analytics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Event reporting ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Amazon Route 53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Netflix eureka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Azure container services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='with Kubernetes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Eureka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Consul ' metadata={'source': 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Privacy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Encryption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authorization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Auditing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authorization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Encryption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authorization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authorization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='a) AWS IoT: In order to manage cloud services,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' such as allowing millions of connected devices to communicate securely and easily with other devices and cloud applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Amazon developed this platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AWS IoT enables customers to build IoT applications that collect, process, analyze, and sense data in order to make appropriate decisions without the need for infrastructure management by utilizing AWS services such as Amazon Kinesis and Amazon Cloud Watch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AWS IoT customers can also always monitor all devices that communicate with their applications[160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' b) Azure IoT Hub: It is a central platform developed by Microsoft for managing bidirectional communication between IoT applications and the devices to which they are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Because of Azure’s extensive capabilities, it enables clients to develop full-featured, scalable IoT solutions that provide secure and reliable communication between the hosted cloud and connected devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Azure is a Microsoft product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' When it comes to controlling IoT connected devices, Azure IoT Hub supports a variety of messaging patterns, including request- response, file upload from devices, and device to cloud telemetry[161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' c) IBM Watson IoT: This platform, which is built on top of the IBM Cloud, allows users to connect and control a variety of IoT appliances, sensors, industries, and home appliances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Using IBM Watson, its clients can create and manage their own IoT applications and appliances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' They can also extract KPIs from their data and use them to control their tools and applications, as well as process their collected data using historical and real-time analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IBM Watson also offers a block chain service[162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' d) Google Cloud IoT: Essentially, it is a fully managed device that is composed of a set of tools that provide a comprehensive solution for secure and easily connecting and processing of data generated, whether they are located in the cloud or at the network edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Google cloud IoT aspires to develop models capable of optimizing client business, anticipating problems, and increasing operational efficiency[154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' e) Oracle IoT: A cloud-based service platform that lets users create a real-time IoT solution to be linked with enterprise applications while leveraging rigorous security cloud capabilities and cutting-edge edge analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Furthermore, it integrates IoT data quickly and easily into customer business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Oracle IoT enables clients to connect their devices to the cloud, which will aid them in making critical strategies and decisions in their businesses[163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) ACTOR BASED MIDDLEWARE FRAMEWORK: In terms of functionality, it is a lightweight middleware that can be implemented at the sensory, gateway, and cloud computing layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Unlike other middleware, the computational operations of this middleware are distributed across multiple layers, including the sensory layer and mobile access layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The sensory swarm is the outermost layer, made up of sensors and actuators, while the mobile access layer, made up of gateways, smart phones, Raspberry Pi, and laptops, is the intermediate layer, and the cloud is the innermost layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The middleware which is also the actor host is designed to be lightweight and can be IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 38 TABLE XIV: COMPARISON OF COMMONLY USED ACTOR-BASED MIDDLEWARE ATTRIBUTES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Actor Based Middleware Calvin NODE-RED Ptolemy Accessor Host Akka Deployment type IaaS PaaS, SaaS Interoperability Actor model (event driven) Web services Accessor Aggregate programming Network connectivity MQTT HTTP, MQTT HTTP,HTML HTTP,HTML Data format supported JSON JSON JSON, XML JSON Programming Language C, Python JavaScript, Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='js C, C++, JavaScript Java, Scala Session persistence Distributed hash table MQTT Local file system Akka persistence library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Stream processing Data flow processing Node-red-contrib-cep Discrete event director Akka hop, Akka stream library and Apache Flink Applications Distributed applications Runtime applications Connecting to IoT Connecting and binding databases Collecting and storing IoT data in event driven applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Finite state machine applications Web applications Real-time streaming Real time applications Building powerful and concurrent Web applications Service discovery Calvin control APIs Bonjour /Avahi Discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='js Discovery function Akka discovery method Kubernetes API AWS Consul Marathon API Security and Privacy Authorization Authentication Authentication Encryption Authentication Encryption Authentication Authorization Encryption embedded in any layer of the application stack as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A storage device, for example, may not be included in the actor-based middleware used on a smart watch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' If the storage device is provided by an actor, it can be downloaded from a cloud repository whenever it is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The most widely used actor-based middleware is discussed here and summarized in the Table XIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' a) Calvin: It is an open source IoT platform developed by Ericsson to be used on energy-constrained smart devices because it offers a portable and lightweight unified programming architecture with input and output ports that define the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Furthermore Calvin can also be used at the edge of IoT ecosystems to reduce long-distance connections, lowering latency and reducing power consumption of IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Their main advantage is its ability to move from one environment to another[164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' b) Node-RED: It is an open source IoT platform developed by IBM and built on the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='js31 programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Because of its small footprint, this platform can be used at the edge of an IoT network, while on the server side, a JavaScript platform with an event-driven module and non- blocking I/O is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It allows users to build IoT applications by dragging and dropping connected blocks that represent IoT components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The platform drawbacks include the fact that it does not support service discovery and only allows for password authentication for security[165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 31 Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='js: It is an open source, cross platform and run time environment for executing JavaScript code on the server side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=" FIGURE 17: Actor based IoT Middleware architecture IEEEAccessLayer ccess Application &Services browserbasedapp,smart health,smart cities Mobile 888 88 smatcty Ptolemy's Swarmlet/Node-Red/Calyin host 84 Cloud Service Actor Host system 04 Sensorsandembeddeddevices(wearables smartwatch." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='camera etcThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': 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+page_content='Publisher-hosting broker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Java management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='extension ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Object oriented API ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Fiware Content broker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Service discovery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Service agents ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='REST ful API ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Security and Privacy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Encryption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Auditing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authorization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Encryption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Authorization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='c) Ptolemy Accessor Host: Professor Edward Lee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='created this open source platform in 1996 to design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' simulate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' and model embedded and real-time devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The underlying concept of this platform is that an IoT system is constructed from software components that interact and communicate with one another through messages sent through interconnected ports on a computer network[166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' c) Akka: It is a collection of open source libraries and a free actor-based platform that was created to allow developers to create distributed and run-time applications in either the Java or Scala programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It enables users to meet business requirements without having to write large low-level code, resulting in high performance, fault tolerance, and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Akka also allows multi-threading, isolates communication between applications and their devices, and provides a clustered architecture with excellent availability[167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) EVENT BASED MIDDLEWARE FRAMEWORK: In order to improve the development of distributed systems, middleware that supports the publish/subscribe paradigm is being developed and implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' According to this definition, this paradigm is a communication infrastructure that aims to provide clients with general- purpose services by assisting them in dealing with the heterogeneity and complexity of large-scale and distributed environments as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The event-based middleware hides some of the complexity of distributed applications from the programmer, which will make it easier to develop and program much different functionality in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The most significant differences between these architectures are their openness to supporting new IoT device types, the types of middleware services or computational units they support, and the locations where the IoT middleware can be embedded or deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' FIGURE 18: Event based IoT Middleware architecture IEEEAccessPublish Publish— Notify ←Subscribe-Subscriber Publish Publish→> Notify→> ←Subscribe-Subscriber Publish Publish Notify→> ←Subscribe-S Subscriber Event Service Broker NetworkThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 40 IoT middleware based on SOA is implemented on servers and in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Because the middleware can be implemented in all tiers and IoT devices can perform computation where it is most advantageous, actor-based delivers the best latency and scalability for large scale linked IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' While these architectures provide some level of security and privacy, cloud-based architecture requires users to place their trust in the cloud provider to protect the privacy and integrity of their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Since a middleware cannot be implemented within the physical device and the data exchanged between physical devices and the middleware can be compromised, there is a weak security link between the physical devices and the middleware in both service and cloud-based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' To ensure QoS, the middleware must have a service discovery component that allows new services to be made accessible on demand and failed services to be dynamically replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The most widely used event-based middleware is discussed below and summarized in the Table XV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' a) Hermes: An event-based, scalable middleware designed to make distributed and large-scale applications easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' To address big size and dynamic situations, Hermes provides self-managed event brokers based on a P2P routing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It features an adaptive solution that takes into account failed event-broker events and routing stacks, all while maintaining the event-broker network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hermes middleware has two versions, both of which share the majority of the codebase and are intended for use in distributed and large-scale systems as well as in communication and implementation among event brokers[155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' b) Gryphon: It is a highly scalable publish/subscribe middleware designed to distribute large amounts of real-time data over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gryphon is a Java- based interface that enables the development of web applications and the creation of a robust, redundant, publish/subscribe, and content-based multi-broker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This middleware is completely secured and offers simple yet efficient routing and event handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It also uses a messaging information flow paradigm (BKS+99) to specify communication between publisher and subscriber[168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' c) Rebeca: This middleware is based on publish/subscribe technology and focuses on the design of efficient routing algorithms and use of professional software engineering methodologies to implement large-scale business applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' To avoid network flooding, Rebeca employs advanced routing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It incorporates interoperability and subscription merging capabilities into its services to facilitate location mobility and reduce the size of the routing table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The event scope function abstracts away the implementation details of a service, such as transmission policies, security, data transmission methods, external and internal interfaces, and notification representation[169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' d) Fi WARE: It enables distributed IoT devices and applications to communicate in an efficient, flexible, secure, and scalable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It was created to facilitate the control and monitoring of a variety of IoT applications, including logistics, retail, and smart cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This platform is comprised of numerous components, including APIs, reusable modules, and massive code, all of which enable a user to create an IoT application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A collection of sensed data from IoT sensors is captured via REST API and later sent to a dedicated server called the broker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' FiWare has developed an API for querying and storing various IoT contents, which enables any application registered as a content consumer to retrieve the necessary data from the broker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This platform has a component called an adapter that is in charge of transferring a certain type of material to subscriber applications[170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' INTERFACING ARCHITECTURE WITH THE EXISTING NETWORK The architecture in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='19 represents the real time interfacing of the existing 5G/B5G network scenario with the next generation 6G all the way through the network slicing phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The slicing aspect comprises of the intelligent cloud slicing, the RAN slicing and the application slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The cloud slicing intends to facilitate the cloud computing and storage access to the edge devices and users via different slices catering to different users simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The RAN slicing is beneficial in connecting and routing the end users and devices to their receiving ends, accordingly making use of techniques like spectrum sharing, beam forming, cognitive antenna and radio transmission, tactile support and transmission system through different slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' All the applications accessed at the user end owing to the slicing phenomenon are furnished by the application slicing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The applications accessed may range from robotics, tactile, haptic and touch based communications to the real time online gaming using the AR/VR/XR with the UHD video streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Other applications may range from the remote industrial application (IIoT), tele-medicine and tele-surgery, smart classroom with UHD video streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The autonomous vehicular and UAV system may be operated by application slice providing the IoT connectivity along with the URLLC of 1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The complete autonomy of the system and devices connected to the internet with the minimum latency and high data rate facilitate an intelligent and autonomous functioning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The smart cities comprising of the smart homes, smart traffic monitoring systems and the rest instill another level of intelligence in the existing wireless network system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The subsequent section highlights some of the future aspects concerning the application of intelligent technologies in the wireless 6G communication system followed by some of the collaborative and ongoing projects IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 41 FIGURE 19: Intelligent interfacing architecture of the existing communication system with the next generation technologies (B5G/6G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='IEEEAccessCLOUD SERVICES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='INTELLIGENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='CLOUD SLICING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart airways ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smartraihw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SMARTHOME ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='WEARABLE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='EDGEDEVICES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='GREENCOMIUNICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Car securi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Smart parking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='RAN SLICING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='回 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='PACKETDATAGATEWAY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='TACTILE SUPPORI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='NE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='USSTOP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='AVNETWORE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SmartParking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='AUTONOMOUSVEHICLE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='AR/VR/XF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='BASESTATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='BO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='IMA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='ROBOTICS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='APPLICATION SLICING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='TELESURGERY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='SMARTCITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='(lloT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='NAN( Neighborhood Area Network) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='5" Generation Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Next Generation Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Technologies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='ScenarioThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 42 FIGURE 20: Touch induced human to robotic interactions with emotional intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' implemented in the B5G/IoT and 6G wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' APPLICABLE USE CASES OF INTELLIGENT TOUCH TECHNOLOGY This section explores the touch technology framework and future application elements, with a focus on tactile- based haptic communication integrated with learning approaches to enable efficient network transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The touch-enabled tele-diagnosis, robotic interaction, and haptic sensation-based shopping experience are just a few of the potential research aspect areas to look into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' CASE 1- ROBOTIC INTERACTION Another application of touch technology could be tactile- based robotic interaction between humans and robots, with the bot system being able to recognize and respond to human emotions [171], [172].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Emotions are interfaced into the machinery by creating a database in the system using AI/ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The following are some examples of such applications: 1) ROBOTIC PETS: It entails the integration of AI algorithms and functionality into robotic systems, as well as the introduction of emotional intelligence (EI) into them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The robotic pet dogs [173] is a dog-shaped robot capable of learning and detecting human gestures, face and eye movement, and responding appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' By introducing a touch-based interface into the system, these robots are capable of reacting to human touch feelings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Those who are elderly or mentally ill and are looking for companionship to relieve their loneliness will find these useful, as they will aid in their emotional development as a result of the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) CLEANING/DOMESTIC ROBOTS: These robotic systems are capable of following user instructions and performing household chores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' These might be driven by popular AI-based home automation systems like Alexa to provide complete automation, allowing them to hear and act on the user’s vocal commands rather than their restricted instruction library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The introduction of touch-based sensation/actuation into the framework enables them to automatically respond and act in response to the sensory simulations provided by the environment, thereby saving time that would otherwise be spent on user instructions, programming, and the interface itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In this way, the touch-enabled robotic-human interaction system with emotional intelligence is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' CASE 2-AR/VR BASED ENTERTAINMENT/ SHOPPING EXPERIENCE AR and VR technology are becoming the two important innovation factors that promote technological progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Here the AR glasses provide a realistic perspective for viewing augmented reality content, while VR headsets provide an immersive sensory experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore AR and VR technology are becoming the two important innovation factors that promote technological progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The virtual reality headsets provide users with an immersive sensory experience by allowing them to view AR content from a realistic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' At the same time, AR/VR technology has opened up a slew of new advertising and marketing possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessTactile support Neaning/DomesticRobots engine gNB Tactilesupport engine RoboticPets gNB Serving gateway Packet data gateway TactileCommunication System Emotional Intelligence Sensation USER Tactile/Touch Human-RobotInteraction FeedbackSystemThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 43 FIGURE 21: Touch technology enabled virtualized shopping experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Around 75 percent of business owners anticipate adopting AR/VR technology in the next two years, and global AR/VR market spending will be more than double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Moreover, AR/VR users are becoming more common in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As consumers increasingly rely on e-commerce and online shopping, there is a high demand for augmented reality content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' WIMI plays a significant role in the AR shopping market and also uses ‘AI+AR’ to support other industries such as advertising, entertainment, and e- commerce[174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Immersive AR/VR solutions help bridge the gap between online and offline purchasing for consumers, and more and more people are seeing AR/VR as a valuable tool for discovering products and getting brand services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AR/VR can currently improve the shopping experience through virtualization, with technology such as a virtual mirror assisting in a virtual garment trial before purchase being implemented[175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence it acts as a future aspect that complements the integration of tactile/haptic-based intelligence with AR/VR technologies to further enrich our virtual shopping experience using touch technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In this regard, the illustration below shows how the proposed methodology can be integrated with existing AR/VR technology to further enhance the real-time user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='21 illustration above depicts a graphical representation of the virtualized shopping experience provided to users via intelligent touch technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' CASE 3-TACTILE AND HAPTIC SENSATION BASED TELEDIAGNOSIS FOR CONTACT FREE COVID-19 CASES EXAMINATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Tactile internet-enabled wireless communication systems have been integrated into the conventional healthcare system to create the smart healthcare system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Tactile- enabled healthcare systems, such as telemedicine, tele- surgery, and remote tele-diagnosis, could all benefit from the suggested touch technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In today’s technological environment, let us consider a practical scenario in which a medical surgeon is working from a smart surface or console that is connected to a telecommunications network in Chennai, India, while the patient is lying on an operation table at Fortis Hospital in New Delhi thousands of miles away[176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The medical surgeon can remotely control the movement of a multi-armed surgical robot to perform gall bladder surgery on the patient by utilizing the smart surface and other communication technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Through the use of a tactile communication network, the doctor can communicate with and instruct the robot, while at the patient’s end, a multi-machinery robot performs operations on the patient in accordance with the doctor’s instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Introducing intelligence into the robotic system, which allows the robot to learn on its own while performing operations, can further enhance the technological benefits of the proposed touch technology as illustrated through Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The procedure will be made easier if the machine is capable of sensing and responding to the user’s touch sensations and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Thus, an attempt is made to incorporate artificial intelligence (AI) into the system while providing instructions through the tactile communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Additionally, if a sense of touch or emotion is introduced in the form of emotional intelligence (EI), the robot will be able to comprehend and execute the user’s instructions without delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This method could be used for both remote and local diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessPacket data gatenay Seninggatewa rtualizedClothestrialbyUser Tactile engine DIG RCADE SHOPPING COMPLEX N AI RTGLTNTELGENG TACTILECOMMUNICATIONSYSTEM INTELLIGENCE Shopping Experience USERDEVICES Virtual MirrorandTrial Room AR/VRinterface with User Device TACTILE/TOUCHFEEDBACKSYSTEMThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 44 FIGURE 22: Touch technology enabled contactless tele-diagnosis and surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In the current covid-19 pandemic, localized tele-diagnosis employing robots could be used to allow on-duty clinicians to undertake contactless diagnosis and testing on infected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' That in turn will lessen their chances of contracting the virus from the patients and thus lessen the strain on them during such a pandemic crisis or emergency situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' RESEARCH CHALLENGES AND FUTURE ASPECTS This section comprehends some of the conclusive future aspects concerning the incorporation of the reconfigurable and intelligent technologies like AI/ML/DL in the B5G/6G and IoT governed wireless communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In the end, intelligent technology machine learning has become one of the promising tools of artificial intelligence for the intelligent integration of wireless communications in the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Their futuristic scope may be steered towards their relevance in channel estimation and detection, inferring user location and behavior, resource allocation, iterative learning, computational intelligence like neural networks and decision making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Some of these have been discussed below: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' RESEARCH CHALLENGES: 1) INTELLIGENT CHANNEL ESTIMATION: The futuristic scope in the ML based channel estimation technique lies in the fact that it may be put to use in direct scenarios without any need for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The only way to be able to learn the channel features of various user environments can be that a system is as smart to understand its parametric needs or in other words that such a generalized system requires a vast amount of pre-collected communication data to be used by ML/DL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) NON/SEMI-AUTONOMOUS DEVICE DISCOVERY: Human intervention in IoT components, such as device discovery, renders these applications non- scalable and prone to error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Because of this limitation, interacting IoT devices like IoT middleware are unsuitable for self-adaptive applications such as M2M communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) HETEROGENOUS ENVIRONMENTS: Since most smart IoT devices and middle wares only handle one or two types of heterogeneous components, this is considered a major issue that must be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Due to the fact that non-autonomous and inflexible services and devices restrict the support of IoT applications, it is critical that new approaches address and resolve the heterogeneity of IoT environments, particularly in large scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) SERVICE LEVEL AGREEMENT: To provide customers with an agreed level of service, three components should be considered: a model that precisely defines all functional and nonfunctional services required by consumers, automatic service to ensure a high level of QoS and adaptation, and a monitoring tool for SLA services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Human intervention in current autonomous networks will be phased out in favor of the development of intelligent IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) QOS LEVELS: Since there is no mechanism in place to guarantee a specific level of QoS for non- functional IoT services, researchers should develop procedures for optimizing and monitoring QoS levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessTactile support engin Packetjlatagatewa Serving gatewa Y appor Doctor(user)attheremotelocation(miles awayfromthepatient) Servinggateway TactileCommunicationsystem Feedbackreceivedbyoperatingdoctor through surgical robot Artificial Intelligence introduced in robotic system Touch Sensation (Feelings/EI)This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 45 6) PRIVACY AND SECURITY: As a result of the resource-constrained devices in IoT environments, the majority of autonomous and semi-autonomous application services restrict security, authorization and authentication mechanisms, among other things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Hence for the intelligent network to communicate between the cloud, gateway, and sensors securely and efficiently, security and privacy must be both end to end and lightweight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' FUTURE RESEARCH ASPECTS: 1) ALGORITHMIC MODELING: Almost all relevant and physical modeling/construction should be seen as an integral step toward algorithmic implementation using applicable ML tools, and DNN is a key technology component in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) ROBOTIC FEELINGS: The implementation of tactile based haptic communication in conjunction with touch enabled gesture recognition is another application that could benefit from this technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This could be used in the inculcation of feelings and emotions in robotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) INTELLIGENT HEALTH MEASURES DURING PANDEMIC LIKE COVID-19: Telemedicine, remote surgery, and treatment using intelligence and haptic sensations may also benefit from it, requiring a suitable algorithm with appropriate ML tools for efficient implementation and risk-free operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' So it may also be useful in the contactless administration to the patients during pandemics such as the one that occurred in COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In this way, the risk of doctors becoming infected with the virus is reduced because the diagnosis will be done by a robotic system that will use haptic and touch sensations that have been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) EFFICIENT NETWORK MANAGEMENT: As a result, a paradigm shift is required for the efficient design of B5G/6G networks in order to leverage AI/ML and take advantage of big data analytics to improve the overall performance of future networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) PROACTIVE NETWORKING: Future networks require a prediction mechanism to help predict and anticipate the future while allocating network resources proactively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' As a result, it aids in the prediction and analysis of traffic patterns while determining off peak times on various spectrum bands so that incoming traffic demands can be properly allocated over a given window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6) BEHAVIORAL LEARNING: Predicting user behaviors will result in better network resource utilization and will allow us to optimally allocate end-to-end network sources in an online fashion, which would be impossible without the assistance of AI and ML techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' RECENT RESEARCH AND PROJECTS CONCERNING 6G WIRELESS COMMUNICATION SYSTEMS Numerous 6G research and development activities have already begun on a global scale and this section summarizes the most significant 6G research activities underway at the moment[68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6G FLAGSHIP (MAY 2018-APRIL 2026) The 6G Flagship[177] is an eight-year research initiative that focuses on the wireless smart society and ecosystem enabled by 6G technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Being funded by the Academy of Finland, this project intends to realize B5G networks from the very outset towards its phase of commercialization, and to develop the new 6G standards for the future digital societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It aims to develop essential technology components of 6G mobile networks in areas such as wireless connectivity, distributed intelligent computing, security, and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In addition to human-to- human communication, the research focuses on communication between devices, processes, and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This, in turn, enables a highly automated, smart society that will permeate all aspects of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Ultimately, the 6G flagship project is to conduct large-scale pilots with a test network using industry and academic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' HEXA-X: A FLAGSHIP FOR 6G VISION AND INTELLIGENT FABRIC OF TECHNOLOGY ENABLERS CONNECTING HUMANS, PHYSICAL AND DIGITAL WORLDS (JAN 2021-JUNE 2023) HEXA-X[178] is the first flagship project of the European Commission for implementing 6G vision and establishing an intelligent fabric of technology enablers for integrating the human, physical, and digital worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' HEXA-X is a European industry-academic collaboration that intends to prepare the way for the next generation of wireless networks through exploratory research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Its objective is to connect humans, with the physical and digital worlds through a fabric of 6G essential enablers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In order to achieve this goal and vision, the HEXA-X project is concentrating on building critical technical enablers in the following areas: 1) High-frequency and high-resolution long-range access using New Radio access technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) Connected intelligence to future networks via AI- powered air interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 3) Network disaggregation and dynamic dependability are enabled by 6G architectural drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TERA FLOW: SECURED AUTONOMIC TRAFFIC MANAGEMENT FOR A TERA OF SDN FLOWS (JAN 2021-JUNE 2023) Tera flow[179] is working on developing a cloud native SDN controller for B5G/6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This novel SDN controller is compatible with the contemporary NFV and MEC frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It will also provide new features for traffic flow aggregation, service layer management, infrastructure layer network equipment integration, AI/ML- based security and forensic evidence for multi-tenancy networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 46 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' DAEMON: NETWORK INTELLIGENCE FOR ADAPTIVE AND SELF LEARNING MOBILE NETWORKS (JAN 2021- JUNE 2023) The major goal of the DAEMON project[180] is to enable high-quality Network Intelligence (NI) for 6G systems, which will completely automate network administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The project includes an end-to-end B5G/6G NI architecture, which can be fully coordinated with the NI- assisted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' DAEMON performs a systematic analysis of each NI task that is solved using AI models and also provides a solid set of guidelines for incorporating machine learning into network functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A major goal of the DAEMON project is to focus on existing B5G network- specific AI methods that go beyond the current trend of integrating AI into network controllers and orchestrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6G BRAINS: BRINGING REINFORCEMENT LEARNING INTO RADIO LIGHTWEIGHT NETWORK FOR MASSIVE CONNECTIONS (JAN 2021-JUNE 2023) The 6G BRAINS[181] project is focused on implementing multi-agent DRL for 6G radio links using AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In order to improve massive connection over D2D assisted highly dynamic cell free networks, a novel comprehensive cross-layer DRL driven resource allocation solution will be required to perform resource allocation for Sub 6GHz/mm- wave/THz/optical wireless communication (OWC) medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This significantly improves the capacity, reliability, and latency of future industrial, intelligent transportation, and e-health networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SOUTH KOREA MSIT 6G RESEARCH PROGRAM The Ministry of Science and ICT (MSIT) [182] in South Korea is working on a bold strategy to be the first country to deploy 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6G services are expected to be commercially available in Korea between 2028 and 2030, according to the South Korean government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The initial deployment is expected in 2028, followed by a mass commercial deployment in 2030, with a total investment of $169 million in R&D for 6G technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The preliminary goal is to deploy a 6G pilot in five important areas, including digital healthcare, immersive content, self-driving cars, smart cities, and smart manufacturing, by 2026[183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The 6G research program’s objectives are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 1) Attain a data rate of 1Tbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) Latency reduction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1ms for wireless networks 3) Increases the range of connectivity coverage to up to 10 km from the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 4) AI must be integrated into the network to ensure that everything is covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 5) By implementing security by design, it is possible to protect the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' JAPAN 6G PROMOTION STRATEGY Japan invests approximately 50 billion dollars in the 6G development project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This initiative aims to strengthen collaboration between the public and private sectors in the field of 6G research and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Furthermore, by 2025, this 6G promotion strategy seeks to establish and exhibit the 6G system’s basic technologies, as well as put new technologies into practice by 2030[184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6TH GENERATION INNOVATION CENTER With the continuation of the 5th Generation Innovation Center (5GIC), the University of Surrey in the United Kingdom launched the 6GIC[185] in 2020 to focus on 6G- related research activities across two themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 1) AMBIENT INFORMATION: There is a use of the advanced wireless technologies, high resolution sensing, and highly accurate geo-location methods to improve the fusion of virtual and physical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This will enable a new level of 6G digital services by better connecting of human senses with ambient and remote data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 2) UBIQUITOUS COVERAGE: It has a role in emphasizing on increasing the quality and range of 6G communication network coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The research will focus on expanding coverage indoors, utilizing intelligent surfaces, and satellite technology to enable 6G services to be available globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' INDUSTRIAL 6G PROGRAMS The Table XVI summarizes several industrial research programs focused on the development and implementation of 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' While Table XVII and Table XVIII give an overall projection of the ongoing recent projects in B5G/IoT and 6G respectively in the Appendix I XIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' CONCLUSION The internet, with the advances in technology, drastically affects the human lifestyle, in profound ways, transforming various facets of life via interactions between the individuals at virtual level throughout most of the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The wireless technologies have thus transformed many elements of life including the business, living standards and the infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In a never-ending quest for elegant solutions to various problems, the society is always on the lookout for new avenues of progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore the motivation behind seamless connectivity has resulted in the evolution of wireless communication from 1G to 5G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TABLE XVI INDUSTRIAL AVENUES ON 6G DEVELOPMENTS Reference Industries Research projects/collaborations [186] Sony, NTT and Intel Cooperative research on 6G technologies to be commercialized by 2030 [187] Huawei 6G wireless technology research at the Canadian Research Center and 13 other universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [188] SK Telecom A 6G-based commercial networking project with Nokia and Ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [189] Samsung Commercialization is anticipated by 2028, along with 6G services that incorporate XR, a high-fidelity mobile hologram, and a digital replica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [190] LG and KAIST32 The opening of a new Research Center aimed at developing the 6G network standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [191] NTT Demonstrating a 100Gbps communication solution using OAM at 28GHz with MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [192] Tektronix Wireless fiber, a 100Gbps communication solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 32 KAIST: Korea Advanced Institute of Science and Technology IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 47 Despite the recent development, work in this field is still underway all over the world with the aim of deploying the 6G communication network by 2030, thus making it one of the most in demand research fields, with the potential to revolutionize personal life of individual and society, business and communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' The real time applications from the individual user’s point of view are perceptible in both professional as well as domestic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' It may vary from e-health, smart appliances to the smart classroom based enhanced learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' From the industrial point of view, the evident outcomes are discernible in departments like automation and industrial manufacturing, logistics, business and process management along with the intelligent transportation of people and goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore the current existing technologies make the IoT concept viable but do not necessarily fits well with the expected scalability and proficiency criteria of the 6G system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Therefore many problems do exist in the technical process and therefore, a societal reform is vital in the technology, universally conceptualizing the IoT connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' A complete interoperability of the integrated devices is required, provided with a high percentage of smartness, maintaining the trust, safety and protection in the system for creation of technologies that emphasize technical requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' However, the most intriguing aspect of the next generation 6G standard is the incorporation of an intelligent interface linking the existing communication standards with the recently researched ones, so as to competently fulfill the high data rate and the stringent latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' In other words, all we lack is an intelligent interface that is capable enough to integrate and enable a tactile based haptic (touch based) communication in the existing B5G network right from its source to its destination, complete with its feedback mechanism, altogether incorporating intelligence in the system so as to satisfy the stringent latency requirement of 1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' This paper therefore provides the preliminary insight to and answers the above mentioned challenges by providing a comprehensive survey of the touch based intelligent communication system using network slicing and TI coupled the intelligence like AI and ML, in 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' APPENDIX I: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' COLLABORATIVE AND ONGOING PROJECTS IN B5G/IOT Table XVII illustrates the recent collaborative projects in B5G and IoT based wireless communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' GLOBAL LEVEL ONGOING PROJECTS FOR 6G DEPLOYMENT Table XVIII summarizes all the ongoing research works and projects pertaining to 6G wireless communication system along with their application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' TABLE XVII COLLABORATIVE AND ONGOING PROJECTS IN B5G AND IOT WIRELESS COMMUNICATION SYSTEMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Ref Research Project/ Group Institution Research area Source [74] 5G Evolution and 6G NTT- DOCOMO Performance enhancement of mobile communication through exploring high frequency bands and improvement in wireless technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='nttdocomo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='co.' 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+page_content='fi/files/nbnfi- fe2019081624413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='pdf [201] Zero touch provisioning CISCO Device configuration, management and orchestration to the local Ethernet in remote locations via dynamic host configuration protocol (DHCP)-IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='cisco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='com/c/en/us/td/docs/switc hes/lan/catalyst3850/software/release/16- 5/configuration_guide/prog/b_165_prog_385 0_cg/zero_touch_provisioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='pdf [79] What should 6G be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' KAUST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Human centric perspective of 6G vision as research channel in post 5G era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' http://hdl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='net/10754/661147 IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 48 TABLE XVIII GLOBAL-LEVEL ONGOING PROJECTS ON 6G ALONGSIDE THEIR ASSOCIATED APPLICATIONS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='No Ongoing Projects Institute/ Organization Enabling Technologies Applications Source 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6G Flagship Academy of Finland To enable THz communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' AI/ML/FL implementation Blockchain/DLT ZSM NTN/3D networking VLC Quantum computing Extended reality Autonomous driving Intelligent healthcare Personalized Body area networks Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 [177] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' HEXA-X European Commission THz communication AI/ML/FL Compressive sensing Swarm networking UAV based networking Internet of everything (IoE) Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 Collaborative robots [178] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Tera flow 5GPPP AI/FL Blockchain/DLT ZSM Autonomous driving IoE [179] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' DAEMON European Commission AI/FL ZSM Compressive sensing UAV connectivity Autonomous driving IoE Intelligent Heathcare [180] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6G BRAINS 5GPPP THz communication AI/FL ZSM Swarm networking Compressive sensing VLC UAV navigation and connectivity Collaborative autonomous driving IoE [181] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' MSIT South Korea THz communication AI/FL Smart surfaces VLC XR Collaborative robots and autonomous driving IoE Smart grid 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 Intelligent Healthcare [182] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' JAPAN JAPAN THz communication AI/FL Smart surfaces Swarm networking VLC Quantum computing UAV mobility XR Autonomous driving IoE Smart grid 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 Intelligent Healthcare [184] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 6GIC University of Surrey, UK THz communication AI/FL Compressive sensing NTN/3D networking IoE Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 Hyper Intelligent IoT [185] ACKNOWLEDGMENT The authors gratefully acknowledge the support provided by 5G and IoT Lab, DoECE, Shri Mata Vaishno Devi University, Katra, Jammu REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Goldsmith, Wireless communications.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 53 [183] ‘South Korea to launch 6G pilot 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Matinmikko-Blue, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Latva-Aho, ‘6Genesis flagship program: Building the bridges towards 6G-enabled wireless smart society and ecosystem’, in 2018 IEEE 10th Latin- American Conference on Communications (LATINCOM), 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' [201] ‘Zero Touch Provisioning’, CISCO, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Accessed: Jun.' metadata={'source': 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received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='E degree in Electronics and Communication Engineering from Jammu University, Jammu and Kashmir, India in 2017 and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Tech Degree in Electronics and Communication Engineering from Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir, India in 2019, where she is pursuing the Ph,D degree in Electronics and Communication Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Her research interest includes the emerging technologies involving the B5G/6G and IoT enabled wireless communication and security network and currently she is doing her research on IoT configured networks in B5G/6G wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' She is a student member of Institute of Electrical and Electronics Engineers (IEEE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Rakesh K Jha (S’10, M’13) is an Associate Professor in the Department of Electronics and Communication Engineering, Indian Institute of Information Technology, Design and Manufacturing, Jabalpur (IIITDM Jabalpur).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He is carrying out his research in wireless communication, power optimizations, wireless security issues, and optical fiber communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has done B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=" Tech (Hon's) in Electronics and Communication Engineering and M." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content="Tech from NIT Jalandhar (Hon's), India in 2008." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' degree from NIT Surat, India in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has completed his 10th exam from govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' High school and Class 12th from Science College.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has published more than 101 Journal Papers out of which more than 61 SCI Journal papers including IEEE Transactions, IEEE Journal, Elsevier, Springer, Taylor & Francis, Hindawi, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has published more than 25 Interference including ITU-T, IEEE ANTS, INDICON, IEEE ANTS, and APAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Jha’s one concept related to the router of Wireless Communication has been accepted by ITU (International Telecommunication Union) in 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='He has received the young scientist author award by ITU in Dec 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has received an APAN fellowship in 2011, 2012- Srilanka, 2016, and in 2017-China, 2018-Singapore,2018- New Zealand, 2019-South Korea, and a student travel grant from COMSNET 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He is a Senior Member of IEEE, GISFI, and SIAM, International Association of Engineers (IAENG), ACCS (Advanced Computing and Communication Society), CSI, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has filed 8 Patents out of which 4 are published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Jha had 10 years of rich academic, Industrial, and research experience in various institutes/University including NIT-Surat, Capgemini India Pvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Ltd and SMVD University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has also served as an organizing member and TPC member for several national and international conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has organized many workshops and has also been invited as a resource person in IEEEAccessThis work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' For more information, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='3148473, IEEE Access M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' : Tactile based Intelligence Touch Technology in IoT configured WCN in B5G/6G-A Survey 54 many workshops organized by prestigious research institutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has guided 05 Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' students, 01 Submitted the thesis, 03 Defended Pre-Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Synopsis and 03 students are presently pursuing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has guided more than 15 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Tech and more than 41 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='Tech students for various projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' More than 4001 citations in his credit in the area of wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' PROF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' SANJEEV JAIN, born at Vidisha in Madhya Pradesh in 1967, obtained his Post Graduate Degree in Computer Science and Engineering from Indian Institute of Technology, Delhi, in 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He later received his Doctorate Degree in Computer Science & Engineering and has over 24 years’ experience in teaching and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has served as Director, Madhav Institute of Technology and Science (MITS), Gwalior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has worked as a vice chancellor at Shri Mata Vaishno Devi University, Katra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Presently he is a Professor in the Computer Science Department, Central University Jammu, Jammu and Kashmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Besides teaching at Post Graduate level Professor Jain has the credit of making significant contribution to R & D in the area of Image Processing and Mobile Adhoc Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He has guided Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' Scholars and has undertaken a number of major R & D projects sponsored by the Government and Private Agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' His work on Digital Watermarking for Image Authentication is highly valued in the research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' He is also a member of Association for Computing Machinery (ACM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} +page_content=' IEEEAccess' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQfHwn2/content/2301.04328v1.pdf'} diff --git a/hNE5T4oBgHgl3EQfFQ4n/content/tmp_files/2301.05420v1.pdf.txt b/hNE5T4oBgHgl3EQfFQ4n/content/tmp_files/2301.05420v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bee743d8e77d221ed42406e97b4e577fb0e97c96 --- /dev/null +++ b/hNE5T4oBgHgl3EQfFQ4n/content/tmp_files/2301.05420v1.pdf.txt @@ -0,0 +1,1008 @@ +Entanglement witness and multipartite quantum +state discrimination +Donghoon Ha and Jeong San Kim +Department of Applied Mathematics and Institute of Natural Sciences, Kyung Hee +University, Yongin 17104, Republic of Korea +E-mail: freddie1@khu.ac.kr +Abstract. +We consider multipartite quantum state discrimination and show that +the minimum-error discrimination by separable measurements is closely related to the +concept of entanglement witness. Based on the properties of entanglement witness, we +establish some necessary and/or sufficient conditions on minimum-error discrimination +by separable measurements. We also provide some conditions on the upper bound of +the maximum success probability over all possible separable measurements. Our results +are illustrated by examples of multidimensional multipartite quantum states. Finally, +we provide a systematic way in terms of EW to construct multipartite quantum state +ensembles showing nonlocality in state discrimination. +1. Introduction +Quantum state discrimination is one of the fundamental concepts used in various +quantum information and computation theory [1–4]. In general, we can always perfectly +discriminate orthogonal quantum states using appropriate measurement. +However, +nonorthogonal quantum states cannot be perfectly discriminated by means of any +measurement. For this reason, various state discrimination strategies have been studied +for optimal discrimination of nonorthogonal quantum states, such as minimum-error +discrimination, unambiguous discrimination and maximum-confidence discrimination +[5–9]. +Entanglement witness(EW) is an important tool to detect the existence of +entanglement inherent in a multipartite quantum state [10–13]. Mathematically, EW +is a Hermitian operator having non-negative mean value for every separable state, but +negative for some entangled states. As EW provides an useful methodology to detect +entanglement that is an important quantum nonlocality, it is natural to ask whether EW +can also be used to characterize other nonlocal phenomenon of multipartite quantum +states. +Quantum nonlocal phenomenon also arises in discriminating multipartite quantum +states; quantum nonlocality occurs when optimal state discrimination cannot be +realized only by local operations and classical communication(LOCC) [14–17]. However, +arXiv:2301.05420v1 [quant-ph] 13 Jan 2023 + +2 +characterizing local discrimination of quantum states is a hard task and very little is +known due to the lack of good mathematical structure for LOCC. +Here, we establish a specific relation between the properties of EW and separable +measurements, a mathematically well-structured set of measurements having LOCC +measurements as a special case. +We show that the minimum-error discrimination +of multipartite quantum states using separable measurements strongly depends on +the existence of EW. More precisely, we establish conditions on minimum-error +discrimination by separable measurements in terms of EW. We also provide conditions +on the upper bound of the maximum success probability over all possible separable +measurements. We illustrate our results using examples of multidimensional multipartite +quantum states. Finally, we provide a systematic way in terms of EW to construct +multipartite quantum state ensembles showing nonlocality in state discrimination. +This paper is organized as follows. In Section 2, we first recall the definitions and +some properties about separable measurements and EW. We also recall the definition +of minimum-error discrimination as well as some useful properties of the optimal +measurements. In Section 3, we provide conditions on the upper bound of the maximum +success probability over all possible separable measurements (Theorems 1 and 2). +In Section 4, we provide conditions on minimum-error discrimination by separable +measurements in terms of EW(Theorems 3 and 4). +Our results are illustrated by +examples of multidimensional multipartite quantum states. In Section 5, we provide +a systematic way in terms of EW to construct multipartite quantum state ensembles +showing nonlocality in state discrimination. We conclude our results in Section 6. +2. Preliminaries +For a multipartite Hilbert space H = �m +k=1 Cdk with m ⩾ 2 and positive integers dk for +k = 1, . . . , m, let H be the set of all Hermitian operators acting on H. We use H+ to +denote the set of all positive-semidefinite operators in H, that is, +H+ = {E ∈ H | ⟨v| E |v⟩ ⩾ 0 ∀ |v⟩ ∈ H}. +(1) +A multipartite quantum state is expressed by a density operator ρ, that is, ρ ∈ H+ +with Tr ρ = 1. A measurement is represented by a positive operator-valued measure +{Mi}i, that is, {Mi}i ⊆ H+ satisfying � +i Mi = 1, where 1 is the identity operator in +H. For the quantum state ρ, the probability of obtaining the measurement outcome +corresponding to Mj is Tr(ρMj). +2.1. Entanglement witness +Definition 1. E ∈ H+ is called separable if it can represented as a sum of positive- +semidefinite product operators, that is, +E = +� +l +m +� +k=1 +El,k, +(2) + +3 +Figure 1. The relationship of the subsets of H. The shaded area SEP∗ \ H+ is the set +of all EWs. +where {El,k}l is a set of positive-semidefinite operators acting on Cdk of H for each +k = 1, . . . , m. Similarly, we say that the set {Ei}i ⊆ H+ is separable if Ei is separable +for all i. +We denote the set of all separable operators in H+ as +SEP = {E ∈ H+ | E : separable}, +(3) +and its dual set as SEP∗, that is, +SEP∗ = {E ∈ H | Tr(EF) ⩾ 0 ∀F ∈ SEP}. +(4) +A Hermitian operator in SEP∗ is also called block positive. Because 1 ∈ SEP, we have +Tr E ⩾ 0 +(5) +for all E ∈ SEP∗, where the equality holds if and only if E is the zero operator in H [18]. +Definition 2. W ∈ H is called an EW if it is block positive but not positive semidefinite, +that is, +W ∈ SEP∗ \ H+. +(6) +For each EW W, it is known that there exists an entangled operator E ∈ H+ \ SEP +satisfying +Tr(WE) < 0, +(7) +thus the term entanglement witness. Figure 1 illustrates the relationship of the subsets +of H. + +SEP4 +2.2. Minimum-error discrimination of multipartite quantum states +For a multipartite quantum state ensemble, +E = {ηi, ρi}n +i=1, +(8) +let us consider the situation that the state ρi is prepared with the probability ηi. To +guess the prepared state, we use the decision rule in terms of a measurement {Mi}n +i=1 +such that the detection of Mi means that the prepared state is guessed to be ρi. The +average probability of correctly guessing the prepared state from E in Eq. (8) with +respect to a measurement {Mi}n +i=1 is +n +� +i=1 +ηi Tr(ρiMi). +(9) +The minimum-error discrimination of E +is to achieve the optimal success +probability, +pG(E) = +max +Measurement +n +� +i=1 +ηi Tr(ρiMi), +(10) +where the maximum is taken over all possible measurements [5]. +We note that a +measurement {Mi}n +i=1 provides the optimal success probability pG(E) if and only if +it satisfies the following condition [19–22], +n +� +j=1 +ηjρjMj − ηiρi ∈ H+ ∀i = 1, . . . , n. +(11) +A measurement {Mi}i is called a separable measurement if {Mi}i ⊆ SEP, and a +measurement is called a LOCC measurement if it can be realized by LOCC. It is known +that every LOCC measurement is a separable measurement [23]. When the available +measurements are limited to separable measurements, we denote the maximum success +probability by +pSEP(E) = +max +Separable +measurement +n +� +i=1 +ηi Tr(ρiMi), +(12) +where the maximum is taken over all possible separable measurements. Similarly, we +denote +pL(E) = +max +LOCC +measurement +n +� +i=1 +ηi Tr(ρiMi), +(13) +where the maximum is taken over all possible LOCC measurements. +From the +definitions, we trivially have +pL(E) ⩽ pSEP(E) ⩽ pG(E). +(14) + +5 +3. Separable measurements and quantum state discrimination +For a multipartite quantum state ensemble E in Eq. (8), we define HSEP(E) as +HSEP(E) = {H ∈ H | H − ηiρi ∈ SEP∗ ∀i = 1, . . . , n}, +(15) +where SEP∗ is defined in Eq. (4). In other words, HSEP(E) is the set of all H such that +H − ηiρi is block positive in H, for all i = 1, . . . , n. We further define +HEW(E) = {H ∈ HSEP(E) | H − ηiρi /∈ H+ for some j ∈ {1, . . . , n}}, (16) +that is, HEW(E) is the set of all H ∈ HSEP(E) such that H − ηjρj is an EW for some +j ∈ {1, . . . , n}. From Definition 2, we can see that +H ∈ HSEP(E) \ HEW(E) +(17) +if and only if +H − ηiρi ∈ H+ ∀i = 1, . . . , n. +(18) +Now, let us consider the minimum quantity +qSEP(E) = +min +H∈HSEP(E) Tr H, +(19) +which is an upper bound of pSEP(E) [24], that is, +pSEP(E) ⩽ qSEP(E). +(20) +The following theorem shows that pSEP(E) in Eq. (12) is equal to qSEP(E) in Eq. (19). +Theorem 1. For a multipartite quantum state ensemble E = {ηi, ρi}n +i=1, +pSEP(E) = qSEP(E). +(21) +Proof. As we already have Inequality (20), it is enough to show that +pSEP(E) ⩾ qSEP(E). +(22) +Let us consider the set, +S(E) = +� +( +n +� +i=1 +ηi Tr(ρiMi) − p, 1 − +n +� +i=1 +Mi) ∈ R × H +��� +p > pSEP(E), Mi ∈ SEP ∀i = 1, . . . , n +� +, +(23) +where R is the set of all real numbers. We note that S(E) is a convex set due to the +convexity of SEP in Eq. (3). Moreover, S(E) does not have the origin (0, 0H) of R × H +otherwise there is a separable measurement {Mi}n +i=1 with +n +� +i=1 +ηi Tr(ρiMi) > pSEP(E), +(24) +and this contradicts the optimality of pSEP(E) in Eq. (12). Here, 0H is the zero operator +in H. We also note that the Cartesian product R × H can be considered as a real vector +space with an inner product defined as +⟨(a, A), (b, B)⟩ = ab + Tr(AB) +(25) + +6 +for (a, A), (b, B) ∈ R × H. +Since S(E) and the single-element set {(0, 0H)} are disjoint convex sets, it follows +from separating hyperplane theorem [25,26] that there is (γ, Γ) ∈ R × H satisfying +(γ, Γ) ̸= (0, 0H), +(26) +⟨(γ, Γ), (r, G)⟩ ⩽ 0 ∀(r, G) ∈ S(E). +(27) +Suppose +Tr Γ ⩽ γpSEP(E), +(28) +Γ − γηiρi ∈ SEP∗ ∀i = 1, . . . , n, +(29) +γ > 0. +(30) +From Conditions (29) and (30), the Hermitian operator H = Γ/γ is an element of +HSEP(E) in Eq. (15). Thus, the definition of qSEP(E) in Eq. (19) leads us to +qSEP(E) ⩽ Tr H. +(31) +Moreover, Condition (28) implies +Tr H ⩽ pSEP(E). +(32) +Inequalities (31) and (32) imply Inequality (22). +To complete the proof, we show the validity of (28), (29) and (30). +Proof of (28). From Eq. (25), Inequality (27) can be rewritten as +Tr Γ − +n +� +i=1 +Tr[Mi(Γ − γηiρi)] ⩽ γp +(33) +for all p > pSEP(E) and all {Mi}n +i=1 ⊆ SEP. +If Mi = 0H for all i = 1, . . . , n, +Inequality (33) becomes Inequality (28) by taking the limit of p to pSEP(E). +Proof of (29). For each j ∈ {1, . . . , n}, let us consider an arbitrary Mj ∈ SEP and +Mi = 0H for all i = 1, . . . , n with i ̸= j. In this case, {Mi}n +i=1 is clearly a subset of SEP, +and Inequality (33) becomes +Tr Γ − Tr[Mj(Γ − γηjρj)] ⩽ γpSEP(E) +(34) +by taking the limit of p to pSEP(E). +Suppose Γ − γηjρj /∈ SEP∗, then there is M ∈ SEP with Tr[M(Γ − γηjρj)] < 0. +We note that M ∈ SEP implies tM ∈ SEP for any t > 0. Thus, {Mi}n +i=1 consisting of +Mj = tM for t > 0 and Mi = 0 for all i = 1, . . . , n with i ̸= j is also a subset of SEP. +Now, Inequality (34) can be rewritten as +Tr Γ − Tr[tM(Γ − γηjρj)] ⩽ γpSEP(E). +(35) +Since Inequality (35) is true for arbitrary large t > 0, γpSEP(E) can also be arbitrary +large. However, this contradicts that both γ and pSEP(E) are finite. Thus, Γ − γηjρj ∈ +SEP∗, which completes the proof of (29). + +7 +Proof of (30). Suppose γ < 0 and consider {Mi}n +i=1 with Mi = 0H for all i = 1, . . . , n. +Since {Mi}n +i=1 ⊆ SEP, Inequality (33) becomes +Tr Γ ⩽ −∞ +(36) +by taking the limit of p to ∞. This contradicts that Γ is bounded, therefore γ ⩾ 0. +Now, let us suppose γ = 0. In this case, Conditions (28) and (29) become +Tr Γ ⩽ 0, +Γ ∈ SEP∗. +(37) +From Condition (37) and the argument with Inequality (5), we have +Γ = 0H, +(38) +which contradicts Condition (26). Thus, γ > 0. +For a given ensemble E = {ηi, ρi}n +i=1, the following theorem provides a necessary +and sufficient condition for a separable measurement {Mi}n +i=1 and H ∈ HSEP(E) to +realize pSEP(E) and qSEP(E), respectively. +Theorem 2. For a multipartite quantum state ensemble E = {ηi, ρi}n +i=1, a separable +measurement {Mi}n +i=1 and H ∈ HSEP(E), {Mi}n +i=1 realizes pSEP(E) and H provides +qSEP(E) if and only if +Tr[Mi(H − ηiρi)] = 0 ∀i = 1, . . . , n. +(39) +Proof. Let us suppose that {M}n +i=1 and H give pSEP(E) and qSEP(E), respectively. +Because Mi ∈ SEP and H − ηiρi ∈ SEP∗ for all i = 1, . . . , n, we have +Tr[Mi(H − ηiρi)] ⩾ 0 ∀i = 1, . . . , n. +(40) +Moreover, we have +n +� +i=1 +Tr[Mi(H − ηiρi)] = Tr H − +n +� +i=1 +ηi Tr(ρiMi) += qSEP(E) − pSEP(E) = 0, +(41) +where the first equality is from �n +i=1 Mi = 1, the second equality is due to the +assumption of H and {Mi}n +i=1, and the last equality is by Theorem 1. Inequality (40) +and Eq. (41) lead us to Condition (39). +Conversely, let us assume that {Mi}n +i=1 and H satisfy Condition (39). +This +assumption implies +qSEP(E) = pSEP(E) ⩾ +n +� +i=1 +ηiTr(ρiMi) += +n +� +i=1 +ηiTr(ρiMi) + +n +� +i=1 +Tr[Mi(H − ηiρi)] += TrH ⩾ qSEP(E), +(42) + +8 +where the first equality is by Theorem 1, the second equality is from Condition (39), the +last equality follows from �n +i=1 Mi = 1, and the first and second inequalities are due to +the definitions of pSEP(E) and qSEP(E), respectively. Inequality (42) leads us to +n +� +i=1 +ηiTr(ρiMi) = pSEP(E), +Tr H = qSEP(E). +(43) +Thus, {Mi}n +i=1 and H provide pSEP(E) and qSEP(E), respectively. +We note that H ∈ HSEP(E) providing qSEP(E) is generally not unique (see Example 2 +in Section 4.2). However, the following corollary states the case that H ∈ HSEP(E) +providing qSEP(E) is unique. +Corollary 1. For a multipartite quantum state ensemble E = {ηi, ρi}n +i=1, we have +pSEP(E) = η1, +(44) +if and only if +η1ρ1 − ηiρi ∈ SEP∗ ∀i = 2, . . . , n. +(45) +In this case, η1ρ1 is the only element of HSEP(E) providing qSEP(E). +Proof. Let {Mi}n +i=1 be the measurement such that +M1 = 1, +M2 = · · · = Mn = 0H. +(46) +We first assume Eq. (44) and consider H ∈ HSEP(E) giving qSEP(E). Since {Mi}n +i=1 is +obviously a separable measurement providing pSEP(E), it follows from Theorem 2 that +Tr(H − η1ρ1) = 0. From H − η1ρ1 ∈ SEP∗ and the argument with Inequality (5), we +have H = η1ρ1. Thus, H ∈ HSEP(E) together with the definition of HSEP(E) leads us to +Condition (45). +Conversely, let us suppose Condition (45) and consider +H = η1ρ1, +(47) +which is in HSEP(E) by Condition (45). The separable measurement {Mi}n +i=1 in Eq. (46) +and H ∈ HSEP(E) in Eq. (47) satisfy Condition (39). Therefore, we have +pSEP(E) = +n +� +i=1 +ηi Tr(ρiMi) = η1, +(48) +where the first equality is by Theorem 2. +When Eq. (44) of Corollary 1 holds, the maximum success probability pSEP(E) can +be achieved without the help of measurement, simply by guessing ρ1 is prepared. As we +can check in the proof of Corollary 1, the choice of ρ1 in Corollary 1 can be arbitrary. +That is, any of {ρi}n +i=1 can be used to play the role of ρ1 in Corollary 1. + +9 +4. Minimum-error discrimination by separable measurements +For a quantum state ensemble E in Eq. (8), the minimum-error discrimination can be +realized by separable measurements if and only if +pSEP(E) = pG(E), +(49) +where pG(E) and pSEP(E) are defined in Eqs. (10) and (12), respectively. In this section, +we provide some necessary and/or sufficient conditions for Eq. (49) in terms of EW. +4.1. Necessary condition for pSEP(E) = pG(E) +Theorem 3. For a multipartite quantum state ensemble E = {ηi, ρi}n +i=1, if there exists +separable measurement {Mi}n +i=1 satisfying +n +� +i=1 +ηiρiMi ∈ HEW(E), +(50) +where HEW(E) is defined in Eq. (16), then +pSEP(E) = +n +� +i=1 +ηi Tr(ρiMi) < pG(E). +(51) +Thus, non-existence of such separable measurement {Mi}n +i=1 satisfying Condition (50) +is a necessary condition for pSEP(E) = pG(E). +Proof. Let us suppose {Mi}n +i=1 is a separable measurement satisfying Condition (50), +and consider +H = +n +� +i=1 +ηiρiMi. +(52) +Equation (51) holds because +n +� +i=1 +ηi Tr(ρiMi) ⩽ pSEP(E) = qSEP(E) ⩽ Tr H = +n +� +i=1 +ηi Tr(ρiMi), +(53) +where the first inequality is due to the definition of pSEP(E), the second inequality follows +from the definition of qSEP(E) together with the condition that H ∈ HEW(E) ⊆ HSEP(E), +the first equality is from Theorem 1, and the second equality is by Eq. (52). This proves +the equality of (51). +If we assume pSEP(E) = pG(E) in (51), {Mi}n +i=1 gives the optimal success probability +pG(E). From the optimality condition in Eq. (11) and the argument with Eq. (17), we +have +n +� +i=1 +ηiρiMi ∈ HSEP(E) \ HEW(E), +(54) +which contradicts Condition (50). Thus, we have the inequality of (51). + +10 +Example 1. For any integers m, d ⩾ 2, let us consider the m-qudit state ensemble +E = {ηi, ρi}d+2 +i=1 consisting of d + 2 states, +ηi = +1 +dm + d, +ρi = |i − 1⟩⟨i − 1|⊗m , i = 1, . . . , d, +ηd+1 = dm − d +dm + d, ρd+1 = +1 +dm − d +� +1 − +d−1 +� +j=0 +|j⟩⟨j|⊗m � +, +ηd+2 = +d +dm + d, ρd+2 = |Φ⟩⟨Φ| , +(55) +where +|Φ⟩ = +1 +√ +d +d−1 +� +i=0 +|i⟩⊗m . +(56) +For a separable measurement {Mi}d+2 +i=1 with +Mi = |i − 1⟩⟨i − 1|⊗m , i = 1, . . . , d, +Md+1 = 1 − +d−1 +� +j=0 +|j⟩⟨j|⊗m , Md+2 = 0H, +(57) +we show that Condition (50) holds with respect to the ensemble in Eq. (55). +It is straightforward to verify that +d+2 +� +j=1 +ηjρjMj − ηiρi = +1 +dm + d(1 − |i⟩⟨i|⊗m ) ∈ H+, i = 1, . . . , d, +d+2 +� +j=1 +ηjρjMj − ηd+1ρd+1 = +1 +dm + d +d−1 +� +j=0 +|j⟩⟨j|⊗m ∈ H+, +d+2 +� +j=1 +ηjρjMj − ηd+2ρd+2 = +1 +dm + d(1 − d |Φ⟩⟨Φ| ) ∈ SEP∗, +(58) +where the last inclusion is from the fact that +d Tr(|Φ⟩⟨Φ| E) ⩽ Tr E ∀E ∈ SEP. +(59) +To show the validity of Inequality (59), we assume that {|e(k) +i ⟩}d +i=1 is an orthonormal +basis of Cd for each k = 1, . . . , m. Since {|e(1) +i1 ⟩⊗· · ·⊗|e(m) +im ⟩}i1,...,im is an orthonormal basis +of the multipartite Hilbert space H, there is a set of complex numbers {ci1,...,im}i1,...,im +such that +|Φ⟩ = +d +� +i1,...,im=1 +ci1,...,im |e(1) +i1 ⟩ ⊗ · · · ⊗ |e(m) +im ⟩ . +(60) +For each i1 ∈ {1, . . . , d}, we have +d +� +i2,...,im=1 +|ci1,...,im|2 = ⟨Φ| +� +|e(1) +i1 ⟩⟨e(1) +i1 | ⊗ ( +d−1 +� +j=0 +|j⟩⟨j| )⊗m−1� +|Φ⟩ = 1 +d, +(61) + +11 +where the first equality follows from Eq. (60) and the second equality is due to Eq. (56). +Equation (61) leads us to +|ci1,...,im|2 ⩽ 1 +d ∀i1, . . . , im = 1, . . . , d. +(62) +Since the choice of {|e(k) +i ⟩}d +i=1 can be arbitrary for each k = 1, . . . , m, we have +d Tr(|Φ⟩⟨Φ|e⟩⟨e|) ⩽ ⟨e|e⟩ +(63) +for any product vector |e⟩ ∈ H. Therefore, Inequality (59) holds. +Now, the inclusions in (58) together with H+ ⊆ SEP∗ imply +d+2 +� +j=1 +ηjρjMj − ηiρi ∈ SEP∗ ∀i = 1, . . . , d + 2. +(64) +Furthermore, a straightforward calculation leads us to +⟨Φ| +� d+2 +� +j=1 +ηjρjMj − ηd+2ρd+2 +� +|Φ⟩ = − d − 1 +dm + d < 0. +(65) +From Eq. (65), we have +d+2 +� +j=1 +ηjρjMj − ηd+2ρd+2 /∈ H+. +(66) +From Eqs. (64) and (66), the ensemble in Eq. (55) and the measurement in Eq. (57) +satisfy Condition (50). Thus, Theorem 3 leads us to +pSEP(E) = +d+2 +� +i=1 +ηi Tr(ρiMi) = +dm +dm + d < pG(E). +(67) +We also note that the separable measurement in Eq. (57) is a LOCC measurement +because it can be implemented by performing the same local measurement {|l⟩⟨l|}d−1 +l=0 +on each party. Thus, we have +pL(E) = pSEP(E) = +dm +dm + d. +(68) +4.2. Necessary and sufficient condition for pSEP(E) = pG(E) +Theorem 4. For a multipartite quantum state ensemble E = {ηi, ρi}n +i=1, pSEP(E) = +pG(E) if and only if there exists H ∈ HSEP(E) such that it provides qSEP(E) but does not +satisfy +H ∈ HEW(E), +(69) +or equivalently, there is H ∈ H satisfying Condition (17) and Tr H = qSEP(E). +Proof. Let {Mi}n +i=1 be a separable measurement giving pSEP(E). +We first suppose +pSEP(E) = pG(E) and consider +H = +n +� +i=1 +ηiρiMi. +(70) + +12 +Since the measurement {Mi}n +i=1 gives the optimal success probability pG(E), the +optimality condition in Eq. (11) leads us to +H − ηiρi = +n +� +j=1 +ηjρjMj − ηiρi ∈ H+ ∀i = 1, . . . , n. +(71) +Therefore, H satisfies Condition (17). Moreover, we have +Tr H = +n +� +i=1 +ηi Tr(ρiMi) = pSEP(E) = qSEP(E), +(72) +where the first equality is from Eq. (70), the second equality is by the assumption of +{Mi}n +i=1, and the last equality is due to Theorem 1. +Conversely, let us assume H is an element of H satisfying Condition (17) and +Tr H = qSEP(E). Thus, the positivenesses in (18) is satisfied in term of H. For each +i = 1, . . . , n, the positive-semidefinite operators Mi and H − ηiρi are orthogonal since +they satisfy Condition (39) from Theorem 2. +The optimality condition in Eq. (11) holds for the measurement {Mi}n +i=1 because +n +� +j=1 +ηjρjMj − ηiρi = +n +� +j=1 +ηjρjMj + +n +� +k=1 +(H − ηkρk)Mk − ηiρi += H − ηiρi ∈ H+ ∀i = 1, . . . , n, +(73) +where the first equality is from the orthogonality of Mi and H −ηiρi for each i = 1, . . . , n +and the second equality is from �n +i=1 Mi = 1. Thus, we have +pG(E) = +n +� +i=1 +ηi Tr(ρiMi) = pSEP(E), +(74) +where the second equality is due to the assumption of {Mi}n +i=1. +If pSEP(E) = pG(E), Theorem 4 implies that there must exist H ∈ HSEP(E)\HEW(E) +providing qSEP(E). In this case, there possibly exists another Hermitian operator H′ +satisfying H′ ∈ HEW(E) and Tr H′ = qSEP(E), which is illustrated in the following +example. +Example 2. For any integers m, d ⩾ 2, let us consider the m-qudit state ensemble +E = {ηi, ρi}dm−d+1 +i=1 +consisting of dm − d + 1 states, +η1 = d +dm, ρ1 = |Φ⟩⟨Φ| , +ηi = 1 +dm, ρi = |βi⟩⟨βi| , i = 2, . . . , dm − d + 1, +(75) +where |Φ⟩ is defined in Eq. (56) and {|βi⟩}dm−d+1 +i=2 +is a set of orthonormal product vectors +orthogonal to |j⟩⊗m for all j = 0, . . . , d − 1. +For a separable measurement {Mi}dm−d+1 +i=1 +with +M1 = +d−1 +� +j=0 +|j⟩⟨j|⊗m , Mi = |βi⟩⟨βi| , i = 2, . . . , dm − d + 1, +(76) + +13 +we can easily see that the success probability obtained from the separable measurement +in discriminating the states from the ensemble E in Eq. (75) is one, that is, +dm−d+1 +� +i=1 +ηi Tr(ρiMi) = 1. +(77) +The success probability in Eq. (77) is a lower bound of pSEP(E) in Eq. (12), therefore +pSEP(E) ⩾ 1. +(78) +Since pG(E) is bounded above by 1, Inequalities (14) and (78) imply +pSEP(E) = pG(E) = 1. +(79) +Furthermore, we have +qSEP(E) = pSEP(E) = 1, +(80) +where the first equality is by Theorem 1 and the second equality is from Eq. (79). +Let us first consider the Hermitian operator +H = +dm−d+1 +� +i=1 +ηiρi. +(81) +Equations (80) and (81) imply +Tr H = 1 = qSEP(E). +(82) +Moreover, a straightforward calculation leads us to +H − ηiρi = +dm−d+1 +� +j=1 +j̸=i +ηjρj ∈ H+ ∀i = 1, . . . , n. +(83) +Due to Eqs. (82) and (83), H is an element of HSEP(E) \ HEW(E) giving qSEP(E). Thus, +we have H satisfying the conditions of Theorem 4. +Now, let us consider the Hermitian operator +Ht = t +dm−d+1 +� +i=1 +ηiρi + 1 − t +dm 1, +(84) +where 0 ⩽ t < 1. Equations (80) and (84) imply +Tr Ht = 1 = qSEP(E) +(85) +for 0 ⩽ t < 1. Moreover, a straightforward calculation leads us to +Ht − η1ρ1 = t +dm−d+1 +� +j=2 +ηjρj + 1 − t +dm (1 − d |Φ⟩⟨Φ|) ∈ SEP∗, +Ht − ηiρi = t +dm−d+1 +� +j=1 +j̸=i +ηjρj + 1 − t +dm (1 − |βi⟩⟨βi|) ∈ H+ ∀i ̸= 1, +(86) + +14 +where the first inclusion follows from Inequality (59). Due to Eq. (86) together with +H+ ⊆ SEP∗, we have +Ht ∈ HSEP(E). +(87) +We also note that +⟨Φ| (Ht − η1ρ1) |Φ⟩ = −(1 − t)(d − 1) +dm +< 0 +(88) +for 0 ⩽ t < 1. Equations (85), (87) and Inequality (88) imply that Ht is an element of +HEW(E) giving qSEP(E). By letting H′ = Ht, we have another Hermitian operator H′, +besides H in Eq. (81), satisfying H′ ∈ HEW(E) and Tr H′ = qSEP(E). +For a multipartite quantum state ensemble E = {ηi, ρi}n +i=1 where H is the only +element of HSEP(E) providing qSEP(E), Theorem 4 tell us that pSEP(E) < pG(E) if and +only if there exists an EW in {H − ηiρi}n +i=1. From Corollary 1, η1ρ1 is the only element +of HSEP(E) providing qSEP(E) when Condition (45) holds. Thus, we have the following +corollary. +Corollary +2. For a multipartite quantum state ensemble E += +{ηi, ρi}n +i=1 with +Condition (45), pSEP(E) < pG(E) if and only if there exists an EW in {η1ρ1 − ηiρi}n +i=2. +Example 3. For any integers m, d ⩾ 2, let us consider the m-qudit state ensemble +E = {ηi, ρi}d+1 +i=1 consisting of d + 1 states, +η1 = 1 +2, +ρ1 = 1 +dm1, +ηi = 1 +2d, ρi = d2 − d +dm − d |Φi⟩⟨Φi| + +dm − d2 +dm(dm − d)1, i = 2, . . . , d + 1, +(89) +where +|Φj⟩ = +1 +√ +d +d−1 +� +k=0 +exp +�i2πjk +d +� +|k⟩⊗m . +(90) +For each i = 2, . . . , d + 1, a straightforward calculation leads us to +η1ρ1 − ηiρi = +d − 1 +2d(dm − d)(1 − d |Φi⟩⟨Φi|) ∈ SEP∗, +(91) +where the inclusion is from the fact that +d Tr(|Φi⟩⟨Φi| E) ⩽ Tr E ∀E ∈ SEP. +(92) +We can show the validity of Inequality (92) in a similar way to that of Inequality (59). +Now, the inclusion in (91) together with Corollary 1 imply +pSEP(E) = η1 = 1 +2. +(93) +Furthermore, a straightforward calculation leads us to +⟨Φi| +� +η1ρ1 − ηiρi +� +|Φi⟩ = − (d − 1)2 +2d(dm − d) < 0 ∀i = 2, . . . , d + 1. +(94) +From Eq. (94), we have +η1ρ1 − ηiρi /∈ H+ ∀i = 2, . . . , d + 1. +(95) + +15 +From Eqs. (91) and (95), η1ρ1−ηiρi is an EW for any i = 2, . . . , d+1. Thus, Corollary 2 +leads us to +pSEP(E) = 1 +2 < pG(E). +(96) +5. Construction of nonlocal quantum state ensemble +In this section, we provide a systematic way in terms of EW to construct multipartite +quantum state ensembles showing nonlocality in state discrimination, that is, pL(E) < +pG(E). For a given EW W, let us consider the multipartite quantum state ensemble +E = {ηi, ρi}2 +i=1 where +η1 = Tr(P + W) +Tr(2P + W), ρ1 = +P + W +Tr(P + W), +η2 = +Tr P +Tr(2P + W), ρ2 = +P +Tr P , +(97) +with any P ∈ H+ satisfying +P + W ∈ H+. +(98) +Since η1ρ1−η2ρ2 is proportional to the EW W, pSEP(E) < pG(E) holds from Corollary 2. +Thus, Inequality (14) leads us to pL(E) < pG(E). +Corollary 2 can also be used to construct a multipartite quantum state ensemble +E = {ηi, ρi}n +i=1 with n > 2 showing nonlocality in quantum state discrimination. +For a set of EWs {Wi}n +i=2, let us consider the multipartite quantum state ensemble +E = {ηi, ρi}n +i=1 where +η1 = +Tr 1 +Tr(n1 − �n +j=2 λjWj), ρ1 = +1 +Tr 1, +ηi = +Tr(1 − λiWi) +Tr(n1 − �n +j=2 λjWj), ρi = +1 − λiWi +Tr(1 − λiWi), i = 2, . . . , n, +(99) +with any set of positive real numbers {λi}n +i=2 satisfying +1 − λiWi ∈ H+ ∀i = 2, . . . , n. +(100) +Because η1ρ1 − ηiρi is proportional to Wi for any i ∈ {2, . . . , n}, pSEP(E) < pG(E) holds +from Corollary 2. Thus, Inequality (14) leads us to pL(E) < pG(E). +6. Conclusions +We have considered multipartite quantum state discrimination and shown that the +minimum-error discrimination by separable measurements strongly depends on the +existence of EW. We have established the necessary and/or sufficient conditions on +minimum-error discrimination by separable measurements, that is, pSEP(E) = pG(E), in +terms of EW (Theorems 3 and 4). We have also provided the conditions on the upper +bound of the maximum success probability over all possible separable measurements +(Theorems 1 and 2). Our results have been illustrated by examples of multidimensional + +16 +multipartite quantum states. Finally, we have provided a systematic way in terms of +EW to construct multipartite quantum state ensembles showing nonlocality in state +discrimination. +Quantum nonlocality is a key ingredient making quantum states outperform +the classical ones in various quantum information processing tasks such as quantum +teleportation and quantum cryptography [27, 28]. +It is also known that quantum +nonlocality plays an important role in quantum algorithms which are more powerful +than any classical ones [29,30]. As the violation of the conditions in Theorem 4 implies +pSEP(E) < pG(E), which consequently means pL(E) < pG(E), our results provides a +useful methodology to guarantee the occurrence of nonlocality in state discrimination. +Our results establish a specific relation between the properties of EW and minimum- +error discrimination by separable measurements, therefore it is natural to investigate the +relationship between EW and other measurements. It is also an interesting future work +to construct good conditions, in terms of EW, for optimal state discrimination in other +state discrimination strategies. +Acknowledgments +This work was supported by Basic Science Research Program(NRF-2020R1F1A1A010501270) +and Quantum Computing Technology Development Program(NRF-2020M3E4A1080088) +through the National Research Foundation of Korea(NRF) grant funded by the Korea +government(Ministry of Science and ICT). +References +[1] Chefles A 2000 Contemp. Phys. 41 401 +[2] Barnett S M and Croke S 2009 Adv. Opt. Photon. 1 238 +[3] Bergou J A 2010 J. Mod. Opt. 57 160 +[4] Bae J and Kwek L-C 2015 J. Phys. A: Math. Theor. 48 083001 +[5] Helstrom C W 1969 J. Stat. Phys. 1 231 +[6] Ivanovic I D 1987 Phys. Lett. A 123 257 +[7] Dieks D 1988 Phys. Lett. A 126 303 +[8] Peres A 1988 Phys. Lett. A 128 19 +[9] Croke S, Andersson E, Barnett S M, Gilson C R and Jeffers J 2006 Phys. Rev. Lett. 96 070401 +[10] Horodecki M, Horodecki P and Horodecki R 1996 Phys. Lett. A 223 1 +[11] Terhal B M 2000 Phys. Lett. A 271 319 +[12] Lewenstein M, Kraus B, Cirac J I and Horodecki P 2000 Phys. Rev. A 62 052310 +[13] Chru´sci´nski D and Sarbicki G 2014 J. Phys. A: Math. Theor. 47 483001 +[14] Peres A and Wootters W K 1991 Phys. Rev. Lett. 66 1119 +[15] Bennett C H, DiVincenzo D P, Fuchs C A, Mor T, Rains E, Shor P W, Smolin J A and Wootters +W K 1999 Phys. Rev. A 59 1070 +[16] Ghosh S, Kar G, Roy A, Sen(De) A and Sen U 2001 Phys. Rev. Lett. 87 277902 +[17] Chitambar E and Hsieh M-H 2013 Phys. Rev. A 88 020302(R) +[18] Ha D and Kim J S 2022 Sci. Rep. 12 14130 +[19] Holevo A S 1974 Probl. Peredachi Inf. 10 51 +[20] Yuen H, Kennedy R and Lax M 1975 IEEE Trans. Inf. Theory 21 125 + +17 +[21] Barnett S M and Croke S 2009 J. Phys. A: Math. and Theor. 42 062001 +[22] Bae J 2013 New J. Phys. 15 073037 +[23] Chitambar E, Leung D, Manˇcinska L, Ozols M and Winter A 2014 Commun. Math. Phys. 328 +303 +[24] Bandyopadhyay S, Cosentino A, Johnston N, Russo V, Watrous J and Yu N 2015 IEEE Trans. +Inf. Theory 61 3593 +[25] Boyd S and Vandenberghe L 2004 Convex Optimization (Cambridge University Press, Cambridge) +[26] When A and B are disjoint convex sets in a real vector space V with an inner product ⟨·, ·⟩, there +exist x ∈ R and ⃗v ∈ V such that ⃗v ̸= ⃗0 and ⟨⃗a,⃗v⟩ ⩽ x ⩽ ⟨⃗b,⃗v⟩ for all ⃗a ∈ A and all ⃗b ∈ B. +[27] Ekert A K 1991 Phys. Rev. Lett. 67 661 +[28] Bennett C H, Brassard G, Cr´epeau C, Jozsa R, Peres A and Wootters W K 1993 Phys. Rev. Lett. +70 1895 +[29] Deutsch D and Jozsa R 1992 Proc. R. Soc. Lond. A 439 553 +[30] Shor P 1994 Proceedings 35th Annual Symposium on Foundations of Computer Science (IEEE) +pp. 124–134 + diff --git a/hNE5T4oBgHgl3EQfFQ4n/content/tmp_files/load_file.txt b/hNE5T4oBgHgl3EQfFQ4n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2396ca1b2488fa8c533786ed2cd3e58c0bc570f6 --- /dev/null +++ b/hNE5T4oBgHgl3EQfFQ4n/content/tmp_files/load_file.txt @@ -0,0 +1,509 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf,len=508 +page_content='Entanglement witness and multipartite quantum state discrimination Donghoon Ha and Jeong San Kim Department of Applied Mathematics and Institute of Natural Sciences, Kyung Hee University, Yongin 17104, Republic of Korea E-mail: freddie1@khu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='kr Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We consider multipartite quantum state discrimination and show that the minimum-error discrimination by separable measurements is closely related to the concept of entanglement witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Based on the properties of entanglement witness, we establish some necessary and/or sufficient conditions on minimum-error discrimination by separable measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We also provide some conditions on the upper bound of the maximum success probability over all possible separable measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Our results are illustrated by examples of multidimensional multipartite quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Finally, we provide a systematic way in terms of EW to construct multipartite quantum state ensembles showing nonlocality in state discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Introduction Quantum state discrimination is one of the fundamental concepts used in various quantum information and computation theory [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In general, we can always perfectly discriminate orthogonal quantum states using appropriate measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' However, nonorthogonal quantum states cannot be perfectly discriminated by means of any measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For this reason, various state discrimination strategies have been studied for optimal discrimination of nonorthogonal quantum states, such as minimum-error discrimination, unambiguous discrimination and maximum-confidence discrimination [5–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Entanglement witness(EW) is an important tool to detect the existence of entanglement inherent in a multipartite quantum state [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Mathematically, EW is a Hermitian operator having non-negative mean value for every separable state, but negative for some entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' As EW provides an useful methodology to detect entanglement that is an important quantum nonlocality, it is natural to ask whether EW can also be used to characterize other nonlocal phenomenon of multipartite quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Quantum nonlocal phenomenon also arises in discriminating multipartite quantum states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' quantum nonlocality occurs when optimal state discrimination cannot be realized only by local operations and classical communication(LOCC) [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' However, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='05420v1 [quant-ph] 13 Jan 2023 2 characterizing local discrimination of quantum states is a hard task and very little is known due to the lack of good mathematical structure for LOCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Here, we establish a specific relation between the properties of EW and separable measurements, a mathematically well-structured set of measurements having LOCC measurements as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We show that the minimum-error discrimination of multipartite quantum states using separable measurements strongly depends on the existence of EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' More precisely, we establish conditions on minimum-error discrimination by separable measurements in terms of EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We also provide conditions on the upper bound of the maximum success probability over all possible separable measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We illustrate our results using examples of multidimensional multipartite quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Finally, we provide a systematic way in terms of EW to construct multipartite quantum state ensembles showing nonlocality in state discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In Section 2, we first recall the definitions and some properties about separable measurements and EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We also recall the definition of minimum-error discrimination as well as some useful properties of the optimal measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In Section 3, we provide conditions on the upper bound of the maximum success probability over all possible separable measurements (Theorems 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In Section 4, we provide conditions on minimum-error discrimination by separable measurements in terms of EW(Theorems 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Our results are illustrated by examples of multidimensional multipartite quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In Section 5, we provide a systematic way in terms of EW to construct multipartite quantum state ensembles showing nonlocality in state discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We conclude our results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Preliminaries For a multipartite Hilbert space H = �m k=1 Cdk with m ⩾ 2 and positive integers dk for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , m, let H be the set of all Hermitian operators acting on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We use H+ to denote the set of all positive-semidefinite operators in H, that is, H+ = {E ∈ H | ⟨v| E |v⟩ ⩾ 0 ∀ |v⟩ ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (1) A multipartite quantum state is expressed by a density operator ρ, that is, ρ ∈ H+ with Tr ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' A measurement is represented by a positive operator-valued measure {Mi}i, that is, {Mi}i ⊆ H+ satisfying � i Mi = 1, where 1 is the identity operator in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For the quantum state ρ, the probability of obtaining the measurement outcome corresponding to Mj is Tr(ρMj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Entanglement witness Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' E ∈ H+ is called separable if it can represented as a sum of positive- semidefinite product operators, that is, E = � l m � k=1 El,k, (2) 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' The relationship of the subsets of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' The shaded area SEP∗ \\ H+ is the set of all EWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' where {El,k}l is a set of positive-semidefinite operators acting on Cdk of H for each k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Similarly, we say that the set {Ei}i ⊆ H+ is separable if Ei is separable for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We denote the set of all separable operators in H+ as SEP = {E ∈ H+ | E : separable}, (3) and its dual set as SEP∗, that is, SEP∗ = {E ∈ H | Tr(EF) ⩾ 0 ∀F ∈ SEP}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (4) A Hermitian operator in SEP∗ is also called block positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Because 1 ∈ SEP, we have Tr E ⩾ 0 (5) for all E ∈ SEP∗, where the equality holds if and only if E is the zero operator in H [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' W ∈ H is called an EW if it is block positive but not positive semidefinite, that is, W ∈ SEP∗ \\ H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (6) For each EW W, it is known that there exists an entangled operator E ∈ H+ \\ SEP satisfying Tr(WE) < 0, (7) thus the term entanglement witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Figure 1 illustrates the relationship of the subsets of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' SEP4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Minimum-error discrimination of multipartite quantum states For a multipartite quantum state ensemble, E = {ηi, ρi}n i=1, (8) let us consider the situation that the state ρi is prepared with the probability ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' To guess the prepared state, we use the decision rule in terms of a measurement {Mi}n i=1 such that the detection of Mi means that the prepared state is guessed to be ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' The average probability of correctly guessing the prepared state from E in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (8) with respect to a measurement {Mi}n i=1 is n � i=1 ηi Tr(ρiMi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (9) The minimum-error discrimination of E is to achieve the optimal success probability, pG(E) = max Measurement n � i=1 ηi Tr(ρiMi), (10) where the maximum is taken over all possible measurements [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We note that a measurement {Mi}n i=1 provides the optimal success probability pG(E) if and only if it satisfies the following condition [19–22], n � j=1 ηjρjMj − ηiρi ∈ H+ ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (11) A measurement {Mi}i is called a separable measurement if {Mi}i ⊆ SEP, and a measurement is called a LOCC measurement if it can be realized by LOCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' It is known that every LOCC measurement is a separable measurement [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' When the available measurements are limited to separable measurements, we denote the maximum success probability by pSEP(E) = max Separable measurement n � i=1 ηi Tr(ρiMi), (12) where the maximum is taken over all possible separable measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Similarly, we denote pL(E) = max LOCC measurement n � i=1 ηi Tr(ρiMi), (13) where the maximum is taken over all possible LOCC measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' From the definitions, we trivially have pL(E) ⩽ pSEP(E) ⩽ pG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (14) 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Separable measurements and quantum state discrimination For a multipartite quantum state ensemble E in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (8), we define HSEP(E) as HSEP(E) = {H ∈ H | H − ηiρi ∈ SEP∗ ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n}, (15) where SEP∗ is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In other words, HSEP(E) is the set of all H such that H − ηiρi is block positive in H, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We further define HEW(E) = {H ∈ HSEP(E) | H − ηiρi /∈ H+ for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n}}, (16) that is, HEW(E) is the set of all H ∈ HSEP(E) such that H − ηjρj is an EW for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' From Definition 2, we can see that H ∈ HSEP(E) \\ HEW(E) (17) if and only if H − ηiρi ∈ H+ ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (18) Now, let us consider the minimum quantity qSEP(E) = min H∈HSEP(E) Tr H, (19) which is an upper bound of pSEP(E) [24], that is, pSEP(E) ⩽ qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (20) The following theorem shows that pSEP(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (12) is equal to qSEP(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a multipartite quantum state ensemble E = {ηi, ρi}n i=1, pSEP(E) = qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (21) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' As we already have Inequality (20), it is enough to show that pSEP(E) ⩾ qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (22) Let us consider the set, S(E) = � ( n � i=1 ηi Tr(ρiMi) − p, 1 − n � i=1 Mi) ∈ R × H ��� p > pSEP(E), Mi ∈ SEP ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n � , (23) where R is the set of all real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We note that S(E) is a convex set due to the convexity of SEP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Moreover, S(E) does not have the origin (0, 0H) of R × H otherwise there is a separable measurement {Mi}n i=1 with n � i=1 ηi Tr(ρiMi) > pSEP(E), (24) and this contradicts the optimality of pSEP(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Here, 0H is the zero operator in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We also note that the Cartesian product R × H can be considered as a real vector space with an inner product defined as ⟨(a, A), (b, B)⟩ = ab + Tr(AB) (25) 6 for (a, A), (b, B) ∈ R × H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Since S(E) and the single-element set {(0, 0H)} are disjoint convex sets, it follows from separating hyperplane theorem [25,26] that there is (γ, Γ) ∈ R × H satisfying (γ, Γ) ̸= (0, 0H), (26) ⟨(γ, Γ), (r, G)⟩ ⩽ 0 ∀(r, G) ∈ S(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (27) Suppose Tr Γ ⩽ γpSEP(E), (28) Γ − γηiρi ∈ SEP∗ ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n, (29) γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (30) From Conditions (29) and (30), the Hermitian operator H = Γ/γ is an element of HSEP(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, the definition of qSEP(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (19) leads us to qSEP(E) ⩽ Tr H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (31) Moreover, Condition (28) implies Tr H ⩽ pSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (32) Inequalities (31) and (32) imply Inequality (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' To complete the proof, we show the validity of (28), (29) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Proof of (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (25), Inequality (27) can be rewritten as Tr Γ − n � i=1 Tr[Mi(Γ − γηiρi)] ⩽ γp (33) for all p > pSEP(E) and all {Mi}n i=1 ⊆ SEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' If Mi = 0H for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n, Inequality (33) becomes Inequality (28) by taking the limit of p to pSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Proof of (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n}, let us consider an arbitrary Mj ∈ SEP and Mi = 0H for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In this case, {Mi}n i=1 is clearly a subset of SEP, and Inequality (33) becomes Tr Γ − Tr[Mj(Γ − γηjρj)] ⩽ γpSEP(E) (34) by taking the limit of p to pSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Suppose Γ − γηjρj /∈ SEP∗, then there is M ∈ SEP with Tr[M(Γ − γηjρj)] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We note that M ∈ SEP implies tM ∈ SEP for any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, {Mi}n i=1 consisting of Mj = tM for t > 0 and Mi = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n with i ̸= j is also a subset of SEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Now, Inequality (34) can be rewritten as Tr Γ − Tr[tM(Γ − γηjρj)] ⩽ γpSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (35) Since Inequality (35) is true for arbitrary large t > 0, γpSEP(E) can also be arbitrary large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' However, this contradicts that both γ and pSEP(E) are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, Γ − γηjρj ∈ SEP∗, which completes the proof of (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' 7 Proof of (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Suppose γ < 0 and consider {Mi}n i=1 with Mi = 0H for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Since {Mi}n i=1 ⊆ SEP, Inequality (33) becomes Tr Γ ⩽ −∞ (36) by taking the limit of p to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' This contradicts that Γ is bounded, therefore γ ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Now, let us suppose γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In this case, Conditions (28) and (29) become Tr Γ ⩽ 0, Γ ∈ SEP∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (37) From Condition (37) and the argument with Inequality (5), we have Γ = 0H, (38) which contradicts Condition (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a given ensemble E = {ηi, ρi}n i=1, the following theorem provides a necessary and sufficient condition for a separable measurement {Mi}n i=1 and H ∈ HSEP(E) to realize pSEP(E) and qSEP(E), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a multipartite quantum state ensemble E = {ηi, ρi}n i=1, a separable measurement {Mi}n i=1 and H ∈ HSEP(E), {Mi}n i=1 realizes pSEP(E) and H provides qSEP(E) if and only if Tr[Mi(H − ηiρi)] = 0 ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (39) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Let us suppose that {M}n i=1 and H give pSEP(E) and qSEP(E), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Because Mi ∈ SEP and H − ηiρi ∈ SEP∗ for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n, we have Tr[Mi(H − ηiρi)] ⩾ 0 ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (40) Moreover, we have n � i=1 Tr[Mi(H − ηiρi)] = Tr H − n � i=1 ηi Tr(ρiMi) = qSEP(E) − pSEP(E) = 0, (41) where the first equality is from �n i=1 Mi = 1, the second equality is due to the assumption of H and {Mi}n i=1, and the last equality is by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Inequality (40) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (41) lead us to Condition (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Conversely, let us assume that {Mi}n i=1 and H satisfy Condition (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' This assumption implies qSEP(E) = pSEP(E) ⩾ n � i=1 ηiTr(ρiMi) = n � i=1 ηiTr(ρiMi) + n � i=1 Tr[Mi(H − ηiρi)] = TrH ⩾ qSEP(E), (42) 8 where the first equality is by Theorem 1, the second equality is from Condition (39), the last equality follows from �n i=1 Mi = 1, and the first and second inequalities are due to the definitions of pSEP(E) and qSEP(E), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Inequality (42) leads us to n � i=1 ηiTr(ρiMi) = pSEP(E), Tr H = qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (43) Thus, {Mi}n i=1 and H provide pSEP(E) and qSEP(E), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We note that H ∈ HSEP(E) providing qSEP(E) is generally not unique (see Example 2 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' However, the following corollary states the case that H ∈ HSEP(E) providing qSEP(E) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a multipartite quantum state ensemble E = {ηi, ρi}n i=1, we have pSEP(E) = η1, (44) if and only if η1ρ1 − ηiρi ∈ SEP∗ ∀i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (45) In this case, η1ρ1 is the only element of HSEP(E) providing qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Let {Mi}n i=1 be the measurement such that M1 = 1, M2 = · · · = Mn = 0H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (46) We first assume Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (44) and consider H ∈ HSEP(E) giving qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Since {Mi}n i=1 is obviously a separable measurement providing pSEP(E), it follows from Theorem 2 that Tr(H − η1ρ1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' From H − η1ρ1 ∈ SEP∗ and the argument with Inequality (5), we have H = η1ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, H ∈ HSEP(E) together with the definition of HSEP(E) leads us to Condition (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Conversely, let us suppose Condition (45) and consider H = η1ρ1, (47) which is in HSEP(E) by Condition (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' The separable measurement {Mi}n i=1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (46) and H ∈ HSEP(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (47) satisfy Condition (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Therefore, we have pSEP(E) = n � i=1 ηi Tr(ρiMi) = η1, (48) where the first equality is by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' When Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (44) of Corollary 1 holds, the maximum success probability pSEP(E) can be achieved without the help of measurement, simply by guessing ρ1 is prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' As we can check in the proof of Corollary 1, the choice of ρ1 in Corollary 1 can be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' That is, any of {ρi}n i=1 can be used to play the role of ρ1 in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Minimum-error discrimination by separable measurements For a quantum state ensemble E in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (8), the minimum-error discrimination can be realized by separable measurements if and only if pSEP(E) = pG(E), (49) where pG(E) and pSEP(E) are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (10) and (12), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In this section, we provide some necessary and/or sufficient conditions for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (49) in terms of EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Necessary condition for pSEP(E) = pG(E) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a multipartite quantum state ensemble E = {ηi, ρi}n i=1, if there exists separable measurement {Mi}n i=1 satisfying n � i=1 ηiρiMi ∈ HEW(E), (50) where HEW(E) is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (16), then pSEP(E) = n � i=1 ηi Tr(ρiMi) < pG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (51) Thus, non-existence of such separable measurement {Mi}n i=1 satisfying Condition (50) is a necessary condition for pSEP(E) = pG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Let us suppose {Mi}n i=1 is a separable measurement satisfying Condition (50), and consider H = n � i=1 ηiρiMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (52) Equation (51) holds because n � i=1 ηi Tr(ρiMi) ⩽ pSEP(E) = qSEP(E) ⩽ Tr H = n � i=1 ηi Tr(ρiMi), (53) where the first inequality is due to the definition of pSEP(E), the second inequality follows from the definition of qSEP(E) together with the condition that H ∈ HEW(E) ⊆ HSEP(E), the first equality is from Theorem 1, and the second equality is by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' This proves the equality of (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' If we assume pSEP(E) = pG(E) in (51), {Mi}n i=1 gives the optimal success probability pG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' From the optimality condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (11) and the argument with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (17), we have n � i=1 ηiρiMi ∈ HSEP(E) \\ HEW(E), (54) which contradicts Condition (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, we have the inequality of (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' 10 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For any integers m, d ⩾ 2, let us consider the m-qudit state ensemble E = {ηi, ρi}d+2 i=1 consisting of d + 2 states, ηi = 1 dm + d, ρi = |i − 1⟩⟨i − 1|⊗m , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d, ηd+1 = dm − d dm + d, ρd+1 = 1 dm − d � 1 − d−1 � j=0 |j⟩⟨j|⊗m � , ηd+2 = d dm + d, ρd+2 = |Φ⟩⟨Φ| , (55) where |Φ⟩ = 1 √ d d−1 � i=0 |i⟩⊗m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (56) For a separable measurement {Mi}d+2 i=1 with Mi = |i − 1⟩⟨i − 1|⊗m , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d, Md+1 = 1 − d−1 � j=0 |j⟩⟨j|⊗m , Md+2 = 0H, (57) we show that Condition (50) holds with respect to the ensemble in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' It is straightforward to verify that d+2 � j=1 ηjρjMj − ηiρi = 1 dm + d(1 − |i⟩⟨i|⊗m ) ∈ H+, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d, d+2 � j=1 ηjρjMj − ηd+1ρd+1 = 1 dm + d d−1 � j=0 |j⟩⟨j|⊗m ∈ H+, d+2 � j=1 ηjρjMj − ηd+2ρd+2 = 1 dm + d(1 − d |Φ⟩⟨Φ| ) ∈ SEP∗, (58) where the last inclusion is from the fact that d Tr(|Φ⟩⟨Φ| E) ⩽ Tr E ∀E ∈ SEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (59) To show the validity of Inequality (59), we assume that {|e(k) i ⟩}d i=1 is an orthonormal basis of Cd for each k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Since {|e(1) i1 ⟩⊗· · ·⊗|e(m) im ⟩}i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=',im is an orthonormal basis of the multipartite Hilbert space H, there is a set of complex numbers {ci1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=',im}i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=',im such that |Φ⟩ = d � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=',im=1 ci1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=',im |e(1) i1 ⟩ ⊗ · · · ⊗ |e(m) im ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (60) For each i1 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d}, we have d � i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=',im=1 |ci1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=',im|2 = ⟨Φ| � |e(1) i1 ⟩⟨e(1) i1 | ⊗ ( d−1 � j=0 |j⟩⟨j| )⊗m−1� |Φ⟩ = 1 d, (61) 11 where the first equality follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (60) and the second equality is due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Equation (61) leads us to |ci1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=',im|2 ⩽ 1 d ∀i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , im = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (62) Since the choice of {|e(k) i ⟩}d i=1 can be arbitrary for each k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , m, we have d Tr(|Φ⟩⟨Φ|e⟩⟨e|) ⩽ ⟨e|e⟩ (63) for any product vector |e⟩ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Therefore, Inequality (59) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Now, the inclusions in (58) together with H+ ⊆ SEP∗ imply d+2 � j=1 ηjρjMj − ηiρi ∈ SEP∗ ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (64) Furthermore, a straightforward calculation leads us to ⟨Φ| � d+2 � j=1 ηjρjMj − ηd+2ρd+2 � |Φ⟩ = − d − 1 dm + d < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (65) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (65), we have d+2 � j=1 ηjρjMj − ηd+2ρd+2 /∈ H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (66) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (64) and (66), the ensemble in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (55) and the measurement in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (57) satisfy Condition (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, Theorem 3 leads us to pSEP(E) = d+2 � i=1 ηi Tr(ρiMi) = dm dm + d < pG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (67) We also note that the separable measurement in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (57) is a LOCC measurement because it can be implemented by performing the same local measurement {|l⟩⟨l|}d−1 l=0 on each party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, we have pL(E) = pSEP(E) = dm dm + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (68) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Necessary and sufficient condition for pSEP(E) = pG(E) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a multipartite quantum state ensemble E = {ηi, ρi}n i=1, pSEP(E) = pG(E) if and only if there exists H ∈ HSEP(E) such that it provides qSEP(E) but does not satisfy H ∈ HEW(E), (69) or equivalently, there is H ∈ H satisfying Condition (17) and Tr H = qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Let {Mi}n i=1 be a separable measurement giving pSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We first suppose pSEP(E) = pG(E) and consider H = n � i=1 ηiρiMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (70) 12 Since the measurement {Mi}n i=1 gives the optimal success probability pG(E), the optimality condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (11) leads us to H − ηiρi = n � j=1 ηjρjMj − ηiρi ∈ H+ ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (71) Therefore, H satisfies Condition (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Moreover, we have Tr H = n � i=1 ηi Tr(ρiMi) = pSEP(E) = qSEP(E), (72) where the first equality is from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (70), the second equality is by the assumption of {Mi}n i=1, and the last equality is due to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Conversely, let us assume H is an element of H satisfying Condition (17) and Tr H = qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, the positivenesses in (18) is satisfied in term of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n, the positive-semidefinite operators Mi and H − ηiρi are orthogonal since they satisfy Condition (39) from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' The optimality condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (11) holds for the measurement {Mi}n i=1 because n � j=1 ηjρjMj − ηiρi = n � j=1 ηjρjMj + n � k=1 (H − ηkρk)Mk − ηiρi = H − ηiρi ∈ H+ ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n, (73) where the first equality is from the orthogonality of Mi and H −ηiρi for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n and the second equality is from �n i=1 Mi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, we have pG(E) = n � i=1 ηi Tr(ρiMi) = pSEP(E), (74) where the second equality is due to the assumption of {Mi}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' If pSEP(E) = pG(E), Theorem 4 implies that there must exist H ∈ HSEP(E)\\HEW(E) providing qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' In this case, there possibly exists another Hermitian operator H′ satisfying H′ ∈ HEW(E) and Tr H′ = qSEP(E), which is illustrated in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For any integers m, d ⩾ 2, let us consider the m-qudit state ensemble E = {ηi, ρi}dm−d+1 i=1 consisting of dm − d + 1 states, η1 = d dm, ρ1 = |Φ⟩⟨Φ| , ηi = 1 dm, ρi = |βi⟩⟨βi| , i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , dm − d + 1, (75) where |Φ⟩ is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (56) and {|βi⟩}dm−d+1 i=2 is a set of orthonormal product vectors orthogonal to |j⟩⊗m for all j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a separable measurement {Mi}dm−d+1 i=1 with M1 = d−1 � j=0 |j⟩⟨j|⊗m , Mi = |βi⟩⟨βi| , i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , dm − d + 1, (76) 13 we can easily see that the success probability obtained from the separable measurement in discriminating the states from the ensemble E in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (75) is one, that is, dm−d+1 � i=1 ηi Tr(ρiMi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (77) The success probability in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (77) is a lower bound of pSEP(E) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (12), therefore pSEP(E) ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (78) Since pG(E) is bounded above by 1, Inequalities (14) and (78) imply pSEP(E) = pG(E) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (79) Furthermore, we have qSEP(E) = pSEP(E) = 1, (80) where the first equality is by Theorem 1 and the second equality is from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Let us first consider the Hermitian operator H = dm−d+1 � i=1 ηiρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (81) Equations (80) and (81) imply Tr H = 1 = qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (82) Moreover, a straightforward calculation leads us to H − ηiρi = dm−d+1 � j=1 j̸=i ηjρj ∈ H+ ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (83) Due to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (82) and (83), H is an element of HSEP(E) \\ HEW(E) giving qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, we have H satisfying the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Now, let us consider the Hermitian operator Ht = t dm−d+1 � i=1 ηiρi + 1 − t dm 1, (84) where 0 ⩽ t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Equations (80) and (84) imply Tr Ht = 1 = qSEP(E) (85) for 0 ⩽ t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Moreover, a straightforward calculation leads us to Ht − η1ρ1 = t dm−d+1 � j=2 ηjρj + 1 − t dm (1 − d |Φ⟩⟨Φ|) ∈ SEP∗, Ht − ηiρi = t dm−d+1 � j=1 j̸=i ηjρj + 1 − t dm (1 − |βi⟩⟨βi|) ∈ H+ ∀i ̸= 1, (86) 14 where the first inclusion follows from Inequality (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (86) together with H+ ⊆ SEP∗, we have Ht ∈ HSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (87) We also note that ⟨Φ| (Ht − η1ρ1) |Φ⟩ = −(1 − t)(d − 1) dm < 0 (88) for 0 ⩽ t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Equations (85), (87) and Inequality (88) imply that Ht is an element of HEW(E) giving qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' By letting H′ = Ht, we have another Hermitian operator H′, besides H in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (81), satisfying H′ ∈ HEW(E) and Tr H′ = qSEP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a multipartite quantum state ensemble E = {ηi, ρi}n i=1 where H is the only element of HSEP(E) providing qSEP(E), Theorem 4 tell us that pSEP(E) < pG(E) if and only if there exists an EW in {H − ηiρi}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' From Corollary 1, η1ρ1 is the only element of HSEP(E) providing qSEP(E) when Condition (45) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a multipartite quantum state ensemble E = {ηi, ρi}n i=1 with Condition (45), pSEP(E) < pG(E) if and only if there exists an EW in {η1ρ1 − ηiρi}n i=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For any integers m, d ⩾ 2, let us consider the m-qudit state ensemble E = {ηi, ρi}d+1 i=1 consisting of d + 1 states, η1 = 1 2, ρ1 = 1 dm1, ηi = 1 2d, ρi = d2 − d dm − d |Φi⟩⟨Φi| + dm − d2 dm(dm − d)1, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d + 1, (89) where |Φj⟩ = 1 √ d d−1 � k=0 exp �i2πjk d � |k⟩⊗m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (90) For each i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d + 1, a straightforward calculation leads us to η1ρ1 − ηiρi = d − 1 2d(dm − d)(1 − d |Φi⟩⟨Φi|) ∈ SEP∗, (91) where the inclusion is from the fact that d Tr(|Φi⟩⟨Φi| E) ⩽ Tr E ∀E ∈ SEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (92) We can show the validity of Inequality (92) in a similar way to that of Inequality (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Now, the inclusion in (91) together with Corollary 1 imply pSEP(E) = η1 = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (93) Furthermore, a straightforward calculation leads us to ⟨Φi| � η1ρ1 − ηiρi � |Φi⟩ = − (d − 1)2 2d(dm − d) < 0 ∀i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (94) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (94), we have η1ρ1 − ηiρi /∈ H+ ∀i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (95) 15 From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (91) and (95), η1ρ1−ηiρi is an EW for any i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, Corollary 2 leads us to pSEP(E) = 1 2 < pG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (96) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Construction of nonlocal quantum state ensemble In this section, we provide a systematic way in terms of EW to construct multipartite quantum state ensembles showing nonlocality in state discrimination, that is, pL(E) < pG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a given EW W, let us consider the multipartite quantum state ensemble E = {ηi, ρi}2 i=1 where η1 = Tr(P + W) Tr(2P + W), ρ1 = P + W Tr(P + W), η2 = Tr P Tr(2P + W), ρ2 = P Tr P , (97) with any P ∈ H+ satisfying P + W ∈ H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (98) Since η1ρ1−η2ρ2 is proportional to the EW W, pSEP(E) < pG(E) holds from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, Inequality (14) leads us to pL(E) < pG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Corollary 2 can also be used to construct a multipartite quantum state ensemble E = {ηi, ρi}n i=1 with n > 2 showing nonlocality in quantum state discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' For a set of EWs {Wi}n i=2, let us consider the multipartite quantum state ensemble E = {ηi, ρi}n i=1 where η1 = Tr 1 Tr(n1 − �n j=2 λjWj), ρ1 = 1 Tr 1, ηi = Tr(1 − λiWi) Tr(n1 − �n j=2 λjWj), ρi = 1 − λiWi Tr(1 − λiWi), i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n, (99) with any set of positive real numbers {λi}n i=2 satisfying 1 − λiWi ∈ H+ ∀i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' (100) Because η1ρ1 − ηiρi is proportional to Wi for any i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' , n}, pSEP(E) < pG(E) holds from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Thus, Inequality (14) leads us to pL(E) < pG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Conclusions We have considered multipartite quantum state discrimination and shown that the minimum-error discrimination by separable measurements strongly depends on the existence of EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We have established the necessary and/or sufficient conditions on minimum-error discrimination by separable measurements, that is, pSEP(E) = pG(E), in terms of EW (Theorems 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' We have also provided the conditions on the upper bound of the maximum success probability over all possible separable measurements (Theorems 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Our results have been illustrated by examples of multidimensional 16 multipartite quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Finally, we have provided a systematic way in terms of EW to construct multipartite quantum state ensembles showing nonlocality in state discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Quantum nonlocality is a key ingredient making quantum states outperform the classical ones in various quantum information processing tasks such as quantum teleportation and quantum cryptography [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' It is also known that quantum nonlocality plays an important role in quantum algorithms which are more powerful than any classical ones [29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' As the violation of the conditions in Theorem 4 implies pSEP(E) < pG(E), which consequently means pL(E) < pG(E), our results provides a useful methodology to guarantee the occurrence of nonlocality in state discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Our results establish a specific relation between the properties of EW and minimum- error discrimination by separable measurements, therefore it is natural to investigate the relationship between EW and other measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' It is also an interesting future work to construct good conditions, in terms of EW, for optimal state discrimination in other state discrimination strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Acknowledgments This work was supported by Basic Science Research Program(NRF-2020R1F1A1A010501270) and Quantum Computing Technology Development Program(NRF-2020M3E4A1080088) through the National Research Foundation of Korea(NRF) grant funded by the Korea government(Ministry of Science and ICT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' References [1] Chefles A 2000 Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE5T4oBgHgl3EQfFQ4n/content/2301.05420v1.pdf'} +page_content=' 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Jeschke4, 3 +1Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster of Excellence ct.qmat, +Universit¨at W¨urzburg, Am Hubland Campus S¨ud, W¨urzburg 97074, Germany +2Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster of Excellence ct.qmat, +Julius-Maximilians-Universit¨at W¨urzburg, Am Hubland Campus S¨ud, W¨urzburg 97074, Germany +3Department of Physics and Quantum Centers in Diamond and Emerging Materials (QuCenDiEM) group, +Indian Institute of Technology Madras, Chennai 600036, India +4Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan +As a highly frustrated model Hamiltonian with an exact dimer ground state, the Heisenberg +antiferromagnet on the maple leaf lattice is of high theoretical interest, and a material realization +is intensely sought after. We determine the magnetic Hamiltonian of the copper mineral bluebellite +using density functional theory based energy mapping. As a consequence of the significant distortion +of the spin S = 1/2 maple leaf lattice, we find two of the five distinct nearest neighbor couplings to be +ferromagnetic. Solution of this Hamiltonian with density matrix renormalization group calculations +points us to the surprising insight that this particular imperfect maple leaf lattice, due to the strongly +ferromagnetic Cu2+ dimer, realizes an effective S = 1 breathing kagome Hamiltonian. In fact, this is +another highly interesting Hamiltonian which has rarely been realized in materials. Analysis of the +effective model within a bond-operator formalism allows us to identify a valence bond solid ground +state and to extract thermodynamic quantities using a low-energy bosonic mean-field theory. We +resolve the puzzle of the apparent one-dimensional character of bluebellite as our calculated specific +heat has a Bonner-Fisher-like shape, in good agreement with experiment. +Triangular motifs in quantum antiferromagnets are +a source of geometric frustration and lead to highly +nontrivial +emergent +phenomena +like +quantum +spin +liquids [1]. +Starting with the triangular lattice, site +depletion leads to new lattices which have lower coor- +dination number but potentially more frustration [2, 3]. +For example, +the kagome lattice is obtained by a +1/4 site-depletion of the triangular lattice and has +coordination number of four; it also hosts some of the +most intensively studied spin liquid candidates [4]. +In +a somewhat more exotic manner, the maple leaf lattice +can be viewed as a one-seventh site-depleted triangular +lattice and has a coordination number of five [5]. The +uniform nearest-neighbor Heisenberg antiferromagnetic +model on this lattice has been solved via exact diag- +onalization and other techniques [6, 7] and found a +magnetically ordered ground state. +Recently, in an +analytic work on the model with a bond anisotropy, +Ghosh et. +al. +established an exact dimer ground +state [8], making the maple leaf lattice the only other +two-dimensional lattice with uniform tiling that admits +an exact dimer ground state besides the widely known +Shastry-Sutherland model [9] which has an extremely +rich phenomenology and phase diagram [10–13]. While +SrCu2(BO3)2 has been found to be an extremely good +representation of the Shastry-Sutherland Hamiltonian, a +material realizing the model proposed by Ghosh et. al. +on the maple leaf lattice has yet to be identified. Can- +didates involving quantum spins are the copper miner- +als [14, 15] spangolite Cu6Al(SO4)(OH)12Cl · 3 H2O [16], +sabelliite +Cu2ZnAsO4(OH)3 +[17], +mo- +javeite +Cu6TeO4(OH)9Cl +[18], +fuetterite +Pb3Cu6TeO6(OH)7Cl5 +[19] +and +finally +bluebellite +Cu6IO3(OH)10Cl [18]. +Magnetic properties have been +characterized experimentally for spangolite [20] and +bluebellite [21], but the magnetic Hamiltonians for any +of these maple leaf compounds remains to be established. +Here, we focus on bluebellite and resolve the most +pressing issues for this prime example of a maple leaf +antiferromagnet: we determine all relevant exchange in- +teractions and solve the resulting Hamiltonian employ- +ing numerical and semi-analytical techniques. +In par- +ticular, we address the question raised by experiment: +why do the susceptibility and specific heat of bluebel- +lite appear to have a Bonner-Fisher type shape, sug- +gestive of one-dimensional systems? An answer to this +question based on order-by-disorder was attempted with- +out knowledge of the Hamiltonian [22]. +Methodolog- +ically, we apply the energy mapping technique which +has proved valuable in extracting the Hamiltonian cou- +plings for many important quantum spin systems [23– +26]. By virtue of a statistical approach and by extracting +more than the apparently important exchange interac- +tions, this method has led to surprising insights and met +with much success for many materials [27–33]. We then +perform density matrix renormalization group (DMRG) +calculations [34] on the resulting maple leaf Hamilto- +nian. +This method has been instrumental in further- +ing the comprehension of the physics of the kagome lat- +tice antiferromagnet [35, 36]. Our DMRG calculations +show that bluebellite has a gapped valence bond solid +ground state. In order to deepen our understanding, we +develop a low-energy theory by implementing the stan- +dard bond operator formalism [37]. This theory permits +one to perform calculations directly in the thermody- +namic limit, obtain static and dynamical structure fac- +arXiv:2301.05224v1 [cond-mat.str-el] 12 Jan 2023 + +2 +−100 +−50 + 0 + 50 + 100 + 150 + 4 + 5 + 6 + 7 + 8 +−30 +−20 +−10 + 0 +bluebellite Cu6IO3(OH)10Cl +U (eV) +Ji (K) +θCW (K) +θCW=−34.7 K +(b) +(a) +(c) +(d) +(e) +J1 +J2 +J3 +J4 +J5 +J6 +J7 +J8 +θCW +FIG. 1. (a) DFT based energy mapping: First eight exchange interactions of bluebellite as function of on-site interaction +strength U, at fixed JH = 1 eV. The vertical line indicates the U value at which the Heisenberg Hamiltonian parameters yield +the experimentally observed Curie-Weiss temperature [21]. (b) bluebellite structure with the five ”nearest neighbor” exchange +paths defining the maple leaf lattice. The lattice vectors are given by a1 = +√ +7/2(ˆx + +√ +3ˆy) and a2 = +√ +7ˆx. (c) Effective S = 1 +breathing kagome model with renormalized interactions. (d) Spin-spin correlations for all nearest-neighbor bonds obtained +from DMRG on a 108 site maple-leaf cluster. The thickness of the bonds indicates the strength of the correlation and the color +red (blue) indicates positive (negative) correlation. Note the clear dimerization in the ground state. (e) Enlargement from (d) +with values of spin-spin correlations. +tors, and assess the behavior of thermodynamic quanti- +ties. +The analytical and numerical calculations reveal +that the bluebellite magnetic interactions very closely +emulate an effective S = 1 kagome Hamiltonian with +a strong breathing anisotropy. So far, the possible S = 1 +kagome candidates, e.g., KV3Ge2O9 [38], NaV6O11 [39], +m–MPYNN · BF4 [40], all undergo lattice distortions at +low temperatures. +By establishing the connection be- +tween maple leaf and kagome, our work paves the path- +way to possible realizations and synthesis of new effec- +tive S = 1 kagome compounds emerging out of S = 1/2 +maple-leaf systems. As an experimental outlook, this en- +ables the study of integer-spin highly frustrated kagome +antiferromagnets, notably magnetization plateaus, exci- +tations, and topological properties. +Model Hamiltonian.- Before we could extract the pa- +rameters of the Heisenberg Hamiltonian H = � +i 0, then q(n) +0:C−1 can be +solved as: +q(n) +0 += +1 +s(n) +C +( ¯S(n) − ¯Y (n)) +C−1 +� +i=1 +z∗ +i +z∗ +i − 1, +(20) +and +q(n) +0:C−1 = ˜η(n)(Λ(n))−1, +(21) + +Mo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +19 +where ˜η(n) = [s(n) +C q(n) +0 η(n) +0 ,s(n) +C q(n) +0 η(n) +1 ,...,s(n) +C q(n) +0 η(n) +C−1] ∈ RC and +Λ(n) = +� +������������������ +s(n) +C +s(n) +C−1 s(n) +C−2 ... s(n) +1 +0 +s(n) +C +s(n) +C−1 ... s(n) +2 +... +0 +s(n) +C +... s(n) +3 +0 +... +0 +... s(n) +4 +0 +0 +... +... +... +0 +0 +0 +... s(n) +C +� +������������������ +∈ RC×C. +(22) +η(n) +j +is the polynomial coefficient of zj in �C−1 +i=0 +� +1 − z +z∗ +i +� +(i.e., �C +j=0 η(n) +j zj := �C−1 +i=0 +� +1 − z +z∗ +i +� +). As +z∗ +i is specified for station n, a superscript n is added to the coefficients. +Note that assuming s(n) +C > 0 in Proposition 4 is not restrictive because otherwise we can reduce +C such that s(n) +C > 0 always holds. +4.3.3. Analytical formulation of mean and variance of queue length and waiting +time +After solving for q(n) +0 ,...,q(n) +C−1, Q(z) is determined. The expectation and variance of the +queue length at station n can be written by definition as: +E[Q(n)] = +∞ +� +k=0 +kq(n) +k += dQ(z) +dz +���� +z=1 +(23) +Var[Q(n)] = E[(Q(n))2] − E[Q(n)]2 = d2Q(z) +dz2 +���� +z=1 ++ E[Q(n)] − E[Q(n)]2. +(24) +Proposition 5. +� +Mean and variance of queue length +� +: ∀ n = 1,..,N, given the distribution of S(n) +and the expression of Y (z), E[Q(n)] and Var[Q(n)] can be calculated as: +E[Q(n)] = +¯¯S(n) + ¯¯Y (n) + ( ¯S(n) − ¯Y (n))[1 + 2( ¯S(n) − C)] − ( ¯S(n) − ¯Y (n))2 +2( ¯S(n) − ¯Y (n)) ++ +C−1 +� +i=1 +1 +1 − z∗ +i +(25) +Var[Q(n)] = +1 +12( ¯S(n) − ¯Y (n))2 +� +− 4( ¯¯¯S(n) − ¯¯¯Y (n))( ¯S(n) − ¯Y (n)) + 3( ¯¯S(n) + ¯¯Y (n))2 +− [6( ¯¯S +(n) − ¯¯Y +(n)) − 1]( ¯S(n) − ¯Y (n))2 − ( ¯S(n) − ¯Y (n))4 +� +− +C−1 +� +i=1 +z∗ +i +(1 − z∗ +i )2 +(26) +where ¯¯S(n) and ¯¯¯S(n) (resp. ¯¯Y (n) and ¯¯¯Y (n)) are the second and third central moments of S(n) (resp. +Y (n)). + +Mo et al.: Evaluation of Public Transit Systems +20 +Article submitted to ; manuscript no. () +Proposition 6. +� +Mean and variance of waiting time +� +: ∀ n = 1,..,N, given the distribution of S(n) +and the expression of Y (z), the mean and variance of waiting time at station n (denoted as W (n)) +is given as: +E[W (n)] = +¯Q(n) +t +λ(n) +(27) +Var[W (n)] = +¯¯Q(n) +t +− ¯Q(n) +t +(λ(n))2 +(28) +where Q(n) +t +is the queue length at an arbitrary time point (as opposed to Q(n) which is the queue +length at the time of vehicle arrival). ¯Q(n) +t +and ¯¯Q(n) +t +are defined as +¯Q(n) +t += E[Q(n)] − ¯Y (n) + 1 +2 +� ¯¯Y (n)/ ¯Y (n) + ¯Y (n) − 1 +� +(29) +¯¯Q(n) +t += Var[Q(n)] − ¯¯Y (n) + +1 +12( ¯Y (n))2 +� +4 ¯Y (n) ¯¯¯Y (n) + 6( ¯Y (n))2 ¯¯Y (n) − ( ¯Y (n))2 + ( ¯Y (n))4 − 3( ¯¯Y (n))2� +(30) +Eq. 27 is the application of Little’s law. Proposition 6 is directly obtained from Powell (1985). +Remark 1. The formulation of E[Q(n)], Var[Q(n)], E[W (n)], and Var[W (n)] in this study are equiv- +alent to Powell (1985) because in his paper the M/G[S]/1 bulk queue model was considered, where +G[S] represents a general (i.e., arbitrary) bulk-service distribution, which includes the service dis- +tribution incorporating random service suspension considered in this study. However, this does not +lower the contribution of this paper because to implement these equations, the formulation of Y (z) +needs to be specified. In the next section, we show how random service suspension introduces a +new distribution for Y (n), which has not been considered in the literature. +4.3.4. Headway distribution +According to Propositions 4 to 6, to calculate q(n) +0:C−1 and the +mean and variance of queue length and waiting time, it is essential to specify Y (z) (i.e., the PGF +of the number of passengers arriving within a headway). According to Eq. 4, taking l → ∞ gives +that Y (n)|H(n) is a Poisson random variable with parameter λ(n)H(n) . Therefore, we first consider +the distribution of H(n) under the random service suspension. +According to the discussion in Section 3.3, the actual headway for vehicle l at station n is +H(n,l) = ¯H + 2·E[I(N,l)] +¯ +F ++ I(n,l) − I(n,l−1), where I(n,l) is the total duration of incidents for vehicle l + +Mo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +21 +during its travel from the transportation hub to station n. Since ¯H and E[I(n,l)] are constants, +obtaining the headway distribution is equivalent to quantifying the distribution of I(n,l) − I(n,l−1). +Notice that I(n,l) and I(n,l−1) are i.i.d for all l by our assumption. It is useful to first consider the +distribution of I(n,l). +Proposition 7. +� +Distribution of incident duration +� +: The total incident duration for vehicle l dur- +ing its travel from the transportation hub to station n (i.e., I(n,l)) follows a compound Poisson- +Exponential distribution with Poisson rate γT (n) and exponential rate θ. Mathematically, +I(n,l) = +K +� +i=1 +Xi +where Xi ∼ Exp(θ) ∀i = 1,...,K, and K ∼ Poi(γT (n)) +(31) +The moment generating function (MGF) of a compound Poisson-Exponential variable can be +written as (Hogg et al. 2010) +MI(n,l)(t) = E[etI(n,l)] = eγT (n)( +θ +θ−t −1) +∀ t < θ +(32) +Similarly, the MGF of −I(n,l−1) is +M−I(n,l−1)(t) = E[e−tI(n,l−1)] = eγT (n)( +θ +θ+t −1) +∀ t > −θ +(33) +From the MGF of I(n,l), we obtain E[I(N,l)] = γT (N) +θ +. Then the headway equation (Eq. 12) becomes +H(n,l) = ¯H + 2γT (N) +θ ¯F ++ I(n,l) − I(n,l−1) +(34) +The following proposition provides the headway distribution: +Proposition 8. +� +MGF of headway +� +: Under the setting of this study, ∀ n = 1,..,N, the MGF of +H(n) can be expressed as +MH(n)(t) = et( ¯ +H+ 2γT (N) +θ ¯ +F +)e +γT (n)( +2t2 +θ2−t2 ) +(35) +From the MGF of H(n), we can obtain the corresponding mean and variance of headway as: +E[H(n)] = ¯H + 2γT (N) +θ ¯F +(36) +Var[H(n)] = 4T (n)γ +θ2 +(37) + +Mo et al.: Evaluation of Public Transit Systems +22 +Article submitted to ; manuscript no. () +Remark 2. The results show that random suspensions can increase the mean and variance of +headway. The impact on mean headway is through the increase in cycle time at the route planning +stage. The headway variance will increase with a higher incident rate (γ) and higher average +incident duration ( 1 +θ). Meanwhile, our model also captures the headway variance propagation along +stations as observed in many previous studies (Andersson and Scalia-Tomba 1981, Hickman 2001): +Var[H(n)] increase with the station index n (due to the increase in T (n)). +However, the support of the derived headway distribution is R, meaning that H(n) can be negative +due to the overtaking of vehicles. The negative value of H(n) will cause problems in the definition of +Y (n) (i.e., the number of arrival passengers within a headway). To address this problem, we assume +that drivers are not allowed to overtake the preceding vehicles. This is true for the subway systems. +Many transit agencies also use this policy for bus operations. Given this assumption, the support +of H(n) becomes [0,+∞]. Whenever H(n) < 0, the actual headway will be 0 since the successor +vehicle will not pass through the predecessor and they will arrive at the station simultaneously (i.e., +bus bunching). Hence, the new truncated headway, denoted as ˆH(n), has a zero-inflation mixture +distribution: +ˆH(n) = +� +� +� +� +� +� +� +0 +if H(n) ≤ 0 +H(n) +otherwise +(38) +The zero-inflation truncated headway distribution is also observed in the previous empirical study +assuming no overtaking (Bellei and Gkoumas 2010). +However, to the best of the author’s knowledge, there is no closed-form MGF for ˆH(n) because the +difference between two compound Poisson-exponential random variables has no closed-form proba- +bility density function. Therefore, to have a tractable headway distribution, we have to approximate +H(n) with other distributions for which the corresponding zero-inflation truncated distribution has +analytical MGF. +In this study, we approximate the distribution of H(n) with normal distribution for two reasons: +1) I(n,l) can be seen as the summation of a large number of i.i.d random variables (Eq. 31) when the + +Mo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +23 +incident frequency is high (i.e., K is large, which is true for this study because we are considering +high-frequency short random disturbance). Hence, from the Central Limit Theorem (CLT), we may +approximate I(n,l) as normally distributed, which leads to H(n) being normally distributed as well. +Approximating the headway disturbance as a normal random variable with the CLT was also used +in Daganzo (2009). 2) After approximating H(n) as a normal random variable (denoted as H(n) +Normal) +with the same mean and variance, the first three moments of H(n) and H(n) +Normal are the same. This +shows that the distribution of H(n) is similar to normal. Appendix J shows detailed derivations +and numerical experiments to validate the approximation. +Now let us consider a zero-inflation truncated distribution of H(n) +Normal with support [0,+∞] and +a probability mass concentrated at zero. Denote the truncated random variable as ˆH(n) +Normal. +Proposition 9. +� +MGF of approximated headway +� +: Under the setting of this study, ∀ n = 1,..,N, +the MGF of ˆH(n) +Normal can be expressed as +M ˆ +H(n) +Normal(t) = Φ +� +−( ¯Hθ + 2γT (N) +¯ +F +) +2 +� +T (n)γ +� ++ et( ¯ +H+ 2γT (N) +θ ¯ +F +)eγT (n)( 2t2 +θ2 ) +� +1 − Φ +� +−( ¯Hθ + 2γT (N) +¯ +F +) +2 +� +T (n)γ +− 2t +� +T (n)γ +θ +�� +(39) +where Φ(·) is the cumulative density function (CDF) of a standard normal distribution. +Based on the MGF of ˆH(n) +Normal, notice that +� +1 − Φ +� −µ +σ +�� += Φ +� µ +σ +� +, we can get the corresponding +mean and variance as follows. +E[ ˆH(n) +Normal] = µ · Φ +�µ +σ +� ++ σ · φ +�−µ +σ +� +(40) +Var[ ˆH(n) +Normal] = µσφ +�−µ +σ +� ++ Φ +�µ +σ +�� +µ2 + σ2� +− +� +µΦ +�µ +σ +� ++ φ +�−µ +σ +� +σ +�2 +(41) +where φ(·) is the probability density function (PDF) of a standard normal distribution. It is not +clear how incidents will affect the mean headway from Eq. 40 directly. However, the following +proposition shows that the mean headway increases as incident frequency (γ) and average incident +duration ( 1 +θ) increase. + +Mo et al.: Evaluation of Public Transit Systems +24 +Article submitted to ; manuscript no. () +Proposition 10. +� +Impact of incidents on headway +� +: The mean of the zero-inflation truncated +headway (i.e, either E[ ˆH(n)] or E[ ˆH(n) +Normal]) increases with the increase in incident intensity (i.e., +increase in γ or 1 +θ, or both). +Proposition 10 is useful for the analysis of system stability with respect to incidents, which is +shown in Section 4.4. +4.3.5. Distribution of Y (n) +The distribution of Y (n) is derived by assuming the headway is +ˆH(n) +Normal (instead of H(n), which may be negative). To derive the PGF of Y (n), the following lemma +is introduced. +Lemma 1. For two arbitrary random variable U and V , assume that +• there is a δ > 0 such that for t in (−δ,δ), the MGF of U|V is MU|V (t) = C1(t)eC2(t)V , where +C1(t) and C2(t) are finite functions of t that do not depend on V , +• and the MGF of V , MV (·), exists and MV [C2(t)] is finite for t in (−δ,δ). +Then the MGF of U is given by +MU(t) = C1(t)MV [C2(t)], +−δ < t < δ. +(42) +The proof of Lemma 1 can be found in Villa and Escobar (2006) Result 1. +Proposition 11. +� +PGF of Y (n)� +: Under the setting of this study, ∀ n = 1,..,N, the PGF of Y (n), +Y (z), can be expressed as +Y (z) = Φ +�−µ +σ +� ++ eµλ(n)(z−1)+ σ2(λ(n)z−λ(n))2 +2 +� +1 − Φ +�−µ +σ − σλ(n)(z − 1) +�� +(43) +where µ = ¯H + 2γT (N) +θ ¯ +F +and σ = +2√ +T (n)γ +θ +are the mean and standard deviation of H(n) +Normal, respectively. +From Eq. 105, we can obtain ¯Y (n), ¯¯Y (n), and ¯¯¯Y (n) by taking corresponding derivatives. The +expression of ¯Y (n) is shown below. The expressions for ¯¯Y (n) and ¯¯¯Y (n) are complicated and thus +omitted. +¯Y (n) = +� +µ · Φ +�µ +σ +� ++ σ · φ +�−µ +σ +�� +· λ(n) +(44) + +Mo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +25 +4.3.6. Solving for the roots +With the expression of Y (z), the only unknown parts for the +queue length calculation (Eq. 25) are z∗ +0,...,z∗ +C−1, which can be obtained by solving the nonlinear +equation Den(z) = 0 (see Section 4.3 for details). It is well known in the queuing literature that +solving for the roots of Den(z) is practically difficult because typical optimization algorithms +usually only find only one root, while we need to find all C roots within the unit circle. This +is especially changeling for Y (z) with complex expressions because the objective function can be +highly nonlinear (such as Y (z) in this study). +In this study, we propose an interpolation-based searching algorithm to efficiently find all roots +of Den(z) within the unit circle. The key idea is to intelligently set up different initial values for a +general root-solving algorithm (such as trust-region and Levenberg-Marquardt algorithms) and let +the algorithm converge to different solutions (i.e., different roots). We elaborate on the algorithm +details in Appendix L. The numerical testing shows that our algorithm is able to find desired roots +for all testing scenarios in Section 5.1. It outperforms the methods in Powell (1985) and Wilson +(2014), where both of them have cases of not being able to find all roots. +4.4. Stability condition +For all the derivations above, we assume that the steady-state distributions of all variables exist. +This triggers the discussion about the stability condition. At the station level, the stability condition +is described in Proposition 12. +Proposition 12. +� +Stability condition +� +: Under the setting of this study, the bulk-service queuing +system at station n is stable if and only if +ρ(n) = +¯Y (n) +¯S(n) = +� +µ · Φ +� µ +σ +� ++ σ · φ +� −µ +σ +�� +· λ(n) +�C +u=0 s(n) +u u += λ(n) · E[ ˆH(n) +Normal] +�C +u=0 s(n) +u u +< 1 +(45) +where ρ(n) is the utilization ratio for station n. +Proposition 12 is intuitive as it indicates that station n is stable if the average number of +passengers arrived within a headway is smaller than the average available capacity for each arrival +vehicle (after alighting). From Proposition 7, we know that a higher rate of incidents (i.e., larger + +Mo et al.: Evaluation of Public Transit Systems +26 +Article submitted to ; manuscript no. () +γ) and higher duration of incidents (i.e., higher 1 +θ) increase E[ ˆH(n) +Normal], which makes the system +more likely to be unstable. Hence, the above result quantifies the throughput loss due to incidents. +There are some remarks for Proposition 12. +Remark 3. As ρ(n) depends on s(n) and s(n) depends on the roots (i.e., z∗ +0,...,z∗ +C−1) at station +n, there is no direct way to judge the stability at station n without iterating the previous n − 1 +stations. But for the first station (n = 1), we have s(1) +C = 1 and s(1) +u = 0 for all u = 0,...,C − 1. Then +Eq. 45 reduces to ρ(1) = +λ(n)·E[ ˆ +H(n) +Normal] +C +, which can be used to assess the stability directly. +Remark 4. Proposition 12 only discusses the stability at the station level. At the route level, a +route is considered stable if “all stations in the route are stable”. Mathematically, a route is stable +if and only if ρ(n) < 1,∀ n = 1,2,...,N. +Remark 5. It is worth discussing the relationship of stability of stations n and n − 1. If station +n−1 is stable, then s(n) can be calculated as described in Section 4.1, and the stability of station n +can be evaluated accordingly. However, if station n−1 is not stable, station n may be stable because +there may be passengers alighting at station n. For this situation, we have v(n−1) +C += 1 and v(n−1) +k += 0 +for all k = 0,1,...,C − 1. Then s(n) is determined by the alighting rate at station n. It is easy to +verify that in this situation ¯S(n) = α(n)C. And the stability condition is ρ(n) = +λ(n)·E[ ˆ +H(n) +Normal] +α(n)C +< 1. +4.5. Summary of calculation procedure +So far, we have derived the calculation process for key variables of interest. Algorithm 1 summarizes +the calculation procedure, which iterates through the N stations of the route. This is more efficient +and provides more analytical insights than a simulation model. +5. Numerical example +5.1. Experimental design +To test the proposed framework, we use an example bus route adapted from Islam et al. (2015) +and Hickman (2001). There are 10 stations and the attributes for each station are shown in Table + +Mo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +27 +Algorithm 1 Performance indicators calculation procedure +1: Initialize v(0) +0 += 1 and v(0) +k += 0 +∀k = 1,...C. +2: for n = 1 : N do +3: +g(n) = v(n−1)A(n) +▷ Eq. 14 +4: +s(n) +k += 1 − g(n) +C−k +∀k = 0,1,...,C +▷ Eq. 13 +5: +Calculate ¯S(n), ¯¯S(n), and ¯¯¯S(n) based on s(n). +6: +Calculate ¯Y (n), ¯¯Y (n), and ¯¯¯Y (n) +▷ Section 4.3.4 +7: +if ¯Y (n) < ¯S(n) then +▷ Station n is stable +8: +Solve the roots z∗ +0,...,z∗ +C−1 for the denominator of Q(z) in Eq. 18 +▷ Section 4.3.1 +9: +Calculate q(n) +0 ,...,q(n) +C−1 based on z∗ +0,...,z∗ +C−1 +▷ Section 4.3.2 +10: +Calculate E[Q(n)],Var[Q(n)],E[W (n)], and Var[W (n)] +▷ Eq. 25 - 28 +11: +v(n) = g(n)B(n) +▷ Eq. 16. B(n) is a function of q(n) +0:C−1 +12: +else +▷ Station n is not stable +13: +q(n) +k += 0 +∀k = 0,1,...,C − 1 +14: +Set E[Q(n)],Var[Q(n)],E[W (n)], and Var[W (n)] to infinity +15: +v(n) +C = 1 and v(n) +k += 0 +∀k = 0,1,...,C − 1 +1. The layout of the bus route is shown in Figure 6, where we assume the no-incident travel time +between two consecutive stations is 5 minutes, the total cycle time without incident is ¯E = 100 +min, and travel time from the transportation hub to the last station is T (N) = 50 minutes. +Table 1 +Example bus system parameters +Station ID λ(n) (passengers/min) α(n) +Station ID λ(n) (passengers/min) α(n) +1 +0.75 +0 +6 +1 +0.8 +2 +1.5 +0 +7 +0.75 +0.5 +3 +0.75 +0.1 +8 +0.5 +0.1 +4 +3 +0.25 +9 +0.2 +0.75 +5 +1.5 +0.25 +10 +0 +1 + +Mo et al.: Evaluation of Public Transit Systems +28 +Article submitted to ; manuscript no. () +Figure 6 +Case study route layout +To test the sensitivity of performance indicators to different parameters, we consider different +values of C, θ, γ, ¯H, and demand (Table 2). The demand is adjusted by a scaling factor that is +applied to the arrival rates λ(n) in Table 1. The fleet size ¯F is determined as +¯ +E +¯ +H . When the sensitivity +testing is conducted for one parameter (e.g., C), other parameters (e.g., θ, γ, ¯H, and the demand +factor) are set to their reference values for comparison. +Table 2 +Scenario design +Parameters +Value space +Reference value +C +{30, 34, 38} +34 +γ (/min) +{0, 1/10, 1/5, 1/3} +1/5 +θ (/min) +{2, 1 ,1/2} +1 +¯H (min) +{2, 4, 7} +6 +Demand factor {0.2, 0.4, 0.6, 0.8, 1} +0.8 +5.2. Performance indicators +The mean and standard deviation of queue length for each station under different testing scenarios +are shown in Figure 7. Generally, for all scenarios, the queue length patterns are consistent with +the congestion patterns we expect given the passenger arrival and alighting rates. That is, the +expected queue length is relatively higher at stations 2 and 8. The expected queue length at the +last station is always zero as its passenger arrival rate is 0. +Figure 7a shows the queue length patterns with respect to bus capacity. The system is not +very sensitive to bus capacity. The reason is that under the reference scenario, the system is not + +Transportation +hub +0 +2 +9 +10 +5 min +5 min +5 min +E +T(N) = 50 min +Half of the cycle time +2Mo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +29 +congested and capacity is not fully utilized. Thus, increasing capacity does not affect the queuing +distribution. Figure 7b shows the impact of incident occurrence rate γ on queue length. When +there is no random suspension in the system (γ = 0), the expected queue length at station 8 is 4.5. +As the frequency of incidents increases, the system becomes more congested with longer expected +queue lengths and higher variances. When the incident frequency increases to 1/3 per minute on +average (γ = 1/3), the expected queue length at station 8 is increased to 8.3 units. Similar results +can be observed for the duration of incidents (Figure 7c). When the average incident duration is 30 +seconds (θ = 2), E(Q(8)) = 5.0. When the average incident duration is 2 minutes (θ = 1/2), E(Q(8)) +increases to 12.6. The impacts of θ and γ on queue length are both more significant at crowded +stations. The impact of ¯H is shown in Figure 7d. As expected, higher headway means a lower +service rate and thus a higher expected queue length. As ¯H increases from 2 minutes to 7 minutes, +the queue length at station 8 increases from 4.1 to 9.7. The impact of the demand factor (Figure +7e) shows similar patterns. As the demand factor increases from 0.5 to 1.0, the queue length at +station 8 increases from 4.1 to 8.3. The impact of ¯H and the demand factor are relatively similar +for crowded and uncrowded stations. +Figure 8 shows the mean and standard deviation of passenger waiting time for the different +scenarios. We observe that the downstream stations generally have higher waiting time expectations +and variances due to the headway variance propagation. For congested stations, such as stations 3 +and 8, extra waiting times are observed due to passengers left behind with capacity constraints. +Figure 8a shows the impact of capacity on waiting time. Similar to the results on queue length, +the impact is not very significant. The impacts of γ and θ on waiting times are shown in Figure 8b +and 8c, respectively. As increases in γ and 1/θ result in an increase in expected headway, the mean +waiting times at all stations are increased. The impacts on crowding stations are more significant. +When γ = 0, there is no incident in the system. In this case, there are no left behind or headway +irregularities at any stations and their expected waiting times are all equal to 2 minutes (i.e., 1 +2 ¯H, +as no incidents mean all stations have the same fixed headway). When γ increases to 1/5, station 3 + +Mo et al.: Evaluation of Public Transit Systems +30 +Article submitted to ; manuscript no. () +(a) Sensitivity on C +(b) Sensitivity on γ +(c) Sensitivity on θ +(d) Sensitivity on ¯H +(e) Sensitivity on demand factor +Figure 7 +Mean and standard deviation of queue length (the shaded part is 0.2×standard deviation) +has left behind passengers and the waiting time is increased to 4.6 minutes. When θ decreases (i.e., +mean incident duration increases) from 2 to 1/2, the expected waiting time at station 8 increases +from 3.0 to 11.8 minutes. Changes in ¯H have the most direct impact on the expected waiting time. +The increase in planned headway causes an increase in waiting time for all stations. There are a few +left-behind passengers observed at stations 3 and 8 when ¯H = 7 min. Finally, as demand increases, +the waiting time increases only if there are left behind (e.g., when demand factor = 1) because it +does not change the headway distribution. At station 3, the increase in the demand factor from +0.5 to 1.0 results in an increase in the expected waiting time from 3.5 to 4.2 minutes. +5.3. Comparison between simulated and theoretical results +To validate the theoretical results, we develop a simulation model to calculate the expectation and +variance of queue length and waiting time. The simulation procedure is shown in Appendix N. +We compare the simulation and theoretical results for the reference parameter setting (Table 2). +A total of 50,000 vehicle runs are simulated. The comparisons of mean and standard deviation for + +Expected queue length when vehicle arrives +14 +H= 2 +H=4 +12 + H= 7 +10 +8 +6 +4 +2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Station ID14 +Expected queue length when vehicle arrives +Demand factor = 0.5 +Demand factor = 0.75 +12 +Demand factor = 1.0 +10 +8 +6 +4 +2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Station ID10 +Expected queue length when vehicle arrives +C = 30 +C = 34 +C = 38 +8 +6 +4 +2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +6 +10 +Station ID12 +Expected queue length when vehicle arrives +y=0 +y= 1/10 +10 +y= 1/5 +y= 1/3 +8 +6 +4 +2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Station IDExpected queue length when vehicle arrives +0 = 1/2 +14 +0 = 1 +0 = 2 +12 +10 +8 +6 +4 +2 +0 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Station IDMo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +31 +(a) Sensitivity on C +(b) Sensitivity on γ +(c) Sensitivity on θ +(d) Sensitivity on ¯H +(e) Sensitivity on demand factor +Figure 8 +Mean and standard deviation of waiting time (the shaded part is 0.2×standard deviation) +queue length and waiting time are shown in Figure 9. We observe that the simulation and theoretical +results match well, validating the theoretical model’s correctness. However, the theoretical results +slightly overestimate the mean and variance of the queue length and waiting time. The main reason +may be the approximation of headway distribution as normal. As shown in Figure 11, the actual +headway has more probability density concentrated at the mean (i.e., more peakedness), implying +that the actual headway has less probability of deviating from the planned one, thus the simulation +scenario may have a smaller queue length and waiting time. +6. Conclusion and discussion +This paper proposes a stochastic framework to evaluate the performance of PTSs under short +random service suspensions. Specifically, we analyze the system stability conditions and derive +closed-form formulations for the mean and variance of queue length and waiting time at each +station. The derived stability conditions are intuitive and imply that the system is more likely to be +unstable with high incident rates, high incident duration, high demand, low service frequency, and + +8 +C = 30 +C = 34 +7 +C = 38 +6 +5 +4 +3 +Expected +2 +1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Station ID12 +y=0 +y= 1/5 +y= 1/10 +y= 1/3 +10 +8 +6 +4 +2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Station ID16 +0 = 1/2 +0=1 +14 +0=2 +12 +10 +8 +6 +4 +2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Station ID12 +H=2 +←H=4 +10 +★H=7 +8 +6 +4 +2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +6 +10 +Station ID8 +Demand factor = 0.5 +Demand factor = 0.8 +7 + passenger waiting time +Demand factor = 1.0 +6 +5 +4 +3 +Expected +2 +1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +6 +10 +Station IDMo et al.: Evaluation of Public Transit Systems +32 +Article submitted to ; manuscript no. () +(a) E[Q(n)] +(b) Std dev[Q(n)] +(c) E[W (n)] +(d) Std dev[W (n)] +Figure 9 +Comparison between simulation and theoretical results (reference scenario) +low vehicle capacity. The proposed model is implemented using an example bus network adapted +from the literature. A sensitivity analysis of different parameters (such as incident rate, incident +duration, vehicle capacity, etc.) was conducted. The results show that congested stations (i.e., +stations with high demand rates) are more vulnerable to random service suspensions. The results +are validated with a simulation model, showing consistency between theoretical and simulation +outcomes. +The proposed model has several potential applications. 1) It can facilitate the design and planning +of PTS with the consideration of random system interruptions, such as the design of headways +and the determination of vehicle capacity. Moreover, the estimated queue length can be used to +evaluate the layout and capacity of congested stations. 2) The model can be used to monitor +system performance and identify critical stations by inputting the historical demand and incident +information. 3) The model can support efficient cost-benefit analysis of approaches to improve +services using estimates of waiting time and queue length. For example, the model can answer that, + +Theory +Simulation +8 +f queue length +6 +of +4 +11 +2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Station IDTheory +Simulation +6 +5 +4 +of +lev. +3 +S +1 +0 +2 +3 +4 +5 +6 +L +8 +9 +Station IDTheory +6 +Simulation +e + time +5 +waiting +4 +of +3 +ation +ctat +2 + 0. 2) when C is even, besides z∗ +0 = 1, there exists another real +root on the negative real axis (denoted as z∗ +C +2 , where −1 ≤ z∗ +C +2 < 0). So, we have +z∗ +C +2 +z∗ +C +2 +−1 > 0, which +leads to �C−1 +i=1 +z∗ +i +z∗ +i −1 > 0. +To validate Eq. 67, consider the fixed capacity situation where s(n) +C = 1 and ¯S(n) = C. Then Eq. +67 reduces to +q(n) +0 +��� +s(n) +C =1 = (C − ¯Y (n)) +C−1 +� +i=1 +z∗ +i +z∗ +i − 1 +(70) +This is the same as Chaudhry et al. (1987). Now we will derive q(n) +1:C−1. Observing that the numerator +of Eq. 66 can be rewritten as +( ¯S(n) − ¯Y (n))(z − 1) +C−1 +� +i=1 +z − z∗ +i +1 − z∗ +i += +1 +s(n) +C +( ¯S(n) − ¯Y (n)) +C−1 +� +i=1 +z∗ +i +z∗ +i − 1 +C−1 +� +i=1 +z∗ +i − z +z∗ +i +(z − 1)s(n) +C += s(n) +C q(n) +0 (z − 1) +C−1 +� +i=1 +� +1 − z +z∗ +i +� += −s(n) +C q(n) +0 +C−1 +� +i=0 +� +1 − z +z∗ +i +� +(71) +Define �C−1 +i=0 +� +1 − z +z∗ +i +� +:= �C +j=0 ηjzj, where ηj is the polynomial coefficient of zj. For the RHS of +Eq. 67, the polynomial coefficient of zC−k is −�C +u=k s(n) +u q(n) +u−k. And from Eq. 71, the polynomial +coefficient of zC−k is −s(n) +C q(n) +0 ηC−k. Matching the coefficient of the same order of z leads to +s(n) +C q(n) +0 ηC−k = +C +� +u=k +s(n) +u q(n) +u−k +k = 1,2,...,C − 1 +(72) +To validate Eq. 72, consider the fixed capacity situation where s(n) +C = 1 and s(n) +k += 0,∀ 0 ≤ k < C. +then Eq. 72 reduces to +q(n) +C−k = q(n) +0 ηC−k +k = 1,2,...,C − 1 +if s(n) +C = 1 +(73) + +Mo et al.: Evaluation of Public Transit Systems +38 +Article submitted to ; manuscript no. () +which is the same as Chaudhry et al. (1987). +Eq. 72 can be expressed in a matrix form by adding s(n) +C q(n) +0 η(n) +0 += s(n) +C q(n) +0 +(note that η(n) +0 += 1 by +definition): +˜η(n) = q(n) +0:C−1Λ(n) +(74) +where ˜η(n) = [s(n) +C q(n) +0 η(n) +0 ,s(n) +C q(n) +0 η(n) +1 ,...,s(n) +C q(n) +0 η(n) +C−1] ∈ RC and +Λ(n) = +� +�������� +s(n) +C +s(n) +C−1 s(n) +C−2 ... s(n) +1 +0 +s(n) +C +s(n) +C−1 ... s(n) +2 +... +0 +s(n) +C +... s(n) +3 +0 +... +0 +... s(n) +4 +0 +0 +... +... +... +0 +0 +0 +... s(n) +C +� +�������� +∈ RC×C +(75) +As s(n) +C > 0 is a known condition, the triangular matrix Λ(n) is invertible. Thus, we have +q(n) +0:C−1 = ˜η(n)(Λ(n))−1 +(76) +Appendix E: Proof of Proposition 5 +The derivation follows the same idea in Powell (1981). These results are equivalent to Powell (1985) +which considered the general bulk-service queue model (but Powell (1985) did not provide detailed +proof in the paper). +Here we try to provide analytical formulations of E[Q(n)] and Var[Q(n)]. The key is to find Q′(1) +and Q′′(1). The derivation follows a similar idea in Powell (1981). +Let A(z) = +( ¯S(n)− ¯Y (n))(z−1) +zC +Y (z) −�C +u=0 s(n) +u +zC−u and Bi(z) = +z−z∗ +i +1−z∗ +i , then Q(z) = A(z)�C−1 +i=1 Bi(z). Based on the fact +that Bi(1) = 1 and Q(z) = 1, we must have A(1) = 1. Hence, +Q′(1) = A′(1)B1(1)...BC−1(1) + A(1)B′ +1(1)...BC−1(1) + ... + A(1)B1(1)...B′ +C−1(1) += A′(1) + +C−1 +� +i=1 +B′ +i(1) +(77) +Since B′ +i(1) = +1 +1−z∗ +i , the problem now becomes finding A′(1). Again, let A(z) = A1(z) +A2(z). Then, +A′(z) = A′ +1(z)A2(z) − A1(z)A′ +2(z) +(A2(z))2 +(78) +Notice that when z → 1, the numerator and denominator of A′(z) approach 0 (because A1(1) = 0 +and A2(1) = 0). Therefore, applying L’Hopital’s rule yields: +A′(z) = A′′ +1(z)A2(z) − A1(z)A′′ +2(z) +2A2(z)A′ +2(z) +(79) + +Mo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +39 +Again we have 0/0 when z → 1 because A′′ +1(z) = 0 and A2(1) = 0. Applying L’Hopital’s rule once +more gives: +A′(z) = −A′ +1(z)A′′ +2(z) − A1(z)A′′′ +2 (z) +2A′ +2(z)A′ +2(z) + 2A2(z)A′′ +2(z) +(80) +Substituting z = 1 leads to +A′(1) = −A′ +1(1)A′′ +2(1) +2(A′ +2(1))2 +(81) +Based on the fact that Y (1) = 1, Y ′(1) = ¯Y (n), Y ′′(1) = E[(Y (n))2] − ¯Y (n), we have +A′ +1(1) = ¯S(n) − ¯Y (n) +(82) +A′ +2(1) = CzC−1Y (z) − Y ′(z)zC +Y (z)2 +− +C +� +u=0 +(C − u)s(n) +u zC−u−1 +����� +z=1 += ¯S(n) − ¯Y (n) +(83) +A′′ +2(1) = C(C − 1)zC−2 +Y (z) +− 2Y ′(z)CzC−1 +Y (z)2 +− Y ′′(z)zC +Y (z)2 ++ 2Y ′(z)2zC +Y (z)3 +− +C +� +u=0 +(C − u)(C − u − 1)s(n) +u zC−u−2 +����� +z=1 += C(C − 1) − 2 ¯Y (n)C − E[(Y (n))2] + 2( ¯Y (n))2 + ¯Y (n) − C2 + C + 2C ¯S(n) − ¯S(n) − E[(S(n))2] += −2 ¯Y (n)C − E[(Y (n))2] + 2( ¯Y (n))2 + ¯Y (n) + 2C ¯S(n) − ¯S(n) − E[(S(n))2] +(84) +Substituting Eq. 82, 83, and 84 into Eq. 81 results in +A′(1) = 2 ¯Y (n)C + E[(Y (n))2] − 2( ¯Y (n))2 − ¯Y (n) − 2C ¯S(n) + ¯S(n) + E[(S(n))2] +2( ¯S(n) − ¯Y (n)) +(85) +Therefore, we have +E[Q(n)] = 2 ¯Y (n)C + E[(Y (n))2] − 2( ¯Y (n))2 − ¯Y (n) − 2C ¯S(n) + ¯S(n) + E[(S(n))2] +2( ¯S(n) − ¯Y (n)) ++ +C−1 +� +i=1 +1 +1 − z∗ +i +(86) +To validate this formulation, let us consider a fixed capacity situation with s(n) +C = 1. Then ¯S(n) = C, +E[(S(n))2] = C2. Then Eq. 86 reduces to +E[Q(n)] +�� +s(n) +C =1 = C − C2 + 2 ¯Y (n)C + E[(Y (n))2] − 2( ¯Y (n))2 − ¯Y (n)+ +2(C − ¯Y (n)) ++ +C−1 +� +i=1 +1 +1 − z∗ +i +(87) +which is equivalent to Powell (1981)’s. +According to Eq. 24, the key to obtain Var[Q(n)] is to calculate Q′′(1). Taking the logarithm of +Q(z) = A(z)�C−1 +i=1 Bi(z) gives +log Q(z) = log A(z) + +C−1 +� +i=1 +log Bi(z) +(88) +Taking derivatives of both sides leads to +Q′(z) +Q(z) = A′(z) +A(z) + +C−1 +� +i=1 +B′ +i(z) +Bi(z) +(89) + +Mo et al.: Evaluation of Public Transit Systems +40 +Article submitted to ; manuscript no. () +Taking derivatives again: +Q′′(z) +Q(z) − Q′(z)2 +Q(z)2 = A′′(z) +A(z) − A′(z)2 +A(z)2 + +C−1 +� +i=1 +�B′′ +i (z) +Bi(z) − B′ +i(z)2 +Bi(z)2 +� +(90) +Solving for Q′′(z) and letting z = 1 gives: +Q′′(1) = E[Q(n)]2 + A′′(1) − A′(1)2 + +C−1 +� +i=1 +� +B′′ +i (1) − B′ +i(1)2� +(91) +Notice that B′′ +i (1) = 0 (∀i = 1,...,C − 1) and E[Q(n)] = Q′(1). Substituting Eq. 77 and 91 into Eq. +24 gives +Var[Q(n)] = A′′(1) − A′(1)2 + A′(1) + +C−1 +� +i=1 +� +B′ +i(1) − B′ +i(1)2� +(92) +Now we only need to solve for A′′(1). The process is similar to finding A′(1). Applying L’Hopital’s +rule five times to Eq. 80 and substituting z = 1 leads to +A′′(1) = −2A′ +2(1)A′′′ +2 (1) + 3A′′ +2(1)2 +6A′ +2(1) +(93) +Notice that the derivation process uses A′′ +1(z) = 0, A1(1) = 0, A2(1) = 0, and A′ +1(1) = A′ +2(1). Details +are omitted due to the tedious mathematical manipulation. To obtain A′′′ +2 (1), taking derivative of +Eq. 84 gives: +A′′′ +2 (1) = +�C(C − 1)(C − 2)zC−3 +Y (z) +− 3Y ′(z)C(C − 1)zC−2 +Y (z)2 +− 3Y ′′(z)CzC−1 +Y (z)2 ++ 4Y ′(z)2CzC−1Y (z) +Y (z)4 +− Y ′′′(z)zC +Y (z)2 ++ 2Y (z)Y ′(z)Y ′′(z)zC +Y (z)4 ++ 4Y ′′(z)Y ′(z)zC + 2CzC−1Y ′(z)2 +Y (z)3 +− 6Y (z)Y ′(z)3zC +Y (z)6 +− +C +� +u=0 +(C − u)(C − u − 1)(C − u − 2)s(n) +u zC−u−3 +� +z=1 += C(C − 1)(C − 2) − 3 ¯Y (n)C(C − 1) − 3Y ′′(1)C + 6( ¯Y (n))2C − Y ′′′(1) + 6 ¯Y (n)Y ′′(1) +− 6( ¯Y (n))3 − (C3 − 3C2 + 2C) + (2 + 3C2 − 6C) ¯S(n) + (3 − 3C)E[(S(n))2] + E[(S(n))3] +(94) +Notice that Y ′′′(1) = E[(Y (n))3] − 3E[(Y (n))2] + 2 ¯Y (n). Hence, +A′′′ +2 (1) = 3C2 ¯S(n) − 3C2 ¯Y (n) − 6C ¯S(n) − 3CE[(S(n))2] − 3CE[(Y (n))2] + 6C( ¯Y (n))2 + 6C ¯Y (n) + 2 ¯S(n) ++ 3E[(S(n))2] + E[(S(n))3] + 6E[(Y (n))2] ¯Y (n) + 3E[(Y (n))2] − E[(Y (n))3] − 6( ¯Y (n))3 − 6( ¯Y (n))2 − 2 ¯Y (n) +(95) +Substituting Eq. 83, 84, and 95 into Eq. 93 results in +A′′(1) = +� +6C2( ¯S(n))2 − 12C2 ( ¯S(n))( ¯Y (n)) + 6C2( ¯Y (n))2 − 6C( ¯S(n))E[(S(n))2] − 6C( ¯S(n))E[(Y (n))2]( ¯Y (n))2 ++ 12C( ¯S(n)) + 6CE[(S(n))2]( ¯Y (n)) + 6CE[(Y (n))2]( ¯Y (n)) − 12C( ¯Y (n))3 − ( ¯S(n))2 − 2( ¯S(n))E[(S(n))3] +− 12( ¯S(n))E[(Y (n))2]( ¯Y (n)) + 2( ¯S(n))E[(Y (n))3] + 12( ¯S(n))( ¯Y (n))3 + 2( ¯S(n))( ¯Y (n)) ++ 3E[(S(n))2]2 + 6E[(S(n))2]E[(Y (n))2] − 12E[(S(n))2]( ¯Y (n))2 + 2E[(S(n))3]( ¯Y (n)) + 3E[(Y (n))2]2 +− 2E[(Y (n))3]( ¯Y (n)) − ( ¯Y (n))2�� +6( ¯S(n) − ¯Y (n)) +(96) + +Mo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +41 +Now with Eq. 96 and 85 we have +A′′(1) − A′(1)2 + A′(1) = +� +( ¯S(n))2 − 4 ¯S(n)E[(S(n))3] − 24 ¯S(n)E[(Y (n))2] ¯Y (n) + 4 ¯S(n)E[(Y (n))3] ++ 24 ¯S(n)( ¯Y (n))3 − 2 ¯S(n) ¯Y (n) + 3E[(S(n))2]2 + 6E[(S(n))2]E[(Y (n))2] − 12E[(S(n))2]( ¯Y (n))2 + 4E[(S(n))3] ¯Y (n) ++ 3E[(Y (n))2]2 + 12E[(Y (n))2]( ¯Y (n))2 − 4E[(Y (n))3] ¯Y (n) − 12( ¯Y (n))4 + ( ¯Y (n))2�� +12( ¯S(n) − ¯Y (n))2 +(97) +Slight manipulation of Eq. 97 leads to +A′′(1) − A′(1)2 + A′(1) += −4( ¯¯¯S(n) − ¯¯¯Y (n))( ¯S(n) − ¯Y (n)) + 3( ¯¯S(n) + ¯¯Y (n))2 − [6( ¯¯S +(n) − ¯¯Y +(n)) − 1]( ¯S(n) − ¯Y (n))2 − ( ¯S(n) − ¯Y (n))4 +12( ¯S(n) − ¯Y (n))2 +(98) +Observe that B′ +i(1) − B′ +i(1)2 = +−z∗ +i +(1−z∗ +i )2 . Therefore, substituting Eq. 97 into 92 gives the final +results: +Var[Q(n)] = +1 +12( ¯S(n) − ¯Y (n))2 +� +−4( ¯¯¯S(n) − ¯¯¯Y (n))( ¯S(n) − ¯Y (n)) + 3( ¯¯S(n) + ¯¯Y (n))2 − +[6( ¯¯S +(n) − ¯¯Y +(n)) − 1]( ¯S(n) − ¯Y (n))2 − ( ¯S(n) − ¯Y (n))4] +� +− +C−1 +� +i=1 +z∗ +i +(1 − z∗ +i )2 +(99) +Appendix F: Proof of Proposition 7 +When there are no incidents in the system, vehicle l reaches station n after T (n) time units. Since +the system can only switch to the incident state from the normal state, the number of incident +occurrences, K, follows a Poisson distribution with rate γT (n). The vehicle stopping time for the +i-th incident, Xi, follows an exponential distribution with rate θ (i.e., mean +1 +θ). Therefore, the +duration of all incidents is I(n,l) = �K +i=1 Xi, where Xi ∼ Exp(θ) ∀i = 1,...,K, and K ∼ Poi(γT (n)) +Appendix G: Proof of Proposition 8 +MH(n,l)(t) = E[etH(n,l)] = E[et( ¯ +H+ 2γT (N) +θ ¯ +F +etI(n,l)e−tI(n,l−1)] += et( ¯ +H+ 2γT (N) +θ ¯ +F +E[etI(n,l)]E[e−tI(n,l−1)] += et( ¯ +H+ 2γT (N) +θ ¯ +F +eγT (n)( +θ +θ−t −1)eγT (n)( +θ +θ+t −1) += et( ¯ +H+ 2γT (N) +θ ¯ +F +)e +γT (n)( +2t2 +θ2−t2 ) +(100) +where Eq. 100 is because of the independence between I(n,l) and I(n,l−1). As this equation holds +for all vehicles l, the MGF of H(n) (i.e., l → ∞) is MH(n)(t) = MH(n,l)(t). + +Mo et al.: Evaluation of Public Transit Systems +42 +Article submitted to ; manuscript no. () +Appendix H: Proof of Proposition 9 +Let µ and σ2 be the mean and variance of H(n) +Normal, respectively, where µ = ¯H + 2γT (N) +θ ¯ +F +and σ = +2√ +T (n)γ +θ +. The MGF of ˆH(n) +Normal can be derived as +M ˆ +H(n) +Normal(t) = E[et ˆ +H(n) +Normal] = P[H(n) +Normal ≤ 0] · e0 + +� +∞ +0 +etz · φH(n) +Normal(z) · dz += Φ(−µ +σ ) + +1 +σ +√ +2π +� +∞ +0 +e +tz+ (z−µ)2 +−2σ2 dz += Φ(−µ +σ ) + eµt+ σ2t2 +2 +� +1 − Φ(−µ +σ − σt) +� +(101) +where Eq. 101 follows the same derivation of a truncated normal distribution (Burkardt 2014). +Subsisting the value of µ and σ completes the proof. +Appendix I: Proof of Proposition 10 +The strict mathematical proof can be done by taking the derivative of E[ ˆH(n) +Normal] in terms of γ or +1 +θ and show that it is always positive. However, in this study, we adopt a more intuitive graphical +proof, which is easier for understanding. +As shown in Figure 10, consider an arbitrary truncated headway distribution (shown in the red +line, denoted the headway as ˆHRed). When the incident intensity increases, according to Eqs. 36 +and 37, both µ and σ increase. Let us first consider the increase in σ and assume µ does not +change (which corresponds to the scenario where ¯F → ∞). Then the distribution will become the +blue curve (denote the corresponding headway as ˆHBlue). Note that ˆHRed and ˆHBlue have the same +peak value, but since ˆHBlue has longer positive tail, we have E[ ˆHBlue] > E[ ˆHRed]. Next, let us also +consider the incident’s impact on the increase in µ as well. The distribution is shown by the green +curve (denoted the headway as ˆHGreen). Since ˆHBlue and ˆHGreen has the same σ, but ˆHGreen has +higher µ (shifted right), we have E[ ˆHGreen] > E[ ˆHBlue]. Hence, E[ ˆHGreen] > E[ ˆHRed], showing that +the increase in incident intensity will increase µ and σ, thus increase the mean of the truncated +headway. + +Mo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +43 +Figure 10 +Illustration for the impact of incidents on expected headway. As the probability mass at zero does not +contribute to the expectation calculation, it is not shown in the figure. +Appendix J: Approximating headway as a normal distributed random variable +We observe that the third central moment of H(n), which is a measure of skewness, is +Skewness[H(n)] = 0, implying that H(n) is symmetric. Moreover, the MGF of the normal distribu- +tion of H(n) +Normal with the same mean and variance is +MH(n) +Normal(t) = et( ¯ +H+ 2γT (N) +θ ¯ +F +)eγT (n)( 2t2 +θ2 ), +(102) +which is very similar to Eq. 35 (the MGF of H(n)). Therefore, it is reasonable to approximate +the distribution of H(n) as a normal distribution with the same mean and variance. Note that the +first three moments of H(n) and H(n) +Normal are the same. And the corresponding forth moments (i.e., +Kurtosis) are: +Kurtosis[H(n)] = 48(T (n)γ)2 + 48T (n)γ +θ4 +(103) +Kurtosis[H(n) +Normal] = 48(T (n)γ)2 +θ4 +(104) +which means that the distribution of H(n) may have heavier tails and peakedness compared to +H(n) +Normal. +Figure 11 empirically compares the distribution of H(n) and H(n) +Normal with various values of +T (n),θ, and γ. The histogram of H(n) is generated by sampling variables from the associated +exponential and Poisson distributions to get the compound distribution. Results show that the +normal distribution approximates the original distribution well. As expected, H(n) shows more +peakedness than H(n) +Normal. + +Probability +Original truncated headway +一 +Incident intensity increase +(assume no change of mean) +Incident intensity increase +(affect both mean and variance) +HeadwayMo et al.: Evaluation of Public Transit Systems +44 +Article submitted to ; manuscript no. () +(a) Example 1 +(b) Example 2 +(c) Example 3 +Figure 11 +Empirical validation for approximating the headway distribution as normal +Appendix K: Proof of Proposition 11 +Recall that Y (n)| ˆH(n) +Normal is a Poisson random variable with parameter λ(n) ˆH(n) +Normal. So, the MGF +of Y (n)| ˆH(n) +Normal is MY (n)| ˆ +H(n) +Normal(t) = exp[λ(n) ˆH(n) +Normal(et − 1)]. Based on Lemma 1, setting C1(t) = 1 +and C2(t) = λ(n)(et − 1), we conclude that the MGF of Y (n) is +MY (n)(t) = Φ +�−µ +σ +� ++ eµλ(n)(et−1)+ σ2(λ(n)et−λ(n))2 +2 +� +1 − Φ +�−µ +σ − σλ(n)(et − 1) +�� +(105) +Substituting t = log z in Eq. 105 completes the proof. +Appendix L: Interpolation-based searching algorithm for root solving +Notice that Den(z) = 0 is equivalent to find z∗ +0,...,z∗ +C−1, such that +1 +Y (z∗ +k) − S(1/z∗ +k) = 0 ⇔ J(z∗ +k) = 1 +∀ k = 0,...,C − 1 +(106) +where J(z) := Y (z)S(1/z). Taking the logarithm of both sides of Eq. 106 and matching the real +and imaginary parts gives: +�Re[log(J(z))] = 0 +Im[log(J(z))] = 0 +(107) +where Re[·] and Im[·] represent the real and imaginary part of a complex number. Eq. 107 can +be solved efficiently with many optimization algorithms (such as trust-region and Levenberg- +Marquardt algorithms). However, as there are C optimal solutions for this problem with |z∗| ≤ 1, +the challenge is how to select different initial values so as to find all solutions. +It can be empirically observed that the distribution of the C solutions has an oval-like shape. +Figure 12 shows some examples of the solution distribution with different values of ρ(n) (where +ρ(n) = ¯Y (n)/ ¯S(n) is the utilization ratio of a bulk service queuing system) and s(n). It is found + +μ(n) +T(n)=30 +Normal +0.08 +H(n) +0=1 +y=1/5 +0.06 +Probability +0.04 +0.02 +0.00 +20 +-10 +0 +10 +20 +30 +Headway(n) +0.040 +T(n)=20 +Normal +H(n) +0=1/2 +0.035 +y=1/3 +0.030 +Probability +0.025 +0.020 +0.015 +0.010 +0.005 +0.000 +-40 +-20 +0 +20 +40 +60 +Headwayμ(n) +T(n)=40 +Normal +H(n) +0=2 +0.20 +y=1/10 +0.15 +Probability +0.10 +0.05 +0.00 +-5 +0 +5 +10 +15 +HeadwayMo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +45 +that the closer ρ(n) is to 1 (resp. 0), the closer the shape of the root distribution is to an ellipse +(resp. circle). The value of s(n) (i.e., available capacity distribution) can also slightly affect the root +distribution. +(a) ρ(n) = 0, C = 40 +(b) ρ(n) = 0.15, C = 40 +(c) ρ(n) = 0.64, C = 40 +(d) ρ(n) = 0.74, C = 40 +Figure 12 +Examples of root distribution +We first express the complex number in polar coordinate system with z = r exp[ϕi], where i = +√−1, r is the length from z to the origin, and ϕ is the angle. Eq. 107 now has C optimal solutions +(r∗ +k,ϕ∗ +k) for k = 0,1,...,C − 1, where 0 ≤ r∗ +k ≤ 1 and 0 ≤ ϕ∗ +k < 2π. Note that z∗ +0 = 1 corresponds to +r∗ +0 = 1 and ϕ∗ +0 = 0. Another property is that the roots must appear as conjugate pairs. Hence, if +(r∗,ϕ∗) is a root and 0 < ϕ∗ < π, then (r∗,2π − ϕ∗) is also a root. +The proposed search algorithm has two steps. The first step is referred to as “clockwise search- +ing”, which is adapted from the numerical method in Powell (1985). The empirical observation +(Figure 12) shows a rough relationship that r∗ +k+1 − r∗ +k ≈ r∗ +k − r∗ +k−1, especially for small ρ(n). This is +equivalent to +r∗ +k+1 ≈ 2r∗ +k − r∗ +k−1 +(108) +Eq. 108 provides a way to determine the initial value for solving for the k + 1-th root when the +k-th and k −1-th roots are available. As we already know r∗ +0 = 1 and ϕ∗ +0 = 0, we first set the initial +value for solving for the second root as rIni +1 = 1−0.5ρ(n) and ϕIni +1 = 3π/C. This is motivated by the +shape of the root distribution with respect to ρ(n). Then rIni +1 +and ϕIni +1 +are used as the initial value +to solve for r∗ +1 and ϕ∗ +1 based on Eq 107. For k ≥ 2, the initial values for solving the for k-th root +are set to rIni +k = r∗ +k−1 + (r∗ +k−1 − r∗ +k−2), ϕIni +k = ϕ∗ +k−1 + (ϕ∗ +k−1 − ϕ∗ +k−2) according to Eq. 108. + +1.0 +p=0.0 +0.5 + part +Imaginary +0.0 +-0.5 +-1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Real part +1.0 +Probability +0.5 +0.0 +0 +10 +20 +30 +40 +Available capacity1.0 +p= 0.15 +0.5 +Imaginary part +0.0 +-0.5 +-1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Real part +Probability +0.1 +0.0 +0 +10 +20 +30 +40 +Available capacity1.0 +p = 0.64 +0.5 +Imaginary part +0.0 +-0.5 +1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Real part +Probability +0.050 +0.025 +0.000 +0 +10 +20 +30 +40 +Available capacity1.0 +p = 0.74 +0.5 +Imaginary part +0.0 +-0.5 +-1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Real part +0.10 +Probability +0.05 +0.00 +0 +10 +20 +30 +40 +Available capacityMo et al.: Evaluation of Public Transit Systems +46 +Article submitted to ; manuscript no. () +However, only performing step 1 (i.e., Powell (1985)’s method) may not find all C distinct roots. +Figure 13 shows some examples of the comparison between roots found in step 1 and all roots. We +observe that when ρ(n) is relatively large (i.e., the system is relatively congested), the clockwise +search does not perform well because the approximate relationship in Eq. 108 does not hold. Even +when ρ(n) is relatively small, it is also possible that some roots do not perfectly fit the oval-like +shape (such as Figure 13a), resulting in the failure of step 1 to find all roots. +(a) ρ(n) = 0.17, C = 40 +(b) ρ(n) = 0.48, C = 36 +(c) ρ(n) = 0.64, C = 40 +(d) ρ(n) = 0.85, C = 40 +Figure 13 +Comparison between roots found with clockwise search and all roots +Therefore, we propose a second step called “interpolation search”. Let the set of found roots +from step 1 be Z(0) = {(r(0) +0 ,ϕ(0) +0 ),(r(0) +1 ,ϕ(0) +1 ),...,(r(0) +M0,ϕ(0) +M0)}, where M0 = |Z(0)| is the number of +roots from step 1. Without loss of generality, assume that the elements in Z(0) are clockwise ranked +(i.e., ϕ(0) +0 +< ϕ(0) +1 +< ... < ϕ(0) +M0). The interpolation search is described in Algorithm 2. The main idea is +to perform interpolation between any two adjacent roots that are already found. The interpolated +points are set as initial values and fed into Eq. 107 to solve for new distinct roots. Then we update +the set of roots with the new distinct roots or perform a finer (i.e., larger L) interpolation if +no distinct roots are found. This process is repeated until there are C distinct roots found. In +Algorithm 2, L is a parameter controlling how many points to interpolate between two known +roots, and ϵ is a predetermined probability threshold to add randomness in the search process. +Appendix M: Proof of Proposition 12 +The stability condition is equivalent to P(Q(n) = 0) = q(n) +0 +> 0. In Eq. 20, we notice that �C−1 +i=1 +z∗ +i +z∗ +i −1 +is always greater than 0 (see Appendix D for details), and s(n) +C > 0 is a known condition. Therefore, +q(n) +0 +> 0 if and only if ¯Y (n) < ¯S(n) (i.e., ρ(n) < 1), which completes the proof. + +1.0 +p=0.17 +All z* +z* after step 1 +0.5 +Imaginary part +0.0 +-0.5 +40 +-1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Real part +Probability +0.1 +0.0 +0 +10 +20 +30 +40 +Available capacity1.0 +p= 0.48 +All z* +z* after step 1 +0.5 +Imaginary part +0.0 +-0.5 +36 +-1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Real part +Probability +0.05 +0.00 +0 +10 +20 +30 +Available capacity1.0 +p= 0.64 +All z* +z* after step 1 +0.5 +Imaginary part +0.0 +-0.5 +40 +-1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Real part +Probability +0.050 +0.025 +0.000 +0 +10 +20 +30 +40 +Available capacity1.0 +p = 0.85 +All z* +z* after step 1 +0.5 +Imaginary part +0.0 +-0.5 +40 +-1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Real part +Probability +0.05 +0.00 +0 +10 +20 +30 +40 +Available capacityMo et al.: Evaluation of Public Transit Systems +Article submitted to ; manuscript no. () +47 +Algorithm 2 Interpolation searching +1: Initialize Z(0), M0, ϵ. Initialize L = 2, k = 0. +2: while Mk < C do +3: +Initialize ZIni as an empty set. +4: +for i = 1 : Mk do +5: +for d = 1 : L − 1 do +6: +rIni = r(k) +i ++ d · +r(k) +i+1−r(k) +i +L +; ϕIni = ϕ(k) +i ++ d · +ϕ(k) +i+1−ϕ(k) +i +L +7: +Draw a random value w uniformly from [0,1) +8: +if w < ϵ then +9: +Draw a random value δ1 uniformly from [− +|r(k) +i+1−r(k) +i +| +2L +, +|r(k) +i+1−r(k) +i +| +2L +] +10: +rIni = rIni + δ1 +11: +Draw a random value δ2 uniformly from [− +|ϕ(k) +i+1−ϕ(k) +i +| +2L +, +|ϕ(k) +i+1−ϕ(k) +i +| +2L +] +12: +ϕIni = ϕIni + δ2 +13: +Add (rIni,ϕIni) into ZIni. +14: +Initialize Ztemp as an empty set. +15: +for all zIni in ZIni do +16: +Solve Eq. 107 using zIni as the initial value, obtaining z∗ +temp. Let its conjugate be ¯z∗ +temp. +17: +If z∗ +temp (¯z∗ +temp) not in Z(k), add it to Ztemp, otherwise do nothing. +18: +Z(k+1) = Z(k) ∪ Ztemp and rank all elements in Z(k+1) clockwise +19: +Denote Z(k+1) as {(r(k+1) +0 +,ϕ(k+1) +0 +),...,(r(k+1) +Mk+1,ϕ(k+1) +Mk+1)} +20: +k = k + 1 +21: +if Mk+1 = Mk then +22: +L = L + 1 +Appendix N: Simulation procedure for comparison +For each vehicle l at each station n, we generate the total duration of incidents I(n,l) as a compound +Poisson exponential variable to get the arrival time. Since no overtaking is allowed, the arrival time + +Mo et al.: Evaluation of Public Transit Systems +48 +Article submitted to ; manuscript no. () +at station n cannot be earlier than vehicle l − 1. When a vehicle arrives at a station, passengers +board based on the first-come-first-serve (FCFS) principle up to the vehicle’s capacity C. Queue +lengths at vehicle arrival and passenger waiting times are recorded during the simulation. To ensure +the system reaches steady-state conditions, the first 10% records are dropped. +Algorithm 3 Simulation procedure +1: Initialize model parameters: C, γ, θ, ¯H, Demand factor. Set the total number of vehicles L. +2: for l = 1:L do +3: +Get vehicle dispatch time as DT (l) +4: +for n = 1 : N do +5: +Sample total incident duration I(n,l) from a compound Poisson exponential distribution +6: +t(n,l) +D += min{DT (l) + T (n) + I(n,l),t(n,l−1) +D +} +7: +Headway for vehicle l at station n is t(n,l) +D +− t(n,l−1) +D +8: +Sample the arrival passengers within the headway as a Poisson process based on λ(n). +9: +Record queue length (including left behind passengers from the last run) +10: +Alight passengers based on the binomial distribution with parameter α(n) +11: +Board passengers based on FCFS principle up to the vehicle capacity +12: +Record left behind passengers and passengers’ waiting time +13: Drop the first 10% records. 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Safety Science 106:230–243. + diff --git a/m9AyT4oBgHgl3EQf__pz/content/tmp_files/load_file.txt b/m9AyT4oBgHgl3EQf__pz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ead0989eff96ef95bb9686fae3d0dd4a7644f4a --- /dev/null +++ b/m9AyT4oBgHgl3EQf__pz/content/tmp_files/load_file.txt @@ -0,0 +1,1650 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf,len=1649 +page_content='Evaluation of Public Transit Systems under Short Random Service Suspensions: A Bulk-Service Queuing Approach Baichuan Mo Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Li Jin* UMich Joint Institute, Shanghai Jiao Tong University, Shanghai 200240, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA Haris N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Koutsopoulos Department of Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' MA 02115,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' USA Zuo-Jun Max Shen Department of Industrial Engineering and Operations Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' CA 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' USA Jinhua Zhao Department of Urban Studies and Planning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' USA This paper proposes a stochastic framework to evaluate the performance of public transit systems under short random service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We aim to derive closed-form formulations of the mean and variance of the queue length and waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' A bulk-service queue model is adopted to formulate the queuing behavior in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The random service suspension is modeled as a two-state (disruption and normal) Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We prove that headway is distributed as the difference between two compound Poisson exponential random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The distribution is used to specify the mean and variance of queue length and waiting time at each station with analytical formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The closed-form stability condition of the system is also derived, implying that the system is more likely to be unstable with high incident rates and long incident duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The proposed model is implemented on a bus network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Results show that higher incident rates and higher average incident duration will increase both the mean and variance of queue length and waiting time, which are consistent with the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Crowding stations are more vulnerable to random service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The theoretical results are validated with a simulation model, showing consistency between the two outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Key words : Bulk service queuing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Random disturbances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Public transit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Introduction Public transit systems (PTSs) play a crucial role in cities worldwide, transporting people to jobs, homes, outings, and other activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, PTSs are usually susceptible to unplanned delays and service disruptions, which happen frequently in PTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' According to Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2022), there are 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='00918v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='PR] 3 Jan 2023 Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 2 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () on average 75 incidents happening in the Chicago urban rail system per day and more than 75% of them are less than 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Causes for these short-term suspensions can be signal system failures, passenger behavior, and infrastructure problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' For this reason, it is important to recognize how a PTS is affected by these short-term service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In this study, we consider two key performance metrics of PTSs: queue length and waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Specifically, we model a PTS as a bulk-service queue and aim to derive the closed-form formulations for the mean and variance of passengers’ queue length and waiting time at a station under random service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' To this end, we derive a stability criterion for the passenger queues under the influence of suspensions, which quantifies the throughput loss due to suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We also characterize the steady-state distribution of passenger queues, which naturally leads to the quantification of the aforementioned metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Queuing behavior at a public transit (PT) station is usually modeled as a bulk-service queue model (Powell 1981, Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2014, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Bulk service means that customers are served in groups rather than individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' At a PT station, with the arrival of vehicles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', buses or trains), a group of passengers will board (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', being served in groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' If the vehicle capacity is less than the number of customers waiting, some customers are left behind (Kahraman and Gosavi 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Most of the previous studies on a bulk service model for PTSs focus on stations (Selvi and Rosenshine 1983, Powell 1985, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2014) used a Markov model to extend the station-level analysis to the route level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, these studies all considered PTSs under normal operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The studies of PTSs under service suspensions using queuing analysis are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Regarding the treatment of service disruptions in bulk-service queue models, Madan (1989) first considered a single channel bulk service queue subject to interruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' They assumed there are two states (work and repair) in the system and derived the probability generating function (PGF) of queue length using steady-state equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Many researchers extended Madan (1989)’s framework by considering more channels (Singh and Ram 1991), more heterogeneous states (Madan 1992, Ayyappan and Karpagam 2020), different service interruption assumptions (Jayaraman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 3 1994), and different repair policies (Tadj and Choudhury 2009, Tadj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, all these studies assumed that the service is offered with a fixed batch size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', fixed capacity), which is not valid for PTSs where the available vehicle capacity for boarding is a random variable depending on the current vehicle load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Besides, all these studies used steady-state equations to derive multiple PGFs of queue length under different system states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', work and repair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Results are usually mathematically tedious and the queue length and waiting time can only be analyzed with a very small service batch size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', Madan (1989) only analyzed the problem with service batch size equal to 1 and 2, for batch size more than 3, the closed-form formulas are hard to derive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Finally, previous studies usually consider the breakdown of servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, there is no straightforward way to map the “breakdown of servers” to a PTS with valid real-world assumptions because, in a PTS, the assumption of an independent server is invalid due to inter-station passenger flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' To fill the research gaps, we propose a bulk queue service-based framework to describe the pas- senger and vehicle dynamics for a PTS and analyze the system performance under short random service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The objective of this study is to derive the stability condition of a PTS and the mean and variance of passengers’ queue length and waiting time for each station under random sus- pensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This analysis provides important insights into how short-term service disruptions impact PTSs’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The results are helpful for the future design of PT’s control and planning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Two building blocks of this study are the work by Powell (1985) and Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Powell (1985) proposed a bulk service queue model for transportation terminals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', station-level) with analytical queue length and waiting time formulations under normal conditions using transform methods (as opposed to steady-state equations methods) and Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2014) extended the analysis from station-level to route-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In this study, we explicitly model the random service suspension in a single-route PTS (in reality, it represents a bus route or one-directional rail line, which is a basic element of more complex PT networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Different from typical service interruption studies where servers may break down, we assume a vehicle in the PTS may suffer from random Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 4 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' A detailed discussion of this assumption is provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2, where we show how it corresponds to many real-world situations and can be seen as the first step toward a general incident representation in PTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Under this assumption, we extend Powell (1985) and Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2014)’s work to obtain the mean and variance of passengers’ queue length and waiting time at each station in the single route PTS by analyzing the headway distribution under random service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The major contribution of this paper is fourfold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This is the first study to explore analytically the bulk-service queuing problem involving short random service suspensions applied to PTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We model the service suspension in PTSs by analyzing vehicles’ speed profiles, which is a novel and practical way to consider “server breakdown” in PTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We prove that the headway under random service suspensions can be represented as the difference between two compound Poisson exponential variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We assume there is no vehicle overtaking and approximate the headway distribution as a zero-inflated truncated normal distri- bution to obtain a closed-form moment-generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Based on this we derive the PGF and corresponding moments of the number of arrival passengers within a headway (these are crit- ical components for the bulk-service queue model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This is a new analytical contribution to the bulk-service queuing theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We introduce a Markov chain model to capture the inter-station passenger flow dynamics based on Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2014)’s work, which extends the typical bulk-service queuing analysis from the station level to the route level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We propose an interpolation-based roots-solving method to find all complex roots for this study’s model specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Roots-solving is an essential step to obtain the queue length and waiting time for the bulk-service queuing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Section 2 reviews the literature on the bulk-service queue problem, random service disruptions, and queuing models for PTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Section 3 presents the model settings for a single-route system with random service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Section 4 shows the Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 5 analysis and derivations of the major results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Section 5 provides numerical examples to illustrate the theoretical results and validates the proposed approach using simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Section 6 concludes the paper and discusses future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Literature review 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Bulk queue models In the bulk service queuing literature, customers are served in a batch of fixed or variable lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The service rate may depend on the number of customers waiting for service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The motivation for this model rises from addressing problems in manufacturing systems, elevators, transport systems, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Bailey (1954) originated the study of bulk queues by considering a system with simple Poisson arrivals at a server that serves, at particular points in time, all waiting customers up to a fixed capacity c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' If no customers are waiting, a zero number of customers are served, implying that the server is never idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The queue, denoted by M/Gc/1, is described using an embedded Markov chain defined at points of service completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Immediately following Bailey (1954), Downton (1955) obtained the waiting time distribution of bulk service queues by considering random arrivals and random service time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Jaiswal (1960) confirmed the results in Downton (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' He derived the waiting time distribution using the embedded Markov-chain approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The general bulk service rule was first introduced by Neuts (1967), where a server, upon finishing a batch, may remain idle if there are fewer than m customers waiting for service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Thus all departing batches from the queue have at least m customers, although no more than the service capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Along with and after those milestone studies, papers have appeared which can be differentiated on the basis of the queuing types (arrival process, service process, number of servers), objectives (queues, waiting times, busy periods, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' ), the time domain of the solution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', steady-state or transient), and the method of solution (transforms or direct numerical methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Chaudhry and Templeton (1983) and Sasikala and Indhira (2016) provide a more complete review of the developments in bulk service queue models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 6 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Random service disruptions The subject of queuing systems wherein the service channel is subject to breakdowns is a popular subject that has received a lot of attention in the past fifty years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' For a recent survey of the related literature, readers can refer to Krishnamoorthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, most of the research on this topic deals with models where the server serves the customers one at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The related literature on bulk service is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Madan (1989) studied a single-channel queueing system with Poisson arrivals and exponential service in batches of fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The system is subject to random interruptions with an operating state and a repairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Both the operating times and the repair times of the service channel are assumed to be exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Madan (1992) generalized the model in Madan (1989) to the case where the repairs are performed in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Singh and Ram (1991) extended the model in Madan (1989) by considering a system with three identical channels, with operating and repair times for all three service channels distributed exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Jayaraman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (1994) considered a single-server queueing system with general bulk service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Arrivals are Poisson but alternate between two modes according to whether the server is operational or in the failed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The duration of the operating and repair periods are exponential and phase-type distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Tadj and Choudhury (2009) analyzed a bulk service queueing system with an unreliable server, Poisson input, and general service and repair times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Tadj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2012) considered a bulk service queuing system where service is provided to groups of customers of fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Service consists of two consecutive phases and may take a vacation following the second phase of service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' While providing service, the server may break down and a delay period precedes the repair period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Queuing models in public transit systems Queuing theory in PTSs is usually conducted at the station level, aiming at obtaining the mean queue length and waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In the case of regular services where headways are equal, assuming that a) passengers arrive at stops according to a Poisson process and b) passengers can be served by the first arriving vehicle, the mean waiting time of passengers (E[W]) is given by: E[W] = H/2, (1) Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 7 where H is the service headway and W is the passenger waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This is the most widely used queuing assumption in transit studies (Dial 1967, Clerq 1972, Wirasinghe 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, in the case where service is not reliable, the assumption of regular service can be problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Numerous models have been proposed to account for the stochastic nature of headways (Welding 1957, Osuna and Newell 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' A well-known model proposed by Osuna and Newell (1972) with Poisson arrival passengers and stochastic headways is E[W] = 1 2 · � E[H] + Var[H] E[H] � , (2) where E[H] and Var[H] are the expectation and variance of headways, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In the case of regular services, the variance is zero and the model reverts to Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, the results in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 1 and 2 do not consider the vehicle capacity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', they assume all passengers can board the first vehicle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In a congested PTS, passengers may be left behind due to limited vehicle capacity, leading to an increase in waiting times (Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Bulk service queue models have been applied in PTSs to capture the effects of capacity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Powell (1981, 1983, 1985) used a bulk service queue model to calculate the passenger queue length and waiting times at public transportation terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The closed-form mean and variance for these two quantities are derived using a transform method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Rapoport et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2010) studied bulk service queues with constant or variable capacity and exogenously determined arrival times (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', passenger arrivals based on smart card data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2014) proposed a bulk service and batch arrival queuing model with reneging behavior to estimate passengers’ waiting for public transport services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' All the aforementioned studies consider the queuing analysis at the station level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The extension of queuing analysis from a station level to a route level is not a trivial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' First, the boarding and alighting behavior at upstream stations affect the available capacity distribution at down- stream stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Second, headways may be correlated across stations, leading to different headway distributions for different stations (Marguier 1985, Hickman 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' To address this problem, Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2014, 2015) proposed a Markov model to combine the Powell (1981) and Hickman (2001)’s Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 8 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () approaches and used a bulk service model to analyze system performance at the route level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' How- ever, a limitation of their research is that the calculation of headway correlation does not consider the vehicle capacity (though the capacity constraint is considered in the queuing behavior), result- ing in the inconsistency of model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Also, they assume that headways follow the Erlang distribution, which leads to model tractability but is not consistent with empirical observations (Bellei and Gkoumas 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Our paper can be seen as an extension of Powell (1985)’s and Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2014)’s work to incorporate random service suspensions in a PTS with more consistent and reasonable assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' And we also characterize the headway distribution under service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Service interruptions in public transit systems Studies on service interruption in PTSs can be categorized into two groups: impact analysis and operations control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Impact analysis studies have used a variety of methods to analyze the impact of service disruptions on performance and level of service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Of these methods, the most common is based on graph theory, surveys, simulation, and empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Graph theory-based methods usually derive resilience or vulnerability indicators based on the network topology (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2016, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2018, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2015, Berdica 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' These methods are effective for understanding high-level network properties related to incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Survey-based methods investigate passenger behavior and opinions during incidents (Currie and Muir 2017, Murray-Tuite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2014, Fukasawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2012, Teng and Liu 2015, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Passengers’ individual-level behavior is analyzed and understood using econometric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Simulation-based methods simulate passenger flows on the transit network under incident scenarios (Balakrishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2008, Suarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2005, Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The empirical data-based methods use smart card and vehicle location data to analyze real- world incident impacts (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2016, Tian and Zheng 2018, Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' These studies can output many metrics of interest such as vehicle load, travel delays caused by incidents, distribution of the impact, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Studies focusing on operations control under service disruptions address aspects including shuttle bus design (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2016, Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2019), vehicle holding (O’Dell and Wilson Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 9 1999), integrating local services (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2014), and timetable adjustment (Kroon and Huisman 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The analysis presented in this paper belongs to the “impact analysis” category, which aims to obtain stability conditions for PTSs and the mean and variance of passengers’ queue length and waiting time of each station under short random suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' None of the previous studies has used the bulk service model for this type of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Single-route public transit system and vehicle movements Consider a single-route PTS with N stations as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Vehicles are dispatched from a transportation hub (also referred to as station 0) and travel from station 1 to station N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' At a specific station n, we assume that passenger arrivals follow a Poisson process with a fixed rate λ(n) during the time period of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When a vehicle arrives at station n, each passenger in the vehicle has a probability of α(n) to alight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Thus, the number of alighting passengers at station n follows a binomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Poisson arrivals and binomial alighting are two common assumptions in much of the PT-related literature (Hickman 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In this study, we do not consider reneging behavior of passengers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', passengers may leave the system if they have waited for too long) since the focus of the paper is on “short” service suspensions and we assume passengers choose to wait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Empirical studies (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2016, Rahimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2019) show that passengers start to leave the system only when delays are large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', 30 minutes or more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Incorporating balking and reneging is outside the scope of this paper and can be a future extension of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Let l = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' be a superscript denoting the vehicle run number (or vehicle ID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Smaller l means vehicles are dispatched at an earlier time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Figure 2 summarizes the vehicle and passenger interactions at station n over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Let t(n,l) A be the time that vehicle l arrives at station n, and t(n,l) D the time that vehicle l departs station n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' H(n,l) is the headway between the preceding vehicle l −1 and vehicle l, as they depart from stop n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', H(n,l) = t(n,l) D − t(n,l−1) D ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When a vehicle arrives at station n, some of the onboard passengers alight first, then the queuing passengers start to Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 10 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () Figure 1 Schematic presentation of a single-route public transit system board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Let Q(n,l) be the number of queuing passengers when vehicle l arrives at station n, R(n,l) the number of left-behind passengers when vehicle l departs station n, and Y (n,l) the number of passengers arriving between t(n,l) D and t(n,l+1) A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' By definition, Q(n,l+1) = R(n,l) + Y (n,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (3) Figure 2 Diagram of vehicles and passengers interaction at station n in the time dimension In this study, we assume that the dwell time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', t(n,l) D − t(n,l) A ) is negligible compared to the vehicle travel time (t(n+1,l) A − t(n,l) D ) such that the number of passengers arriving during the dwell Transportation hub and Station 1 Station 2 Station N "Station 0" : : Boarding .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' No Alighting No boarding All passengers passengers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' alighting : passengers passengers alight 2(1) α(1) α(N) Alighting probability 2(2) α(2) Inflow rate a(N) Route Empty Vehicle l Vehicle l Vehicle l vehicle lMo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 11 time is zero (same assumption as in Powell (1981)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Then, given the headway H(n,l+1), Y (n,l)|H(n,l+1) follows a Poisson distribution with parameter λ(n)H(n,l+1): Y (n,l) | H(n,l+1) ∼ Poi(λ(n)H(n,l+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (4) In other words, Y (n,l) can be seen as the number of arriving passengers within a headway (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', H(n,l+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' From the vehicle’s perspective, let S(n,l) be the number of available spaces after passengers alighting from vehicle l at station n, G(n,l) the number of remaining passengers on vehicle l after passengers alighting at station n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' By definition, G(n,l) = C − S(n,l), (5) where C is the capacity of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Denote V (n,l) as the vehicle load (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', number of onboard passengers) when vehicle l departs station n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', the vehicle load when it arrives at station n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Then, the number of alighting passengers from vehicle l at station n given V (n−1,l) follows a binomial distribution: � V (n−1,l) − G(n,l)� | V (n−1,l) ∼ Bin(V (n−1,l),αn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (6) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Random service suspensions and vehicle speed profile Let us assume that there are random service suspensions when a vehicle travels in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Given these disturbances, the speed curve of vehicle l from station n to n + 1 can be described by the red line in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Every random incident causes a speed reduction or stop of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In reality, these incidents can be caused by many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' For example, in a bus system, they may be caused by traffic congestion or accidents, drivers’ or passengers’ behavior, vehicle engine issues, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In a rail system, the reasons may be signal failures, infrastructure problems, and drivers’ or pas- sengers’ behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The speed curve is a general representation of different incidents, interruptions, suspensions, or disruptions that impede the vehicle’s movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 12 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () Figure 3 Schematic speed curve of vehicle l traveling from station n to n + 1 The actual vehicle speed profile under interruptions can be complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' To facilitate math- ematical modeling, we assume that the speed of a vehicle under random interruptions can be approximated by an impulse function (blue line in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The impulse function separates the vehicle trajectory into traveling and stopping phases, denoted as the normal and disruption states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In the normal state, a vehicle travels at a constant speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Once an incident happens, the vehicle stops immediately and enters the disruption state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We assume that, in a sufficiently small time interval, ∆, the probability of incident occurrence is γ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Furthermore, the duration of an incident follows an exponential distribution with rate θ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', mean of 1 θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Then, the state of a vehicle is a two-state Markov process (Figure 4) with the state space of {Normal, Disruption}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Figure 4 Transition diagram of vehicle states For the two-state Markov process, the duration of disruption and normal states follows the exponential distribution with rates θ and γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Approximating the actual speed curve as an impulse function can be seen as the first step toward a general incident representation in PTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Actually, any type of incident can be represented as Station n Stationn + 1 Speed Incident 1 Incident 2 Normal Disruption Normal Disruption Normal state state state state state Actual A fix speed Approximated 0 Time (n,l) +(n+1,l) DMo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 13 a mixture of different types of normal and disruption states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The normal and disruption states can be defined with heterogeneous occurrence probabilities and duration for different categories of incidents, which results in a more sophisticated speed curve representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Headway under random service suspensions Under the assumption of an impulse-function speed profile, all vehicles have the same fixed travel speed under the normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Therefore, if there is no incident in the system, all stations have the same deterministic nominal headway (denoted as ¯H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The relationship among ¯H, route cycle time ¯E (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', the time that a vehicle travels from the transportation hub to the last station and returns to the hub), and fleet size (denoted as ¯F) for the route is ¯H = ¯E ¯F (7) With random service suspensions, the route cycle time would increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' There are two possible responses for the transit agency: 1) To maintain the same planned headway ¯H, the agency needs to increase the fleet size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', number of vehicles) for the route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2) With the same fleet size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', limited resources), the agency would have to increase the planned headway ¯H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In this paper, we consider the second scenario because it reflects incidents’ impact on headway and service performance, which is more relevant to this paper’s topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Therefore, we assume that at the route planning stage, transit agencies have an estimate of the average delay in the cycle time, ¯D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Let I(n,l) be the total duration of all incidents happening during the vehicle l’s travel time from the transportation hub to station n (a random variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Then E[I(N,l)] is the expected incident duration for a vehicle traveling from the transportation hub to the last station N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Assuming the road conditions for two directions of the route are the same, then the total estimated delay for the cycle trip is ¯D = 2 · E[I(N,l)] (8) It is worth noting that some transit agencies may plan the headway by assuming a larger delay (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', not the mean, but the 85% percentile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Hence, we may also formulate ¯D as a general function of Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 14 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () E[I(N,l)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In this study, we adopt Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 8 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Then, the incident-adjusted planned headway (denoted as ¯HAdj) is ¯HAdj = ¯E + ¯D ¯F = ¯H + 2 · E[I(N,l)] ¯F (9) where 2·E[I(N,l)] ¯ F is the planned headway adjustment term due to incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Note that we assume I(N,l) are identically distributed for all l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' So the incident-adjusted planned headway is not affected by vehicle ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In this study, we assume that the single-route PTS will dispatch vehicles based on the incident-adjusted planned headway ¯HAdj and that all dispatches are on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Let T (n) be the travel time for vehicles from the transportation hub to station n when there is no incident ( a fixed constant in this study due to the fixed speed assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Without loss of generality, let us assume vehicle (l −1) departs from the transportation hub at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Considering random service suspensions, vehicle (l − 1)’s departure time from station n is: t(n,l−1) D = T (n) + I(n,l−1) (10) Note that the dwell time is ignored as we assumed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Given that the incident-adjusted planned headway is ¯H + 2·E[I(N,l)] ¯ F , the planned departure time of vehicle l from station n is t(n,l) D = ¯H + 2 · E[I(N,l)] ¯F + T (n) + I(n,l) (11) Therefore, with random incidents, the actual headway of vehicle l at station n is H(n,l) = t(n,l) D − t(n,l−1) D = ¯H + 2 · E[I(N,l)] ¯F + I(n,l) − I(n,l−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (12) In this study, we assume I(n,l) and I(n,l−1) are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This assumption facilitates closed- form derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In reality, if the incidents are caused by road congestion or infrastructure issues, it is possible that the incident durations for two consecutive vehicles passing through the same route segment are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, addressing the correlation is not a trivial problem in the bulk service queue model (Powell 1981) and is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Analysis The objective of this study is to derive the stability conditions of a PTS (Proposition 12) and the mean and variance of passengers’ queue length and waiting (Propositions 5 and 6) time at each station under random service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Figure 5 shows how the distributions of different random variables (particularly, S(n,l),V (n,l),Q(n,l)) are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The major calculation consists of three parts: Given the distribution of V (n−1,l) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', vehicle load when vehicle l departs station n − 1), calculate the distribution of S(n,l) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', the number of available space after passengers alighting from vehicle l at station n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The details are shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1 Given the distribution of S(n,l), calculate the distribution of Q(n,l) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', the number of queuing passengers at station n when vehicle l arrives) and the mean and variance of queue length and waiting time at station n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Given the distribution of S(n,l) and Q(n,l), calculate the distribution of V (n,l), which is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2 Figure 5 Analysis framework With the three components, we can derive the distribution of S(n,l),Q(n,l),V (n,l) for all n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',N given the distribution of V (0,l) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', vehicle load when vehicle l arrives at the first station, it is always zero by definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Note that, in this section, we focus on the steady-state distribution of these variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', l → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3 Mean and variance of queue length and waiting time at station n Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1 Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2 P(V(n-1,l) = P(s(n,l)) P(S(n,l)) = P(Q(n,l) P(Q(n,l), P(S(n,l)) = P(V(n,l)) n=n+1Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 16 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () After obtaining the corresponding distributions, we discuss the stability conditions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='4 and summarize the approach in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Available vehicle space steady-state distribution In this section, we aim to derive the steady-state distribution of S(n,l) given the steady-state dis- tribution of V (n−1,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Define v(n,l) k := P(V (n,l) = k), s(n,l) k := P(S(n,l) = k), and g(n,l) k := P(G(n,l) = k) for all k = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Assuming that the steady state probabilities for all variables exist (the stabil- ity condition will be discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='4), we have v(n) k := liml→∞ v(n,l) k = P(V (n) = k), s(n) k := liml→∞ s(n,l) k = P(S(n) = k), and g(n) k := liml→∞ g(n,l) k = P(G(n) = k), where V (n) = liml→∞ V (n,l), S(n) = liml→∞ S(n,l), and G(n) = liml→∞ G(n,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � Distribution of S(n,l) given P[V (n−1,l)] � : ∀ n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='.,N, given the distribu- tion of V (n−1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', v(n) := [v(n−1) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',v(n−1) C ] ∈ RC+1), the distribution of S(n) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', s(n) := [s(n−1) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',s(n−1) C ] ∈ RC+1) is given as: s(n) k = g(n) C−k ∀k = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',C, (13) where g(n) := [g(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',g(n) C ] ∈ RC+1 and g(n) = v(n−1)A(n), (14) A(n) is a (C + 1) × (C + 1) matrix with the element in row i and column j equal to a(n) ij , and a(n) ij is defined as a(n) ij = � � � � � � � � � � � � � � � � � 1, if i = 0 and j = 0 � i i − j � (α(n))i−j(1 − α(n))j, if i ≥ j and i,j ̸= 0 0, otherwise ∀ i,j = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',C (15) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Vehicle load steady-state distribution In this section, we derive the steady-state distribution of V (n,l) given the steady-state distribution of G(n,l) and Q(n,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Define q(n) k := liml→∞ q(n,l) k = P(Q(n) = k), where Q(n) = liml→∞ Q(n,l) and q(n,l) k = P(Q(n,l) = k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Denote the first C elements of the steady-steady queue length distribution as q(n) 0:C−1, where q(n) 0:C−1 = [q(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',q(n) C−1] ∈ RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 17 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � Distribution of V (n,l) given P[G(n,l)] and P[Q(n,l)] � : ∀ n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='.,N, given the dis- tribution of G(n) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', g(n)) and q(n) 0:C−1, the distribution of V (n) can be expressed as: v(n) = g(n)B(n) (16) where B(n) is a matrix with the element in row i and column j equal to b(n) ij : b(n) ij = � � � � � � � � � � � � � � � � � � � � � � � � � � � q(n) j−i, if 0 ≤ i ≤ j < C 1 − C−i−1 � k=0 q(n) k , if j = C and 0 ≤ i < C 1, if i = j = C 0, otherwise ∀ i,j = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',C (17) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Queuing analysis at a station In this section, assuming that we know the distribution of S(n) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', s(n) = [s(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',s(n) C ] ∈ RC+1), our goal is to derive q(n,l) 0:C−1 and the mean and variance of passenger queue length and waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Probability generating function of queue length We start with deriving the prob- ability generating function (PGF) for Q(n), where Q(n) = liml→∞ Q(n,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � PGF of Q(n)� : ∀ n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='.,N, given the distribution of S(n) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', s(n)), the PGF of Q(n) can be expressed as: Q(z) = �C u=0 s(n) u ��u i=0 q(n) i (zC − zC−u+i) � zC Y (z) − �C u=0 s(n) u zC−u , (18) where Y (z) is the PGF of Y (n) and Y (n) = liml→∞ Y (n,l) is the number of arrival passengers at station n within a headway at the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 18, there are C unknown variables, q(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',q(n) C−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Note that q(n) C does not appear in Q(z) because when u = C and i = C, we have q(n) C (zC −zC−u+i) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' To quantify Q(z), Rouche’s theorem is used (Beardon 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Let Num(z) and Den(z) be the numerator and denominator of Q(z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', Q(z) = Num(z) Den(z) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' As shown in Powell (1981), one can prove that Den(z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', zC Y (z) − �C u=0 s(n) u zC−u) Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 18 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () has exactly C complex roots within (or on) the unit circle on a complex plane using Rouche’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Notice that for any z ∈ C that satisfies |z| ≤ 1, where C is the set of complex numbers, the generating function Q(z) must be analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Therefore, if z∗ is the root of Den(z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', Den(z∗) = 0), it should also be the root of Num(z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', Num(z∗) = 0) such that Q(z) is analytic (Rudin 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Hence, one can solve for q(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',q(n) C−1 using the following two steps: Step 1: Solve Den(z) = 0 for C different roots z∗ 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',z∗ C−1 ∈ C that satisfy |z∗ i | ≤ 1, ∀ 0 ≤ i ≤ C − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Note that z = 1 is always a root of Den(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' But it does not give information about q(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',q(n) C−1 as Num(1) = 0 is naturally satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Hence, we adopt the convention that z∗ 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Step 2: Combining Num(z∗ i ) = 0 (∀ 1 ≤ i ≤ C − 1) and Q(1) = 1, solve for q(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',q(n) C−1 (there are C system equations and C unknown variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Note that when z → 1, both Num(z) and Den(z) approach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Therefore, using L’Hopital’s rule, lim z→1Q(z) = lim z→1 Num′(z) Den′(z) = �C u=0 s(n) u ��u i=0 q(n) i (u − i) � ¯S(n) − ¯Y (n) = 1 (19) where ¯S(n) = �C u=0 us(n) u = E[S(n)], ¯Y (n) = Y ′(1) = E[Y (n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 19 is the equation used to solve for q(n) 0:C−1 (instead of directly using Q(1) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Queue length distribution Though q(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',q(n) C−1 can be obtained by solving C system equations as mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1, we provide a simpler way to calculate q(n) 0:C−1, which is known as matching the polynomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' [Distribution of Q(n) given P[S(n)]]: ∀ n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='.,N, given the distribution of S(n) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', s(n)), all complex roots of Den(z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', z∗ 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',z∗ C−1), and ¯Y (n), if s(n) C > 0, then q(n) 0:C−1 can be solved as: q(n) 0 = 1 s(n) C ( ¯S(n) − ¯Y (n)) C−1 � i=1 z∗ i z∗ i − 1, (20) and q(n) 0:C−1 = ˜η(n)(Λ(n))−1, (21) Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 19 where ˜η(n) = [s(n) C q(n) 0 η(n) 0 ,s(n) C q(n) 0 η(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',s(n) C q(n) 0 η(n) C−1] ∈ RC and Λ(n) = � ������������������ s(n) C s(n) C−1 s(n) C−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' s(n) 1 0 s(n) C s(n) C−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' s(n) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 0 s(n) C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' s(n) 3 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' s(n) 4 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' s(n) C � ������������������ ∈ RC×C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (22) η(n) j is the polynomial coefficient of zj in �C−1 i=0 � 1 − z z∗ i � (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', �C j=0 η(n) j zj := �C−1 i=0 � 1 − z z∗ i � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' As z∗ i is specified for station n, a superscript n is added to the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Note that assuming s(n) C > 0 in Proposition 4 is not restrictive because otherwise we can reduce C such that s(n) C > 0 always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Analytical formulation of mean and variance of queue length and waiting time After solving for q(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',q(n) C−1, Q(z) is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The expectation and variance of the queue length at station n can be written by definition as: E[Q(n)] = ∞ � k=0 kq(n) k = dQ(z) dz ���� z=1 (23) Var[Q(n)] = E[(Q(n))2] − E[Q(n)]2 = d2Q(z) dz2 ���� z=1 + E[Q(n)] − E[Q(n)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (24) Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � Mean and variance of queue length � : ∀ n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='.,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' given the distribution of S(n) and the expression of Y (z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' E[Q(n)] and Var[Q(n)] can be calculated as: E[Q(n)] = ¯¯S(n) + ¯¯Y (n) + ( ¯S(n) − ¯Y (n))[1 + 2( ¯S(n) − C)] − ( ¯S(n) − ¯Y (n))2 2( ¯S(n) − ¯Y (n)) + C−1 � i=1 1 1 − z∗ i (25) Var[Q(n)] = 1 12( ¯S(n) − ¯Y (n))2 � − 4( ¯¯¯S(n) − ¯¯¯Y (n))( ¯S(n) − ¯Y (n)) + 3( ¯¯S(n) + ¯¯Y (n))2 − [6( ¯¯S (n) − ¯¯Y (n)) − 1]( ¯S(n) − ¯Y (n))2 − ( ¯S(n) − ¯Y (n))4 � − C−1 � i=1 z∗ i (1 − z∗ i )2 (26) where ¯¯S(n) and ¯¯¯S(n) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' ¯¯Y (n) and ¯¯¯Y (n)) are the second and third central moments of S(n) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Y (n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 20 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � Mean and variance of waiting time � : ∀ n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='.,N, given the distribution of S(n) and the expression of Y (z), the mean and variance of waiting time at station n (denoted as W (n)) is given as: E[W (n)] = ¯Q(n) t λ(n) (27) Var[W (n)] = ¯¯Q(n) t − ¯Q(n) t (λ(n))2 (28) where Q(n) t is the queue length at an arbitrary time point (as opposed to Q(n) which is the queue length at the time of vehicle arrival).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' ¯Q(n) t and ¯¯Q(n) t are defined as ¯Q(n) t = E[Q(n)] − ¯Y (n) + 1 2 � ¯¯Y (n)/ ¯Y (n) + ¯Y (n) − 1 � (29) ¯¯Q(n) t = Var[Q(n)] − ¯¯Y (n) + 1 12( ¯Y (n))2 � 4 ¯Y (n) ¯¯¯Y (n) + 6( ¯Y (n))2 ¯¯Y (n) − ( ¯Y (n))2 + ( ¯Y (n))4 − 3( ¯¯Y (n))2� (30) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 27 is the application of Little’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 6 is directly obtained from Powell (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The formulation of E[Q(n)], Var[Q(n)], E[W (n)], and Var[W (n)] in this study are equiv- alent to Powell (1985) because in his paper the M/G[S]/1 bulk queue model was considered, where G[S] represents a general (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', arbitrary) bulk-service distribution, which includes the service dis- tribution incorporating random service suspension considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, this does not lower the contribution of this paper because to implement these equations, the formulation of Y (z) needs to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In the next section, we show how random service suspension introduces a new distribution for Y (n), which has not been considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Headway distribution According to Propositions 4 to 6, to calculate q(n) 0:C−1 and the mean and variance of queue length and waiting time, it is essential to specify Y (z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', the PGF of the number of passengers arriving within a headway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4, taking l → ∞ gives that Y (n)|H(n) is a Poisson random variable with parameter λ(n)H(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Therefore, we first consider the distribution of H(n) under the random service suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' According to the discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3, the actual headway for vehicle l at station n is H(n,l) = ¯H + 2·E[I(N,l)] ¯ F + I(n,l) − I(n,l−1), where I(n,l) is the total duration of incidents for vehicle l Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 21 during its travel from the transportation hub to station n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Since ¯H and E[I(n,l)] are constants, obtaining the headway distribution is equivalent to quantifying the distribution of I(n,l) − I(n,l−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Notice that I(n,l) and I(n,l−1) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='d for all l by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' It is useful to first consider the distribution of I(n,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � Distribution of incident duration � : The total incident duration for vehicle l dur- ing its travel from the transportation hub to station n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', I(n,l)) follows a compound Poisson- Exponential distribution with Poisson rate γT (n) and exponential rate θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Mathematically, I(n,l) = K � i=1 Xi where Xi ∼ Exp(θ) ∀i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',K, and K ∼ Poi(γT (n)) (31) The moment generating function (MGF) of a compound Poisson-Exponential variable can be written as (Hogg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2010) MI(n,l)(t) = E[etI(n,l)] = eγT (n)( θ θ−t −1) ∀ t < θ (32) Similarly, the MGF of −I(n,l−1) is M−I(n,l−1)(t) = E[e−tI(n,l−1)] = eγT (n)( θ θ+t −1) ∀ t > −θ (33) From the MGF of I(n,l), we obtain E[I(N,l)] = γT (N) θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Then the headway equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 12) becomes H(n,l) = ¯H + 2γT (N) θ ¯F + I(n,l) − I(n,l−1) (34) The following proposition provides the headway distribution: Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � MGF of headway � : Under the setting of this study, ∀ n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='.,N, the MGF of H(n) can be expressed as MH(n)(t) = et( ¯ H+ 2γT (N) θ ¯ F )e γT (n)( 2t2 θ2−t2 ) (35) From the MGF of H(n), we can obtain the corresponding mean and variance of headway as: E[H(n)] = ¯H + 2γT (N) θ ¯F (36) Var[H(n)] = 4T (n)γ θ2 (37) Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 22 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The results show that random suspensions can increase the mean and variance of headway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The impact on mean headway is through the increase in cycle time at the route planning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The headway variance will increase with a higher incident rate (γ) and higher average incident duration ( 1 θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Meanwhile, our model also captures the headway variance propagation along stations as observed in many previous studies (Andersson and Scalia-Tomba 1981, Hickman 2001): Var[H(n)] increase with the station index n (due to the increase in T (n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, the support of the derived headway distribution is R, meaning that H(n) can be negative due to the overtaking of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The negative value of H(n) will cause problems in the definition of Y (n) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', the number of arrival passengers within a headway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' To address this problem, we assume that drivers are not allowed to overtake the preceding vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This is true for the subway systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Many transit agencies also use this policy for bus operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Given this assumption, the support of H(n) becomes [0,+∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Whenever H(n) < 0, the actual headway will be 0 since the successor vehicle will not pass through the predecessor and they will arrive at the station simultaneously (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', bus bunching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Hence, the new truncated headway, denoted as ˆH(n), has a zero-inflation mixture distribution: ˆH(n) = � � � � � � � 0 if H(n) ≤ 0 H(n) otherwise (38) The zero-inflation truncated headway distribution is also observed in the previous empirical study assuming no overtaking (Bellei and Gkoumas 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, to the best of the author’s knowledge, there is no closed-form MGF for ˆH(n) because the difference between two compound Poisson-exponential random variables has no closed-form proba- bility density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Therefore, to have a tractable headway distribution, we have to approximate H(n) with other distributions for which the corresponding zero-inflation truncated distribution has analytical MGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In this study, we approximate the distribution of H(n) with normal distribution for two reasons: 1) I(n,l) can be seen as the summation of a large number of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='d random variables (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 31) when the Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 23 incident frequency is high (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', K is large, which is true for this study because we are considering high-frequency short random disturbance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Hence, from the Central Limit Theorem (CLT), we may approximate I(n,l) as normally distributed, which leads to H(n) being normally distributed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Approximating the headway disturbance as a normal random variable with the CLT was also used in Daganzo (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2) After approximating H(n) as a normal random variable (denoted as H(n) Normal) with the same mean and variance, the first three moments of H(n) and H(n) Normal are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This shows that the distribution of H(n) is similar to normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Appendix J shows detailed derivations and numerical experiments to validate the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Now let us consider a zero-inflation truncated distribution of H(n) Normal with support [0,+∞] and a probability mass concentrated at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Denote the truncated random variable as ˆH(n) Normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � MGF of approximated headway � : Under the setting of this study, ∀ n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='.,N, the MGF of ˆH(n) Normal can be expressed as M ˆ H(n) Normal(t) = Φ � −( ¯Hθ + 2γT (N) ¯ F ) 2 � T (n)γ � + et( ¯ H+ 2γT (N) θ ¯ F )eγT (n)( 2t2 θ2 ) � 1 − Φ � −( ¯Hθ + 2γT (N) ¯ F ) 2 � T (n)γ − 2t � T (n)γ θ �� (39) where Φ(·) is the cumulative density function (CDF) of a standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Based on the MGF of ˆH(n) Normal, notice that � 1 − Φ � −µ σ �� = Φ � µ σ � , we can get the corresponding mean and variance as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' E[ ˆH(n) Normal] = µ · Φ �µ σ � + σ · φ �−µ σ � (40) Var[ ˆH(n) Normal] = µσφ �−µ σ � + Φ �µ σ �� µ2 + σ2� − � µΦ �µ σ � + φ �−µ σ � σ �2 (41) where φ(·) is the probability density function (PDF) of a standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' It is not clear how incidents will affect the mean headway from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 40 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, the following proposition shows that the mean headway increases as incident frequency (γ) and average incident duration ( 1 θ) increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 24 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � Impact of incidents on headway � : The mean of the zero-inflation truncated headway (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e, either E[ ˆH(n)] or E[ ˆH(n) Normal]) increases with the increase in incident intensity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', increase in γ or 1 θ, or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 10 is useful for the analysis of system stability with respect to incidents, which is shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Distribution of Y (n) The distribution of Y (n) is derived by assuming the headway is ˆH(n) Normal (instead of H(n), which may be negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' To derive the PGF of Y (n), the following lemma is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' For two arbitrary random variable U and V , assume that there is a δ > 0 such that for t in (−δ,δ), the MGF of U|V is MU|V (t) = C1(t)eC2(t)V , where C1(t) and C2(t) are finite functions of t that do not depend on V , and the MGF of V , MV (·), exists and MV [C2(t)] is finite for t in (−δ,δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Then the MGF of U is given by MU(t) = C1(t)MV [C2(t)], −δ < t < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (42) The proof of Lemma 1 can be found in Villa and Escobar (2006) Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � PGF of Y (n)� : Under the setting of this study, ∀ n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='.,N, the PGF of Y (n), Y (z), can be expressed as Y (z) = Φ �−µ σ � + eµλ(n)(z−1)+ σ2(λ(n)z−λ(n))2 2 � 1 − Φ �−µ σ − σλ(n)(z − 1) �� (43) where µ = ¯H + 2γT (N) θ ¯ F and σ = 2√ T (n)γ θ are the mean and standard deviation of H(n) Normal, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 105, we can obtain ¯Y (n), ¯¯Y (n), and ¯¯¯Y (n) by taking corresponding derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The expression of ¯Y (n) is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The expressions for ¯¯Y (n) and ¯¯¯Y (n) are complicated and thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' ¯Y (n) = � µ · Φ �µ σ � + σ · φ �−µ σ �� λ(n) (44) Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Solving for the roots With the expression of Y (z), the only unknown parts for the queue length calculation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 25) are z∗ 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',z∗ C−1, which can be obtained by solving the nonlinear equation Den(z) = 0 (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' It is well known in the queuing literature that solving for the roots of Den(z) is practically difficult because typical optimization algorithms usually only find only one root, while we need to find all C roots within the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This is especially changeling for Y (z) with complex expressions because the objective function can be highly nonlinear (such as Y (z) in this study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In this study, we propose an interpolation-based searching algorithm to efficiently find all roots of Den(z) within the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The key idea is to intelligently set up different initial values for a general root-solving algorithm (such as trust-region and Levenberg-Marquardt algorithms) and let the algorithm converge to different solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', different roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We elaborate on the algorithm details in Appendix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The numerical testing shows that our algorithm is able to find desired roots for all testing scenarios in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' It outperforms the methods in Powell (1985) and Wilson (2014), where both of them have cases of not being able to find all roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Stability condition For all the derivations above, we assume that the steady-state distributions of all variables exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This triggers the discussion about the stability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' At the station level, the stability condition is described in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' � Stability condition � : Under the setting of this study, the bulk-service queuing system at station n is stable if and only if ρ(n) = ¯Y (n) ¯S(n) = � µ · Φ � µ σ � + σ · φ � −µ σ �� λ(n) �C u=0 s(n) u u = λ(n) · E[ ˆH(n) Normal] �C u=0 s(n) u u < 1 (45) where ρ(n) is the utilization ratio for station n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 12 is intuitive as it indicates that station n is stable if the average number of passengers arrived within a headway is smaller than the average available capacity for each arrival vehicle (after alighting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' From Proposition 7, we know that a higher rate of incidents (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', larger Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 26 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () γ) and higher duration of incidents (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', higher 1 θ) increase E[ ˆH(n) Normal], which makes the system more likely to be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Hence, the above result quantifies the throughput loss due to incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' There are some remarks for Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' As ρ(n) depends on s(n) and s(n) depends on the roots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', z∗ 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',z∗ C−1) at station n, there is no direct way to judge the stability at station n without iterating the previous n − 1 stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' But for the first station (n = 1), we have s(1) C = 1 and s(1) u = 0 for all u = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',C − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 45 reduces to ρ(1) = λ(n)·E[ ˆ H(n) Normal] C , which can be used to assess the stability directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Proposition 12 only discusses the stability at the station level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' At the route level, a route is considered stable if “all stations in the route are stable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Mathematically, a route is stable if and only if ρ(n) < 1,∀ n = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' It is worth discussing the relationship of stability of stations n and n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' If station n−1 is stable, then s(n) can be calculated as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1, and the stability of station n can be evaluated accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, if station n−1 is not stable, station n may be stable because there may be passengers alighting at station n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' For this situation, we have v(n−1) C = 1 and v(n−1) k = 0 for all k = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',C − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Then s(n) is determined by the alighting rate at station n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' It is easy to verify that in this situation ¯S(n) = α(n)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' And the stability condition is ρ(n) = λ(n)·E[ ˆ H(n) Normal] α(n)C < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Summary of calculation procedure So far, we have derived the calculation process for key variables of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Algorithm 1 summarizes the calculation procedure, which iterates through the N stations of the route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' This is more efficient and provides more analytical insights than a simulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Numerical example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Experimental design To test the proposed framework, we use an example bus route adapted from Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' (2015) and Hickman (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' There are 10 stations and the attributes for each station are shown in Table Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 27 Algorithm 1 Performance indicators calculation procedure 1: Initialize v(0) 0 = 1 and v(0) k = 0 ∀k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2: for n = 1 : N do 3: g(n) = v(n−1)A(n) ▷ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 14 4: s(n) k = 1 − g(n) C−k ∀k = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',C ▷ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 13 5: Calculate ¯S(n), ¯¯S(n), and ¯¯¯S(n) based on s(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 6: Calculate ¯Y (n), ¯¯Y (n), and ¯¯¯Y (n) ▷ Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='4 7: if ¯Y (n) < ¯S(n) then ▷ Station n is stable 8: Solve the roots z∗ 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',z∗ C−1 for the denominator of Q(z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 18 ▷ Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1 9: Calculate q(n) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',q(n) C−1 based on z∗ 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',z∗ C−1 ▷ Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2 10: Calculate E[Q(n)],Var[Q(n)],E[W (n)], and Var[W (n)] ▷ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 25 - 28 11: v(n) = g(n)B(n) ▷ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' B(n) is a function of q(n) 0:C−1 12: else ▷ Station n is not stable 13: q(n) k = 0 ∀k = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',C − 1 14: Set E[Q(n)],Var[Q(n)],E[W (n)], and Var[W (n)] to infinity 15: v(n) C = 1 and v(n) k = 0 ∀k = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=',C − 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The layout of the bus route is shown in Figure 6, where we assume the no-incident travel time between two consecutive stations is 5 minutes, the total cycle time without incident is ¯E = 100 min, and travel time from the transportation hub to the last station is T (N) = 50 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Table 1 Example bus system parameters Station ID λ(n) (passengers/min) α(n) Station ID λ(n) (passengers/min) α(n) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='75 0 6 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='8 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5 0 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1 4 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='25 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='75 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='25 10 0 1 Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 28 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () Figure 6 Case study route layout To test the sensitivity of performance indicators to different parameters, we consider different values of C, θ, γ, ¯H, and demand (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The demand is adjusted by a scaling factor that is applied to the arrival rates λ(n) in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The fleet size ¯F is determined as ¯ E ¯ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When the sensitivity testing is conducted for one parameter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', C), other parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', θ, γ, ¯H, and the demand factor) are set to their reference values for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Table 2 Scenario design Parameters Value space Reference value C {30, 34, 38} 34 γ (/min) {0, 1/10, 1/5, 1/3} 1/5 θ (/min) {2, 1 ,1/2} 1 ¯H (min) {2, 4, 7} 6 Demand factor {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='8, 1} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Performance indicators The mean and standard deviation of queue length for each station under different testing scenarios are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Generally, for all scenarios, the queue length patterns are consistent with the congestion patterns we expect given the passenger arrival and alighting rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' That is, the expected queue length is relatively higher at stations 2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The expected queue length at the last station is always zero as its passenger arrival rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Figure 7a shows the queue length patterns with respect to bus capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The system is not very sensitive to bus capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The reason is that under the reference scenario, the system is not Transportation hub 0 2 9 10 5 min 5 min 5 min E T(N) = 50 min Half of the cycle time 2Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 29 congested and capacity is not fully utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Thus, increasing capacity does not affect the queuing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Figure 7b shows the impact of incident occurrence rate γ on queue length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When there is no random suspension in the system (γ = 0), the expected queue length at station 8 is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' As the frequency of incidents increases, the system becomes more congested with longer expected queue lengths and higher variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When the incident frequency increases to 1/3 per minute on average (γ = 1/3), the expected queue length at station 8 is increased to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Similar results can be observed for the duration of incidents (Figure 7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When the average incident duration is 30 seconds (θ = 2), E(Q(8)) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When the average incident duration is 2 minutes (θ = 1/2), E(Q(8)) increases to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The impacts of θ and γ on queue length are both more significant at crowded stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The impact of ¯H is shown in Figure 7d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' As expected, higher headway means a lower service rate and thus a higher expected queue length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' As ¯H increases from 2 minutes to 7 minutes, the queue length at station 8 increases from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The impact of the demand factor (Figure 7e) shows similar patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' As the demand factor increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='0, the queue length at station 8 increases from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='1 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The impact of ¯H and the demand factor are relatively similar for crowded and uncrowded stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Figure 8 shows the mean and standard deviation of passenger waiting time for the different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We observe that the downstream stations generally have higher waiting time expectations and variances due to the headway variance propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' For congested stations, such as stations 3 and 8, extra waiting times are observed due to passengers left behind with capacity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Figure 8a shows the impact of capacity on waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Similar to the results on queue length, the impact is not very significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The impacts of γ and θ on waiting times are shown in Figure 8b and 8c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' As increases in γ and 1/θ result in an increase in expected headway, the mean waiting times at all stations are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The impacts on crowding stations are more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When γ = 0, there is no incident in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' In this case, there are no left behind or headway irregularities at any stations and their expected waiting times are all equal to 2 minutes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', 1 2 ¯H, as no incidents mean all stations have the same fixed headway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When γ increases to 1/5, station 3 Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 30 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () (a) Sensitivity on C (b) Sensitivity on γ (c) Sensitivity on θ (d) Sensitivity on ¯H (e) Sensitivity on demand factor Figure 7 Mean and standard deviation of queue length (the shaded part is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2×standard deviation) has left behind passengers and the waiting time is increased to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='6 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' When θ decreases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', mean incident duration increases) from 2 to 1/2, the expected waiting time at station 8 increases from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='0 to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='8 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Changes in ¯H have the most direct impact on the expected waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The increase in planned headway causes an increase in waiting time for all stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' There are a few left-behind passengers observed at stations 3 and 8 when ¯H = 7 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Finally, as demand increases, the waiting time increases only if there are left behind (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', when demand factor = 1) because it does not change the headway distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' At station 3, the increase in the demand factor from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='0 results in an increase in the expected waiting time from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Comparison between simulated and theoretical results To validate the theoretical results, we develop a simulation model to calculate the expectation and variance of queue length and waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The simulation procedure is shown in Appendix N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We compare the simulation and theoretical results for the reference parameter setting (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' A total of 50,000 vehicle runs are simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The comparisons of mean and standard deviation for Expected queue length when vehicle arrives 14 H= 2 H=4 12 H= 7 10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 Station ID14 Expected queue length when vehicle arrives Demand factor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5 Demand factor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='75 12 Demand factor = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='0 10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 Station ID10 Expected queue length when vehicle arrives C = 30 C = 34 C = 38 8 6 4 2 0 1 2 3 4 5 6 7 8 6 10 Station ID12 Expected queue length when vehicle arrives y=0 y= 1/10 10 y= 1/5 y= 1/3 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 Station IDExpected queue length when vehicle arrives 0 = 1/2 14 0 = 1 0 = 2 12 10 8 6 4 2 0 2 3 4 5 6 7 8 9 10 Station IDMo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () 31 (a) Sensitivity on C (b) Sensitivity on γ (c) Sensitivity on θ (d) Sensitivity on ¯H (e) Sensitivity on demand factor Figure 8 Mean and standard deviation of waiting time (the shaded part is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='2×standard deviation) queue length and waiting time are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' We observe that the simulation and theoretical results match well, validating the theoretical model’s correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' However, the theoretical results slightly overestimate the mean and variance of the queue length and waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The main reason may be the approximation of headway distribution as normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' As shown in Figure 11, the actual headway has more probability density concentrated at the mean (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', more peakedness), implying that the actual headway has less probability of deviating from the planned one, thus the simulation scenario may have a smaller queue length and waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Conclusion and discussion This paper proposes a stochastic framework to evaluate the performance of PTSs under short random service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Specifically, we analyze the system stability conditions and derive closed-form formulations for the mean and variance of queue length and waiting time at each station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The derived stability conditions are intuitive and imply that the system is more likely to be unstable with high incident rates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' high incident duration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' high demand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' low service frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' and 8 C = 30 C = 34 7 C = 38 6 5 4 3 Expected 2 1 0 1 2 3 4 5 6 7 8 9 10 Station ID12 y=0 y= 1/5 y= 1/10 y= 1/3 10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 Station ID16 0 = 1/2 0=1 14 0=2 12 10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 Station ID12 H=2 ←H=4 10 ★H=7 8 6 4 2 0 1 2 3 4 5 6 7 8 6 10 Station ID8 Demand factor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='5 Demand factor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='8 7 passenger waiting time Demand factor = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='0 6 5 4 3 Expected 2 1 0 1 2 3 4 5 6 7 8 6 10 Station IDMo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' : Evaluation of Public Transit Systems 32 Article submitted to ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' () (a) E[Q(n)] (b) Std dev[Q(n)] (c) E[W (n)] (d) Std dev[W (n)] Figure 9 Comparison between simulation and theoretical results (reference scenario) low vehicle capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The proposed model is implemented using an example bus network adapted from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' A sensitivity analysis of different parameters (such as incident rate, incident duration, vehicle capacity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=') was conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The results show that congested stations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=', stations with high demand rates) are more vulnerable to random service suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The results are validated with a simulation model, showing consistency between theoretical and simulation outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' The proposed model has several potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 1) It can facilitate the design and planning of PTS with the consideration of random system interruptions, such as the design of headways and the determination of vehicle capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' Moreover, the estimated queue length can be used to evaluate the layout and capacity of congested stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 2) The model can be used to monitor system performance and identify critical stations by inputting the historical demand and incident information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 3) The model can support efficient cost-benefit analysis of approaches to improve services using estimates of waiting time and queue length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' For example, the model can answer that, Theory Simulation 8 f queue length 6 of 4 11 2 0 1 2 3 4 5 6 7 8 9 Station IDTheory Simulation 6 5 4 of lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQf__pz/content/2301.00918v1.pdf'} +page_content=' 3 S 1 0 2 3 4 5 6 L 8 9 Station IDTheory 6 Simulation e time 5 waiting 4 of 3 ation ctat 2 0. So uniqueness of the base 3/2 representation implies im- +mediately uniqueness of the 1/2·3/2 representation AFS(N). This observation obviously extends to base +p/q. +Next we consider the question whether also the sequence y3/2, the sum of digits function modulo 2 of +the base 1/2·3/2 representation, is fixed point of a 2-block substitution. This is indeed the case, and this +2-block substitution is given by Rigo and Stipulanti in [8]. +Theorem 10. ([8]) y3/2 is the fixed point with prefix 00 of the 2-3-block substitution +κ′ : + + + + + + + +00 → 001 +01 → 000 +10 → 111 +11 → 110 + + + + + + + +In [8] the proof of Theorem 10 is based on a generalization of Cobham’s theorem to what are called +S-automatic sequences built on tree languages with a periodic labeled signature. Here we consider a more +direct route, based on a simple closure property of p-q-block-substitutions. Recall that a coding is a letter +to letter map from one alphabet to another. +3 + +Theorem 11. Let x = (x(N)) be a fixed point of a p-q-block substitution. Let r be a positive integer. Then +y = (x(rN)) is a coding of a p-q-block substitution. +Proof. If x is a fixed point of a p-q-block substitution, then x is also fixed point of a pr-qr-block substitution. +As new alphabet, take the words of length r occurring in x. On this alphabet, the pr-qr-block substitution +induces a p-q-block substitution in an obvious way. Mapping each word of length r to its first letter is a +coding that gives the result. +Alternative proof of Theorem 10. We apply Theorem 11 with r = 2. The 4-6-block-substitution is given by +0010 �→ 010101, 0100 �→ 010010, 0101 �→ 010010, 0110 �→ 010101, +1001 �→ 101010, 1010 �→ 101101, 1011 �→ 101101, 1101 �→ 101010. +Coding 00 �→ a, 01 �→ b, 10 �→ c, 11 �→ d, this induces the 2-3-block substitution +ac �→ bbb, ba �→ bac, bb �→ bac, bc �→ bbb, cb �→ ccc, cc �→ cdb, cd �→ cdb, db �→ ccc. +If we code further a, b �→ 0, and c, d �→ 1, then we obtain κ′ from Theorem 10. +Acknowledgement +I am grateful to Jean-Paul Allouche for several useful comments. +References +[1] S. Akiyama, C. Frougny, and J. Sakarovitch, Powers of rationals modulo 1 and rational base number +systems, Israel J. Math. 168 (2008), 53–91. +[2] F. M. Dekking, What Is the Long Range Order in the Kolakoski Sequence?, in: The mathematics of +long-range aperiodic order (Waterloo, ON, 1995), 115-125, NATO Adv. Sci. Inst. Ser. C Math. Phys. +Sci. 489, (1997), Kluwer Acad. Publ., Dordrecht. +[3] F. M. Dekking and M. Keane, Two-block substitutions and morphic words, arXiv:2202.13548 [math.CO], +2022. +[4] T. Edgar, H. Olafson, and J. Van Alstine, Some combinatorics of rational base representations, preprint, +2014. +Available at https://community.plu.edu/ edgartj/preprints/basepqarithmetic.pdf +[5] C. Frougny and K. Klouda, Rational base number systems for p-adic numbers, RAIRO Theor. Inform. +Appl. 46 (2019), 87–106. +[6] On-Line Encyclopedia of Integer Sequences, founded by N. J. A. Sloane, electronically available at +http://oeis.org. +[7] J. Propp, How do you write one hundred in base 3/2? +https://mathenchant.wordpress.com/2017/09/17/How-do-you-write-one-hundred-in-base-3/2? +Ac- +cessed in January 2023. +[8] M. Rigo, M. Stipulanti, Automatic sequences: from rational bases to trees, Discrete Mathematics and +Theoretical Computer Science DMTCS 24 (2022), #25. +2010 Mathematics Subject Classification: Primary 11B85, Secondary 68R15 +Keywords: Base 3/2, Thue-Morse sequence, sum of digits, two-block substitution +4 + diff --git a/odFRT4oBgHgl3EQfcTcd/content/tmp_files/load_file.txt b/odFRT4oBgHgl3EQfcTcd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d50272e15be614ddae71b0e940c14167d90769d4 --- /dev/null +++ b/odFRT4oBgHgl3EQfcTcd/content/tmp_files/load_file.txt @@ -0,0 +1,205 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf,len=204 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='13563v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='CO] 31 Jan 2023 The Thue-Morse sequence in base 3/2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Dekking CWI and Delft University of Technology Faculty EEMCS, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Box 5031 2600 GA Delft, The Netherlands F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='Dekking@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='nl Abstract We discuss the base 3/2 representation of the natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' We prove that the sum of digits function of the representation is a fixed point of a 2-block substitution on an infinite alphabet, and that this implies that sum of digits function modulo 2 of the representation is a fixed point x3/2 of a 2-block substitution on {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' We prove that x3/2 is mirror invariant, and present a list of conjectured properties of x3/2, which we think will be hard to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Finally, we make a comparison with a variant of the base 3/2 representation, and give a general result on p-q-block substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' 1 Introduction A natural number N is written in base 3/2 if N has the form N = � i≥0 di �3 2 �i , (1) with digits di = 0, 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Base 3/2 representations are also known as sesquinary representations of the natural numbers (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=', [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' We write these expansions as SQ(N) = dR(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' d1(N)d0(N) = dR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' d1d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' We have, for example, SQ(7) = 211, since 2 · (9/4) + (3/2) + 1 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' See A024629 in [6] for the continuation of the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' N 0 1 2 3 4 5 6 7 8 9 10 SQ(N) 0 1 2 20 21 22 210 211 212 2100 2101 Ignoring leading 0’s, the base 3/2 representation of a number N is unique (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Let for N ≥ 0 s3/2(N) := i=R � i=0 di(N) be the sum of digits function of the base 3/2 expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' We have (see A244040 in [6]) s3/2 = 0, 1, 2, 2, 3, 4, 3, 4, 5, 3, 4, 5, 5, 6, 7, 4, 5, 6, 5, 6, 7, 7, 8, 9, 5, 6, 7, 5, 6, 7, 7, 8, 9, 8, 9, 10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' In this note we study the base 3/2 analogue of the Thue-Morse sequence (where the base equals 2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=', the sequence (see A357448 in [6]) (x3/2(N)) := (s3/2(N) mod 2) = 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' 1 The Thue Morse sequence is the fixed point starting with 0 of the substitution 0 → 01, 1 → 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' This might be called a 1-2-block substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' A 2-3-block substitution κ on an alphabet A replaces blocks ab of length 2 by words κ(ab) of length 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Its action extends to infinite sequences x by defining κ : x �→ y by y3k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' y3(k+1)−1 = κ(x2kx2k+1), for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' The sequence x3/2 is a fixed point of the 2-3-block substitution κ : \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 00 → 010 01 → 010 10 → 101 11 → 101 \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe Theorem 1 will be proved in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' 2 Sum of digits function and Thue-Morse in base 3/2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='1 Sum of digits function in base 3/2 Let s3/2 = (0, 1, 2, 2, 3, 4, 3, 4, 5, 3, 4, 5, 5, 6, 7, 4, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=') be the sum of digits function of the base 3/2 expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' To describe this sequence we extend the notion of a p-q-block substitution to alphabets of infinite cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' The sequence s3/2 is the fixed point starting with 0 of the 2-3-block substitution given by a, b �→ a, a + 1, a + 2 for a = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' and b = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='. Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' We have d(0) = 0, d(1) = 1 and from the uniqueness of the base 3/2 expansions it follows immediately that d(3N + r) = d(2N) + r for N ≥ 0 and r = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Thus s3/2(3N) = s3/2(2N), s3/2(3N + 1) = s3/2(2N) + 1, and s3/2(3N + 2) = s3/2(2N) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' This gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='2 Thue-Morse in base 3/2 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' This follows directly from Theorem 2 by taking a and b modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' □ Although iterates of κ : 00 → 010, 01 → 010, 10 → 101, 11 → 101 are undefined, we can generate the fixed point x3/2 by iteration of a map κ′ defined by κ′(w) = κ(w) if w has even length, and κ′(v) = κ(w) if v = w0 or v = w1 has odd length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' The fact that the iterates of κ are undefined causes difficulty in establishing properties of x3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' This is similar to the lack of progress in the last 25 years to prove the conjectures on the Kolaskoski sequence, which is also fixed point of a 2-block substitution (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' [2], [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Here is a property that is open for the Kolakoski sequence, but can be proved for x3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' If a word w occurs in x3/2, then its binary complement w∗ defined by 0∗ = 1, 1∗ = 0, also occurs in x3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' First one checks this for all 16 words of length 6 that occur in x3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Note that then also w∗ occurs for all w with |w| ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Here |w| denotes the length of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Let u be a word of length m ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' By adding at most 3 letters at the beginning and/or end of u one can obtain a word v with |v| = 3n that occurs in x3/2 at a position 0 modulo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' But then Theorem 1 gives that v = κ(w) for at least one word w occurring in x3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' The length of w is |w| = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Since κ(w∗) = (κ(w))∗ the result follows by induction on m = |u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' For example, for |u| = m = 7, one has |v| = 9, and so |w| = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Here are some conjectured properties of x3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' 2 Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' x3/2 is reversal invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=', if the word w = w1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' wm occurs in x3/2 then ←− w = wm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' w1 occurs in x3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' x3/2 is uniformly recurrent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=', each word that occurs in x3/2 occurs infinitely often, with bounded gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' The frequencies µ[w] of the words w occurring in x3/2 exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Two conjectured values: µ[00] = 1/10, µ[01] = 4/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' µ is mirror invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=', µ[w] = µ[w∗] for all words w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Conjecture 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' µ is reversal invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=', µ[w] = µ[←− w ] for all words w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Conjecture 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Shallit) The critical exponent (=largest number of repeated blocks) of x3/2 is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' 3 Base 3/2 and base 1/2·3/2 Many authors refer to the paper [1] from Akiyama, Frougny, and Sakarovitch for the properties of base 3/2 expansions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=', [7], [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' However, the p/q expansions studied in paper [1] are different from the 3/2 expansions that are usually considered as in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' In paper [1]: N = � i≥0 di 1 q �p q �i , (2) with digits di = 0, 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' We write AFS(N) for the expansion of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Here is the table given in [1] for the case p = 3, q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' N 0 1 2 3 4 5 6 7 8 9 10 AFS(N) ε 2 21 210 212 2101 2120 2122 21011 21200 21202 These expansions will not even be found in the OEIS (at the moment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' The situation is clarified in the paper [5] by Frougny and Klouda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Here both representations are considered and called respectively base p/q and base 1/q·p/q representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' A combination of the results in [1] and [5] yields a proof of the uniqueness of the base 3/2 expansions (QS(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' There is also a direct proof of uniqueness in [4], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Note that AFS(N) = QS(2N) for N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' So uniqueness of the base 3/2 representation implies im- mediately uniqueness of the 1/2·3/2 representation AFS(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' This observation obviously extends to base p/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Next we consider the question whether also the sequence y3/2, the sum of digits function modulo 2 of the base 1/2·3/2 representation, is fixed point of a 2-block substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' This is indeed the case, and this 2-block substitution is given by Rigo and Stipulanti in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' ([8]) y3/2 is the fixed point with prefix 00 of the 2-3-block substitution κ′ : \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 00 → 001 01 → 000 10 → 111 11 → 110 \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe In [8] the proof of Theorem 10 is based on a generalization of Cobham’s theorem to what are called S-automatic sequences built on tree languages with a periodic labeled signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Here we consider a more direct route, based on a simple closure property of p-q-block-substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Recall that a coding is a letter to letter map from one alphabet to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' 3 Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Let x = (x(N)) be a fixed point of a p-q-block substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Let r be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Then y = (x(rN)) is a coding of a p-q-block substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' If x is a fixed point of a p-q-block substitution, then x is also fixed point of a pr-qr-block substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' As new alphabet, take the words of length r occurring in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' On this alphabet, the pr-qr-block substitution induces a p-q-block substitution in an obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Mapping each word of length r to its first letter is a coding that gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Alternative proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' We apply Theorem 11 with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' The 4-6-block-substitution is given by 0010 �→ 010101, 0100 �→ 010010, 0101 �→ 010010, 0110 �→ 010101, 1001 �→ 101010, 1010 �→ 101101, 1011 �→ 101101, 1101 �→ 101010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Coding 00 �→ a, 01 �→ b, 10 �→ c, 11 �→ d, this induces the 2-3-block substitution ac �→ bbb, ba �→ bac, bb �→ bac, bc �→ bbb, cb �→ ccc, cc �→ cdb, cd �→ cdb, db �→ ccc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' If we code further a, b �→ 0, and c, d �→ 1, then we obtain κ′ from Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Acknowledgement I am grateful to Jean-Paul Allouche for several useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' References [1] S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Dekking and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Keane, Two-block substitutions and morphic words, arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='13548 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='CO], 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Edgar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Olafson, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Van Alstine, Some combinatorics of rational base representations, preprint, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Available at https://community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='plu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='edu/ edgartj/preprints/basepqarithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='pdf [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Frougny and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Klouda, Rational base number systems for p-adic numbers, RAIRO Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' 46 (2019), 87–106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' [6] On-Line Encyclopedia of Integer Sequences, founded by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Sloane, electronically available at http://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Propp, How do you write one hundred in base 3/2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' https://mathenchant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='wordpress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content='com/2017/09/17/How-do-you-write-one-hundred-in-base-3/2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Ac- cessed in January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Rigo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' Stipulanti, Automatic sequences: from rational bases to trees, Discrete Mathematics and Theoretical Computer Science DMTCS 24 (2022), #25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} +page_content=' 2010 Mathematics Subject Classification: Primary 11B85, Secondary 68R15 Keywords: Base 3/2, Thue-Morse sequence, sum of digits, two-block substitution 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFRT4oBgHgl3EQfcTcd/content/2301.13563v1.pdf'} diff --git a/pdFRT4oBgHgl3EQfdTeD/content/2301.13567v1.pdf b/pdFRT4oBgHgl3EQfdTeD/content/2301.13567v1.pdf new file mode 100644 index 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Alexander Borzunov 1 2 +Abstract +Many deep learning applications benefit from +using large models with billions of parameters. +Training these models is notoriously expensive +due to the need for specialized HPC clusters. In +this work, we consider alternative setups for train- +ing large models: using cheap “preemptible” in- +stances or pooling existing resources from multi- +ple regions. We analyze the performance of exist- +ing model-parallel algorithms in these conditions +and find configurations where training larger +models becomes less communication-intensive. +Based on these findings, we propose SWARM +parallelism1, a model-parallel training algorithm +designed for poorly connected, heterogeneous +and unreliable devices. SWARM creates tem- +porary randomized pipelines between nodes that +are rebalanced in case of failure. +We empiri- +cally validate our findings and compare SWARM +parallelism with existing large-scale training ap- +proaches. Finally, we combine our insights with +compression strategies to train a large Trans- +former language model with 1B shared param- +eters (≈13B before sharing) on preemptible T4 +GPUs with less than 200Mb/s network. +1. Introduction +For the past several years, the deep learning community has +been growing more reliant on large pretrained neural net- +works. The most evident example of this trend is natural lan- +guage processing, where the parameter count of models has +grown from hundreds of millions (Vaswani et al., 2017; Rad- +ford et al., 2018; Devlin et al., 2019) to billions (Narayanan +et al., 2021; Raffel et al., 2020; Wang & Komatsuzaki, 2021; +Sun et al., 2021) to hundreds of billions (Brown et al., 2020; +*Equal contribution +1HSE University 2Yandex 3University +of +Washington. +Correspondence +to: +Max +Ryabinin +. +1SWARM parallelism is a backronym for Stochastically Wired +Adaptively Rebalanced Model Parallelism. +Fedus et al., 2021; Chowdhery et al., 2022; Rae et al., 2021) +with consistent gains in quality (Kaplan et al., 2020). Like- +wise, many models in computer vision are reaching the +billion-parameter scale (Ramesh et al., 2021; Zhai et al., +2021; Dai et al., 2021; Dhariwal & Nichol, 2021). +At this scale, the models no longer fit into a single accelera- +tor and require specialized training algorithms that partition +the parameters across devices (Krizhevsky et al., 2012; Dean +et al., 2012). While these model-parallel algorithms use dif- +ferent partitioning strategies, they all share the need to per- +form intensive device-to-device communication (Narayanan +et al., 2019; 2021). Also, if a single device fails, it will +cause the entire training process to break down. As a re- +sult, model-parallel algorithms are typically deployed in +dedicated high-performance computing (HPC) clusters or +supercomputers (Shoeybi et al., 2019; Rajbhandari et al., +2020; Narayanan et al., 2021). +This kind of infrastructure is notoriously expensive to build +and operate, which makes it available only to a few well- +resourced organizations (Larrea et al., 2019; Strohmaier +et al., 2021; Langston, 2020). Most researchers cannot +afford the experiments necessary for a proper evaluation +of their ideas. This ultimately limits the scientific progress +for many important research areas, such as solving NLP +problems in “non-mainstream” languages. +Several recent works propose more cost-efficient distributed +training strategies that leverage fleets of temporary “pre- +emptible” instances that can be dynamically allocated in +regions with low demand for hardware and electricity, mak- +ing them 2–10 times cheaper than their dedicated counter- +parts (Harlap et al., 2017). Another solution is to train in +“collaborations” by pooling together preexisting resources +or using the help of volunteers (Diskin et al., 2021; Atre +et al., 2021; Ryabinin & Gusev, 2020; Yuan et al., 2022). +However, training in either of those setups requires special- +ized algorithms that can adapt to the changing number of +workers, utilize heterogeneous devices and recover from +hardware and network failures. While there are several prac- +tical algorithms for unreliable hardware (Kijsipongse et al., +2018; Lin et al., 2020; Ryabinin et al., 2021), they can only +train relatively small models that fit into the memory of the +arXiv:2301.11913v1 [cs.DC] 27 Jan 2023 + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +smallest device. This limits the practical impact of cost- +efficient strategies, because today’s large-scale experiments +often involve models with billions of parameters. +In this work, we aim to find a practical way of training large +neural networks using unreliable heterogeneous devices +with slow interconnect. We begin by studying the impact +of model size on the balance between communication and +computation costs of pipeline-parallel training. Specifically, +increasing the size leads computation costs to grow faster +than the network footprint, thus making household-grade +connection speeds more practical than one might think. +This idea inspires the creation of SWARM parallelism, a +pipeline-parallel approach designed to handle peer failures +by prioritizing stable peers with lower latency. In addition, +this approach periodically rebalances the pipeline stages, +which allows handling devices with different hardware and +network speeds. +In summary, we make the following contributions: +• We analyze the existing model-parallel training tech- +niques and formulate the “Square-Cube Law” of dis- +tributed training: a counterintuitive observation that, +for some methods, training larger models can actually +decrease the network overhead. +• We develop SWARM parallelism, a decentralized +model-parallel algorithm2that leverages randomized +fault-tolerant pipelines and dynamically rebalances +nodes between pipeline stages. To the best of our +knowledge, this is the first decentralized algorithm +capable of billion-scale training on heterogeneous un- +reliable devices with slow interconnect. +• Combining +insights from +the +square-cube law, +SWARM parallelism, and 8-bit compression, we show +that it is possible to train a billion-scale Transformer +language model on preemptible servers with low-power +GPUs and the network bandwidth of less than 200Mb/s +while achieving high training throughput. +2. Background & Related Work +2.1. Model-parallel training +Over the past decade, the deep learning community has +developed several algorithms for training large neural net- +works. Most of them work by dividing the model between +multiple workers, which is known as model parallelism. +The exact way in which these algorithms divide the model +determines their training performance and the maximum +model size they can support. +2The +code +for +our +experiments +can +be +found +at +github.com/yandex-research/swarm +Traditional model parallelism. +Historically, the first +general strategy for training large models was to assign +each device to compute a subset of each layer (e.g., a +subset of neurons), then communicate the results between +each other (Krizhevsky et al., 2012; Ben-Nun & Hoefler, +2019; Tang et al., 2020). Since each device stores a frac- +tion of layer parameters, this technique can train models +with extremely wide layers that would not fit into a single +GPU. However, applying traditional model parallelism to +deep neural networks comes at a significant performance +penalty, as it requires all-to-all communication after each +layer. As a result, while intra-layer parallelism is still widely +used (Shazeer et al., 2018; Rajbhandari et al., 2020), it is +usually applied within one physical server in combination +with other strategies (Krizhevsky, 2014; Chilimbi et al., +2014; Jia et al., 2019; Narayanan et al., 2021). +Pipeline parallelism +circumvents the need for expensive +all-to-all communication by assigning each device with one +or several layers (Huang et al., 2019). During the forward +pass, each stage applies its subset of layers to the inputs +supplied by the previous stage, then sends the outputs of +the last layer to the next stage. For the backward pass, +this process is reversed, with each pipeline stage passing the +gradients to the device that supplied it with input activations. +To better utilize the available devices, the pipeline must +process multiple microbatches per step, allowing each stage +to run in parallel on a different batch of inputs. In prac- +tice, the number of microbatches is limited by the device +memory: this results in reduced device utilization when +processing the first and the last microbatches, known as +the “bubble” overhead (Huang et al., 2019). To combat this +issue, subsequent studies propose using activation check- +pointing, interleaved scheduling, and even asynchronous +training (Narayanan et al., 2019; 2021; Huang et al., 2019; +Shoeybi et al., 2019; Yang et al., 2019). +Aside from model parallelism, there two more strategies +for training large models: data parallelism with dynamic +parameter loading (Rajbhandari et al., 2020) and model- +specific algorithms such as Mixture-of-Experts (Shazeer +et al., 2017). We discuss these algorithms in Appendix B +and compare the performance of offloading with SWARM +in Section 4.2 and Appendix E. +2.2. Distributed training outside HPC +The techniques described in Section 2.1 are designed for +clusters of identical devices with rapid and reliable commu- +nication, making them a natural fit for the HPC setup. As we +discussed earlier, such infrastructure is not always available, +and a more cost-efficient alternative is to use “preemptible” +instances (Li et al., 2019; Zhang et al., 2020; Harlap et al., +2017) or volunteer computing (Kijsipongse et al., 2018; +Ryabinin & Gusev, 2020; Atre et al., 2021; Diskin et al., +2021). However, these environments are more difficult for + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +distributed training: each machine can disconnect abruptly +due to a failure or preemption. Besides, since there is a +limited number of available instances per region, training at +scale often requires operating across multiple locations or +using different instance types. +To handle unstable peers and heterogeneous devices, the +research community has proposed elastic and asynchronous +training methods, correspondingly. Moreover, training large +models over heterogeneous devices can be optimized with +global scheduling (Yuan et al., 2022). We describe these +methods in more detail in Appendix B; importantly, neither +of them are unable to satisfy all the constraints of our setup. +By contrast, the largest models have billions of parameters, +which exceeds the memory limits of most low-end comput- +ers. However, model-parallel algorithms are not redundant, +which makes them more vulnerable to hardware and net- +work failures. There exist two methods that allow training +large models with unreliable devices (Ryabinin & Gusev, +2020; Thorpe et al., 2022): however, the first one supports +only specific architectures and requires at least 1Gb/s band- +width, whereas the second one has no publicly available +implementations, relies on redundant computations for fault +tolerance and considers only the homogeneous setup. +2.3. Communication efficiency and compression +In this section, we discuss techniques that address training +with limited network bandwidth or high latency, such as +gradient compression or overlapping computation with com- +munication phases. These techniques are often necessary +for distributed training without high-speed connectivity, be- +cause otherwise the performance of the system becomes +severely bottlenecked by communication. +Efficient gradient communication. +Data-parallel train- +ing requires synchronization of gradients after each back- +ward pass, which can be costly if the model has many pa- +rameters or the network bandwidth is limited. There exist +several methods that approach this problem: for example, +Deep Gradient Compression (Lin et al., 2018) sparsifies the +gradients and corrects the momentum after synchronization, +while PowerSGD (Vogels et al., 2019) factorizes the gradi- +ents and uses error feedback to reduce the approximation +error. Recently, Wang et al. (2022) proposed to compress +the changes of model activations, achieving high-speed com- +munication for finetuning models of up to 1.5B parameters. +Alternatively, Dettmers (2015) uses 8-bit quantization to +compress gradients before communication. We evaluate +it along with compression-aware architectures, leaving the +exploration of more advanced approaches to future work. +Besides gradient compression, another effective technique +is to use layer sharing (Lan et al., 2020), which reduces the +number of aggregated gradients by a factor of how many +times each layer is reused. +Overlapping +communication +and +computation. +Model, pipeline, and data parallelism all have syn- +chronization points and require transfer of gradients or +activations. +One way to reduce the transfer cost is to +overlap communication with computation, hiding the +synchronization latency. +This overlap can be achieved +by combining parallelism techniques (Krizhevsky, 2014; +Rajbhandari et al., 2020), by synchronizing gradients +layer-by-layer in lockstep with backpropagation (Paszke +et al., 2019), or by using pure pipeline parallelism (Huang +et al., 2019; Narayanan et al., 2019). +However, pure +pipeline parallelism requires many stages to effectively hide +the latency. To overcome this problem, we study inter-layer +compression techniques that work well even with relatively +few pipeline stages. +3. Communication-efficient model parallelism +In this section, we outline our approach for training large +models with heterogeneous unreliable poorly-connected de- +vices. To that end, the section is organized as follows: +• Section 3.1 analyzes how existing model-parallel al- +gorithms scale with model size and shows conditions +where training increasingly larger models leads to less +intense network usage; +• Section 3.2 describes SWARM parallelism — a decen- +tralized algorithm for training large models under the +conditions outlined in Section 2.2. +3.1. The square-cube law of distributed training +To better understand the general scaling properties of model +parallelism, we need to abstract away from the application- +specific parameters, such as model architecture, batch size, +and system design. To that end, we first consider a simplified +model of pipeline parallelism. Our “pipeline” consists of +k stages, each represented by n×n matrices. Intuitively, +the first matrix represents the input data and all subsequent +matrices are linear “layers” applied to that data. This model +abstracts away from application-specific details, allowing us +to capture general relationships that hold for many models. +During “training”, stages iteratively perform matrix multi- +plication and then send the output to the subsequent pipeline +stage over a throughput-limited network. These two opera- +tions have different scaling properties. The compute time +for naïve matrix multiplication scales as O(n3). While this +can be reduced further in theory (Coppersmith & Winograd, +1990; Alman & Williams, 2021), it is only used for very +large matrices (Zhang & Gao, 2015; Fatahalian et al., 2004; +Huang et al., 2020). Therefore, deep learning on GPUs +typically relies on O(n3) algorithms. +In turn, the communication phase requires at most O(n2) +time to transfer a batch of n×n activations or gradients. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +“input” +n×n +⇒ +× += +⇒ +× +“input” +n×n +“weight” +n×n +“output” +n×n +O(n3) computation +O(n2) communication +Stage k +Stage k+1 +Figure 1: (Left) An intuitive explanation of the square-cube law, (Right) Rela- +tive device utilization for Transformer layers using Tesla V100 and 500Mb/s +network bandwidth. See Section 4.1 and Appendix F for a detailed setup. +base +768 units +1 layer +xxlarge +4096 units +1 layer +gpt-3 +12288 units +1 layer +ours +4096 units +12 layers +0% +50% +100% +18% +82% +32% +68% +82% +18% +89% +11% +Processing time +Idle time +Therefore, as we increase the model size, the computation +time grows faster than communication time, regardless of +which matrix multiplication algorithm we use. We refer +to this idea as the square-cube law after the eponymous +principle in physics (Galileo, 1638; Allen, 2013). +This principle applies to many real-world neural network ar- +chitectures, albeit with some confounding variables. In con- +volutional neural networks (Fukushima, 1980), the compu- +tation time scales as O(BHWC2) and the communication +is O(BHWC), where B, H, W and C stand for batch size, +height, width and the number of channels. Recurrent neural +networks (Rumelhart et al., 1986; Hochreiter & Schmid- +huber, 1995) need O(BLH2) compute in terms of batch +size, sequence length, and hidden size, respectively, and +O(BLH) or O(BH) communication, depending on the ar- +chitecture. With the same notation, Transformers (Vaswani +et al., 2017) require O(BL2H) compute for attention lay- +ers, O(BLH2) compute for feedforward layers, but only +O(BLH) communication. +Based on these observations, we conclude that pipeline par- +allelism naturally grows more communication-efficient with +model size. More precisely, increasing the hidden dimen- +sion will reduce the communication load per device per unit +of time, making it possible to train the model efficiently +with lower network bandwidth and higher latency3. While +the exact practical ramifications depend on the use case, +Section 4.1 demonstrates that some of the larger models +trained with pipeline parallelism can already train at peak +efficiency with only hundreds of Mb/s bandwidth. +In theory, the square-cube principle also applies to intra- +layer parallelism, but using this technique at 500 Mb/s +would become practical only for layer sizes of more than +216 units. Data-parallel training with sharding or offload- +ing (Ren et al., 2021) does not scale as well, as its communi- +cation time scales with the size of model parameters instead +of activations. However, it may be possible to achieve simi- +lar scaling with gradient compression algorithms. +3Latency slows the communication down by a constant factor +that also grows less important with model size. +3.2. SWARM parallelism +Traditional pipeline parallelism can be communication- +efficient, but this alone is not enough for our setups. Since +training devices can have different compute and network +capabilities, a pipeline formed out of such devices would +be bottlenecked by the single “weakest link”, i.e., the par- +ticipant with the smallest training throughput. As a result, +the more powerful nodes along the pipeline would be un- +derutilized due to either lack of inputs or slow subsequent +stages. On top of that, if any node fails or leaves training +prematurely, it will stall the entire training procedure. +To overcome these two challenges, we replace the rigid +pipeline structure with temporary “pipelines” that are built +stochastically on the fly during each iteration. Each par- +ticipant can send their outputs to any peer that serves the +next pipeline stage. Thus, if one peer is faster than others, +it can process inputs from multiple predecessors and dis- +tribute its outputs across several weaker peers to maximize +utilization. Also, if any participant disconnects, its predeces- +sors can reroute their requests to its neighbors. New peers +can download up-to-date parameters and optimizer statistics +from remaining workers at the chosen stage. This allows +the training to proceed as long as there is at least one active +participant per stage: we elaborate on the fault tolerance of +SWARM parallelism in Appendix A. +The resulting system consists of several consecutive swarms, +as depicted in Figure 2. Peers within one swarm serve the +same pipeline stage (i.e., the same subset of layers with the +same parameters). We assume that the model consists of +similar “blocks” and thus partition it into evenly sized stages, +leaving the study of better strategies (Huang et al., 2019; +Narayanan et al., 2019) as future work. During the forward +pass, peers receive inputs from predecessors (determined on +each iteration) and send activations to peers in the next stage. +For the backward pass, peers receive gradients for outputs, +compute gradients for layer inputs and accumulate gradients +for parameters. Once enough gradients are accumulated, +peers form groups, run All-Reduce to average gradients +within their pipeline stages and perform the optimizer step. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +T4 +T4 +T4 +T4 +T4 +T4 +T4 +T4 +T4 +T4 +T4 +T4 +A100 +T4 +T4 +T4 +T4 +STAGE 1 +SWARM +STAGE 2 +SWARM +STAGE 3 +SWARM +2 +1 +3 +Pipeline stages +Activation links +Normal +Failure +Rewired +A100 +T4 +device switches stage +state +Workers +Load balancing +move +Alive Dead +Figure 2: An overview of SWARM parallelism, illustrating both normal operation, device failures and adaptive rebalancing. +One of the workers at stage 2 leaves; another peer from stage 3 takes its place by downloading the latest stage 2 parameters +and statistics from peers. +SWARM parallelism can also use Delayed Parameter Up- +dates (DPU) (Ren et al., 2021) to further improve hardware +utilization by performing the optimizer step in parallel with +processing the next batch. While it is technically asyn- +chronous, DPU was shown to achieve similar per-iteration +convergence as fully synchronous training, both theoreti- +cally (Stich & Karimireddy, 2020; Arjevani et al., 2020) and +empirically (Ren et al., 2021; Diskin et al., 2021). +Each peer has queues for incoming and outgoing requests +to maintain high GPU utilization under latency and to com- +pensate for varying network speeds. Similarly to other +pipeline implementations (Huang et al., 2019; Narayanan +et al., 2021), SWARM parallelism uses activation check- +pointing (Griewank & Walther, 2000; Chen et al., 2016) to +reduce the memory footprint. +Stochastic wiring. +To better utilize heterogeneous de- +vices and recover from faults, we dynamically “wire” each +input through each stage and pick devices in proportion to +their training throughput. To achieve this, SWARM peers +run “trainer” processes that route training data through the +“stages” of SWARM, balancing the load between peers. +For each pipeline stage, trainers discover which peers +currently serve this stage via a Distributed Hash Table +(DHT, Maymounkov & Mazieres, 2002). Trainers then +assign a microbatch to one of those peers based on their +performance. If that peer fails, it is temporarily banned +and the microbatch is sent to another peer within the same +stage. Note that trainers themselves do not use GPUs and +have no trainable parameters, which makes it possible to +run multiple trainers per peer. +Each trainer assigns data independently using the Inter- +leaved Weighted Round-Robin (Katevenis et al., 1991; +Tabatabaee et al., 2020) scheduler. Our specific implementa- +tion of IWRR uses a priority queue: each peer is associated +with the total processing time over all previous requests. A +training minibatch is then routed to the node that has the +smallest total processing time. Thus, for instance, if device +A takes half as long to process a sample as device B, the +routing algorithm will choose A twice as often as B. Fi- +nally, if a peer does not respond or fails to process the batch, +trainer will “ban” this peer until it reannounces itself in the +DHT, which is done every few minutes. For a more detailed +description of stochastic wiring, please refer to Appendix C. +Curiously, different trainers can have different throughput es- +timates for the same device because of the network topology. +For instance, if training nodes are split between two cloud +regions, a given peer’s trainer will have a higher throughput +estimate for peers in the same data center. In other words, +trainers automatically adjust to the network topology by +routing more traffic to peers that are “nearby”. +Adaptive swarm rebalancing. +While stochastic wiring +allows for automatic rebalancing within a stage, addi- +tional cross-stage rebalancing may be required to maximize +throughput, especially when devices are very unreliable. +As we described in Section 2.2, our workers can join and +leave training at any time. If any single pipeline stage loses +too many peers, the remaining ones will face an increased +processing load, which will inevitably form a bottleneck. +SWARM parallelism addresses this problem by allowing +peers to dynamically switch between “pipeline stages” to +maximize the training throughput. Every T seconds, peers +measure the utilization rate of each pipeline stage as the +queue size. Peers from the most underutilized pipeline +stage will then switch to the most overutilized one (see +Figure 2 for an overview and Appendix D for a formal +description and complexity analysis), download the latest +training state from their new neighbors and continue training. +Similarly, if a new peer joins midway through training, it is +assigned to the optimal pipeline stage by following the same +protocol. As a side effect, if one pipeline stage requires +more compute than others, SWARM will allocate more +peers to that stage. In Appendix I, we evaluate our approach +to dynamic rebalancing in realistic conditions. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +4. Experiments +4.1. Communication efficiency at scale +Before we can meaningfully evaluate SWARM parallelism, +we must verify our theoretical observations on communi- +cation efficiency. Here we run several controlled experi- +ments that measure the GPU utilization and network usage +for different model sizes, using the Transformer architec- +ture (Vaswani et al., 2017) that has been widely adopted +in various fields (Lin et al., 2021). To decouple the perfor- +mance impact from other factors, we run these experiments +on homogeneous V100 GPU nodes that serve one pipeline +stage over the network with varying latency and bandwidth. +We use a batch size of 1 and sequences of 512 tokens; the +complete configuration is deferred to Appendix F. +First, we measure how the model size affects the compu- +tation to communication ratio at 500 Mb/s network band- +width in both directions. We consider 4 model configura- +tions: the base configuration from the BERT paper (De- +vlin et al., 2019), “xxlarge" (“large” with dmodel=4096), +which is used in several recent works (Lan et al., 2020; Sun +et al., 2021; He et al., 2020), and a GPT-3-scale model with +dmodel=12288 (Brown et al., 2020). We also evaluate a +modified Transformer architecture (“Ours”) as defined in +Section 4.3 with dmodel=4096, 3 layers per pipeline stage +and 8-bit quantized activations. As we demonstrate in Ap- +pendix J, this compression strategy can significantly reduce +network usage with little effect on convergence. In the first +three configurations, the model consists of 12 Transformer +layers placed on 12 servers with a single GPU; in the last +one, there are 4 servers, each hosting 3 layers. Appendix F +contains FLOP and parameter counts of each configuration. +As depicted in Figure 1 (right) and Figure 3, larger models +achieve better GPU utilization rate in the same network +conditions, since their communication load grows slower +than computation. More importantly, even at 500 Mb/s, the +resulting GPU idle time can be pushed into the 10–20% +range, either naturally for GPT-3-sized models or through +base +768 units +1 layer +xxlarge +4096 units +1 layer +gpt-3 +12288 units +1 layer +ours +4096 units +12 layers +0ms +2000ms +4000ms +6000ms +568ms +1204ms +5317ms +1158ms +7059ms +832ms +66ms/298ms +GPU Computation +Waiting for network +Figure 3: Pipeline computation and idle time per batch at +500 Mb/s bandwidth. +Table 1: Relative device utilization at 500 Mb/s bandwidth +and varying network latency. +Latency +(RTT) +Relative GPU utilization +(100% - idle time) +base +xxlarge +GPT-3 +Ours +none +18.0% +32.1% +82.1% +89.5% +10ms +11.8% +28.9% +79.3% +87.2% +50ms +4.88% +20.1% +70.3% +79.5% +100ms +2.78% +14.9% +60.2% +71.5% +200ms +1.53% +10.1% +48.5% +59.2% +activation compression for smaller models. In addition, +large models maintain most of their training efficiency at +the 100ms latency (Table 1), which is roughly equivalent to +training on different continents (Verizon, 2021). +4.2. Detailed performance comparison +Here we investigate how SWARM parallelism compares to +existing systems for training large models: GPipe (Huang +et al., 2019) and ZeRO-Offload (Ren et al., 2021). The pur- +pose of this section is to compare the training throughput in +“ideal” conditions (with homogeneous reliable devices and +balanced layers), as deviating from these conditions makes +it infeasible to train with baseline systems. We benchmark +individual SWARM components in preemptible setups in +Appendices I and H. +We evaluate training performance for sequences of 4 Trans- +former layers of identical size distributed over 16 workers. +The pipeline does not contain embeddings or language mod- +eling heads, as it would result in imbalance between the +stages. Similarly to Section 4.1, we use two layer configu- +rations: “xxlarge” (dmodel=4096, dFFN=16384, 32 heads) +and “GPT-3” (dmodel=12288, dFFN=49152, 96 heads). The +microbatch size is 4 for “xxlarge” and 1 for “GPT-3”, and +the sequence length is 512. +To provide a more detailed view of the training performance, +we measure two separate performance statistics: the training +throughput and the All-Reduce time. The training through- +put measures the rate at which the system can process train- +ing sequences, i.e., run forward and backward passes. In +turn, the All-Reduce time is the time each system spends +to aggregate those accumulated gradients across devices. +The total time per step can be computed as batch_size +/ throughput + all_reduce_time. +Intuitively, +training with small batch sizes is more sensitive to the All- +Reduce time (since the algorithm needs to run All-Reduce +more frequently) and vice versa. +Hardware setup: Each worker uses a V100-PCIe GPU +with 16 CPU threads (E5 v5-2660v4) and 128 GB RAM. +The only exception is for ZeRO-Offload with “GPT-3” lay- +ers, where we had to double the RAM size because the +system required 190GB at peak. Similarly to Section 4.1, + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +Table 2: Training performance for different model sizes. +System +Throughput, samples/s All-Reduce time, s/round +No latency +Latency +No latency +Latency +“GPT-3” +SWARM +0.619 +0.558 +441.7 +455.4 +GPipe +0.633 +0.477 +403 +469.6 +1F1B +0.638 +0.482 +Offload +0.382 +0.382 +1527.9 +1635.4 +“xxlarge” +SWARM +2.358 +2.161 +45.36 +51.269 +GPipe +2.541 +0.957 +44.17 +64.828 +1F1B +2.550 +0.987 +Offload +3.08 +3.08 +168.71 +252.26 +each worker can communicate at a 500 Mb/s bandwidth +for both upload and download for a total of 1 Gb/s. In +terms of network latency, we consider two setups: with no +latency, where workers communicate normally within the +same rack, and with latency, where we introduce additional +100 ± 50ms latency directly in the kernel4. +GPipe configuration: We use a popular PyTorch-based +implementation of GPipe5. The model is partitioned into 4 +stages repeated over 4 model-parallel groups. To fit into the +GPU memory for the “GPT-3” configuration, we offload the +optimizer into RAM using ZeRO-Offload. Before averaging, +we use PyTorch’s built-in All-Reduce to aggregate gradients. +We evaluate both the standard GPipe schedule and the 1F1B +schedule (Narayanan et al., 2019). +ZeRO-Offload configuration: Each worker runs the entire +model individually, then exchanges gradients with peers. +For “xxlarge”, we use the official implementation from (Ren +et al., 2021). However, for “GPT-3”, we found that optimizer +offloading still does not allow us to fit 4 layers into the GPU. +For this reason, we also offload the model parameters using +the offload_param option. +In turn, when training smaller models, ZeRO-Offload outper- +forms both SWARM and GPipe. This result aligns with our +earlier observations in Figure 1, where the same model spent +most of the time waiting for the communication between +pipeline stages. +We also observe that ZeRO-Offload takes longer to aggre- +gate gradients, likely because each peer must aggregate +the entire model, whereas in SWARM and GPipe, peers +aggregate a single pipeline stage. The variation between +All-Reduce time in GPipe and SWARM is due to implemen- +tation differences. Overall, SWARM is competitive to HPC +baselines even in an idealized homogeneous environment. +4More specifically, tc qdisc add dev <...> root +netem delay 100ms 50ms +5The source code is available at https://github.com/ +kakaobrain/torchgpipe +4.3. Large-scale distributed training +To verify the efficiency of SWARM parallelism in a practi- +cal scenario, we conduct a series of large-scale distributed +experiments using preemptible (unreliable) cloud T4 and +A100 GPUs over a public cloud network. +We train a Transformer language model with the architecture +similar to prior work (Brown et al., 2020; Wang & Komat- +suzaki, 2021; Black et al., 2021) and 1.01 billion parameters +in total. Our model consists of 3 stages, each containing a +single Transformer decoder block with dmodel = 4096 and +16 layers per pipeline stage. All workers within a stage serve +the same group of layers, and all layers within each group +use the same set of parameters, similarly to ALBERT (Lan +et al., 2020). On top of this, the first stage also contains +the embedding layer, and the last stage includes the lan- +guage modeling head. Because of layer sharing, this model +is equivalent to a 13B model from (Brown et al., 2020) in +terms of compute costs. +We use 8-bit compression (Dettmers et al., 2021) for activa- +tions and gradients to reduce the communication intensity. +Additional training setup details are covered in Appendix G. +SWARM nodes run rebalancing every T = 300 seconds, +and trainers measure peer performance using a moving aver- +age with α = 0.1. However, as we show in Appendix I, the +throughput of SWARM is not very sensitive to the choice of +these hyperparameters. +First, to verify that model parallelism with asynchronous +updates does not have significant convergence issues, we +train the model on the Pile (Gao et al., 2020) dataset with +400 preemptible T4 instances, each hosting one accelerator. +As a baseline, we use regular data-parallel training with +offloading on 128 A100 GPUs. We run both experiments +for approximately 4 weeks and compare the learning curves. +Figure 4 shows the results of this experiment: it can be seen +that the training dynamics of two approaches are indeed +similar, which demonstrates the viability of SWARM paral- +lelism for heterogeneous and poorly-connected devices. +0 +10B +20B +30B +40B +50B +Tokens processed +2 +3 +4 +5 +6 +7 +Training loss +SWARM +DDP +Figure 4: Training convergence comparison. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +Table 3: Pipeline throughput, layer sharing. +Hardware +setup +Throughput, +samples/s +Optimal +bandwidth, Mb/s +Actual Best-case Upload Download +T4 +17.6 +19.2 +317.8 +397.9 +A100 +16.9 +25.5 +436.1 +545.1 +T4 & A100 +27.3 +— +— +— +Table 4: Pipeline throughput, default Transformer. +Hardware +setup +Throughput, +samples/s +Actual Best-case +T4 +8.8 +288.1 +A100 +8.0 +382.5 +T4 & A100 +13.4 +— +In the next experiment, we aim to measure the pipeline +throughput in different hardware conditions and to com- +pare it with an estimate of best-case pipeline performance. +We consider several setups: first, we use the same 400 pre- +emptible T4 nodes; in another setup, we use 7 instances +with 8 A100 GPU each; finally, we combine these fleets to +create a heterogeneous setup. We examine the performance +of the pipeline both with weight sharing and with standard, +more common, Transformer blocks. +We measure the number of randomly generated samples +processed by the pipeline both in our infrastructure and the +ideal case that ignores all network-related operations (i.e., +has infinite bandwidth and zero latency). The ideal case is +emulated by executing a single pipeline stage 3 times locally +on a single server and multiplying the single-node estimates +by the number of nodes. +As demonstrated in the left two columns of Table 3 and +Table 4, asynchronous training of compute-intensive models +with 8-bit compressed activations regardless of the architec- +ture specifics allows us to achieve high performance without +a dedicated networking solution. Furthermore, the load bal- +ancing algorithm of SWARM allows us to dynamically and +efficiently utilize different hardware without being bottle- +necked by slower devices. +Next, we use the same load testing scenario to estimate +the bandwidth required to fully utilize each device type in +the above infrastructure. For this, we measure the aver- +age incoming and outgoing bandwidth on the nodes that +serve the intermediate stage of the pipeline. We summarize +our findings in the right two columns of Table 3: it turns +out that with layer sharing and 8-bit compression, medium- +performance GPUs (such as T4) can be saturated even with +moderate network speeds. Based on our main experiment, +the optimal total bandwidth is roughly 100Mb/s higher than +the values reported in Table 3 due to gradient averaging, +loading state from peers, maintaining the DHT and stream- +0 +2 +4 +6 +8 +10 12 14 16 18 20 22 24 26 28 30 32 +Time, hours +12 +13 +14 +15 +16 +17 +Samples/second +No rebalancing +SWARM, T=60 +SWARM, T=300 +Optimal +Figure 5: Throughput of rebalancing methods over time. +ing the training data. Although training over the Internet +with more efficient hardware might indeed underutilize the +accelerator, this issue can be offset by advanced compres- +sion strategies such as compression-aware architectures or +layer sharing, as shown in Table 3. +4.4. Adaptive rebalancing evaluation +Lastly, we validate the efficiency of the peer rebalancing +algorithm proposed in Section 3.2. We use statistics of the +number of active T4 nodes from the 32-hour segment of the +experiment described in the beginning of this section. We +compare our strategy with a baseline that has no rebalancing +and with an always optimal strategy. Appendix I contains +further details and analysis of the results shown in Figure 5 +and Table 5. Notably, our strategy provides a significant +improvement over the baseline that grows over time, and +this improvement persists even with infrequent rebalancing. +Table 5: Relative throughput comparison of pipeline rebal- +ancing methods. +Rebalancing +% of optimal +Overall First 1h Last 1h +None +82.7 +99.0 +45.4 +T = 300 +95.8 +99.4 +88.9 +T = 60 +97.6 +99.8 +91.7 +5. Conclusion +In this work, we evaluate the feasibility of high-throughput +training of billion-scale neural networks on unreliable peers +with low network bandwidth. We find that this is feasible +by training very large models with pipeline parallelism. 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Answers to common questions +Why not just use data parallelism with offloading? +Regular data parallelism requires all-reduce steps where +peers exchange gradients, which can be prohibitively ex- +pensive for large models. For example, a 1 billion param- +eter model with 16-bit gradients requires 2 GB of data to +be synchronized between all n devices. We need at least +n messages to perform this synchronization. If we have +100 devices with bidirectional communication, each client +would need to send 2 GB of data to finish the synchroniza- +tion. Thus, with slow interconnects, such synchronizations +are not practical. +Why not just use fully sharded data parallelism with +elasticity? +Sharded data parallelism requires all-to-all +communication of parameter buffers at each layer. Each +of these communications can be done in parallel and has a +size of parameter count divided by n; in total, n messages +are required. Thus, for 1B parameters in 16-bit precision, a +total of 2 GB need to be synchronized for both the forward +and backward pass. For low-bandwidth devices with 100 +Mb/s speed, this would entail an overhead of 5.5 minutes per +forward/backward pass, which is difficult to overlap with +computation. This is exacerbated further, because all-to-all +communication latency is determined by the slowest peer. +Thus, sharded data parallelism can be particularly inefficient +for setups where peers have different network bandwidths. +Should I use SWARM in a supercomputer? +By default, +SWARM is worse than traditional parallelism due to ex- +tra complexity (see experiments in Section 4.2. However, +SWARM can be useful in case of supercomputers that have +heterogeneous devices. +ZeRO-Offload allows one to train 13B parameters on a +single V100, so why do I need SWARM? +Using ZeRO- +Offload can slow down training due to the slow data transfer +between external memory and the accelerator. Training +with SWARM can accelerate training while also allowing +the training of large models. See Appendix E for a more +detailed comparison. +Is it worth using preemptible instances and SWARM +from an economic standpoint? +Due to a significantly +smaller cost per hour, one can leverage a larger amount of +computation when using spot instances compared to on- +demand cloud VMs or dedicated HPC setups. See Ap- +pendix K and Table 9 for a comparison of both hourly and +total costs for an example large-scale pretraining task. +When should I avoid using SWARM? +SWARM is very +efficient at training large models with more than 1B pa- +rameters. For smaller models, a sharded data-parallel ap- +proach can be more optimal. For HPC environments with +homogeneous networking, standard sharded data-parallel +or pipeline-parallel training will be more efficient than +SWARM because the environment is stable and predictable, +so rebalancing is not required. For HPC environments which +are so extensive that failure of a node is likely, the practical- +ity of SWARM depends on how many nodes are expected to +fail. Elastic sharded data parallelism is better than SWARM +if the number of expected failures is relatively low. +Can I use SWARM without layer sharing or quantiza- +tion? +Yes, SWARM can still be effective in these scenar- +ios. Our bandwidth experiments in the main paper give some +idea what the network overhead is. By using no quantization, +which means using regular 16-bit activations, the network +overhead increases roughly by a factor of two. Without layer +sharing, the overhead within each pipeline stage to synchro- +nize the gradients is increased by the number of layers not +being shared. As such, a rough estimate of the efficiency of +SWARM in these scenarios can be estimated by taking our +model size and network bandwidth requirements data and +multiplying it by the relevant factor. +How many pipeline stages can SWARM have? +While it +might theoretically work with any number of pipeline stages, +using long pipelines can result in reduced training through- +put. Similarly to traditional pipeline parallelism, SWARM +suffers from the pipeline “bubble” problem (Huang et al., +2019). More specifically, at the beginning of initial batch +processing, peers near the end of the “pipeline” will be +waiting for inputs. Likewise, early layers will be idle after +processing the final microbatch. In theory, this can be cir- +cumvented using asynchronous updates (Narayanan et al., +2019; Yang et al., 2019), but we did not investigate them in +this work due to potential convergence issues. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +How much failure can SWARM handle? +As long as +there is at least one operational peer at every pipeline stage +and at least one trainer, SWARM will work without any +issues. The key factors defining the training run state at a +given SGD step are the model parameters, the optimizer +statistics, the data loader state, and the step number (re- +quired for proper scheduling). The up-to-date parameters +and optimizer statistics, as well as the step number, are nat- +urally located on all active nodes of a given stage, since +they are required for training. Thus, when a peer joins the +network, it can download the checkpoint corresponding to +the current training state from other peers. +As we mention in Section 3.2, peer failures do not affect +forward and backward passes as long as there is at least one +peer at the required stage: because of rewiring, it is possible +to resend activations or gradients to another worker that has +identical model weights by construction. Similarly, the data +loader state can be recomputed from the last known SGD +step. However, we do not track the order of examples sam- +pled within the same batch; because of the i.i.d. assumption +in the large-scale training setup, the distribution of gradients +is expected to be the same. Hence, if the peer leaves from +the pipeline stage, other workers can compute gradients and +replace those accumulated by the disconnected peer, so that +the number of examples for an SGD step stays the same. +Do the compression-aware architecture modifications +apply only to Transformers? +Bottleneck and maxout +compression are general compression techniques that can +be applied to any layer in any architecture. However, their +effectiveness may vary depending on where in the model +they are applied and what kind of model these are applied +to (for example, CNNs vs. RNNs vs. Transformers). +Some configurations in Section 4.1 measure less than +20% GPU idle time, while many HPC systems only +achieve ≈ 80% GPU utilization. Does this mean that +SWARM is 30% faster? +No, because these are differ- +ent measurement types. (Narayanan et al., 2021) measures +GPU utilization as a fraction of theoretical peak FLOP/s of +their GPUs. In contrast, we only measure what fraction of +time the GPU is running the model, regardless of efficiency. +Since no deep learning workload can achieve 100% peak +FLOP/s, 20% GPU idle time for SWARM means that it can +reach ≈ 0.8x the training throughput compared to training +with an infinitely fast network. As a rule of thumb, one +can say that SWARM will run at a 20% slower speed than +systems described by (Narayanan et al., 2021) using the +infrastructure that is several times cheaper. +B. Additional Related Work +Dynamic parameter loading. +Several recent studies pro- +pose alternative execution algorithms that allow training +large models with data parallelism. +Since neural net- +works typically use a small fraction of weights at any +given moment, the remaining “inactive” parameters can be +sharded (Rajbhandari et al., 2020) or offloaded to external +memory (Pudipeddi et al., 2020; Ren et al., 2021; Rajbhan- +dari et al., 2021). In sharded data parallelism (Rajbhandari +et al., 2020), inactive tensors are distributed across all n de- +vices such that each device stores 1 +nth of all parameters. For +active layers, the shards are gathered such that each device +holds the entire tensor just-in-time for computation. After +the computation, the parameters’ memory is freed so that +only the sharded memory remains ( 1 +nth per device). This +makes it very memory efficient to store model and optimizer +states for inactive layers if many devices are available. Sim- +ilarly to tensor parallelism, these algorithms can support +arbitrary models without the need for layer partitioning and +can, in principle, run a large model on a single GPU, which +is useful for finetuning and inference. +Architecture-specific methods. +Finally, some distributed +training algorithms take advantage of specific layers, such +as locally connected layers (Dean et al., 2012; Coates et al., +2013), Mixture-of-Experts (Jacobs et al., 1991; Shazeer +et al., 2017; Lepikhin et al., 2020), Switch layers (Fedus +et al., 2021) or Product Key Memory (Lample et al., 2019). +These layers contain many near-independent parts that can +be assigned to different devices. They can easily scale to +an extremely large number of parameters with a relatively +small increase in compute (Shazeer et al., 2017). However, +they are also less parameter-efficient (Fedus et al., 2021) +and may not apply to all architectures. +Optimal scheduling for distributed training. +When the +configuration of each peer is known, it is possible to sig- +nificantly optimize the pipeline scheduling by going be- +yond the greedy approach with global optimization tech- +niques (Zheng et al., 2022; Tarnawski et al., 2021), even +with heterogeneous hardware (Yuan et al., 2022). How- +ever, we consider a setup in which this is not possible: pre- +emptible and volunteer peers can join at any point of the +experiment, and dynamically rescheduling and orchestrating +them in a centralized manner is technically difficult because +of the communication and reliability constraints. +Elastic training. +To train with a dynamic number of work- +ers, deep learning practitioners have developed elastic train- +ing algorithms (TorchElastic; ElasticHorovod). If a worker +leaves or fails during training, these algorithms rebalance +the load between the remaining nodes and continue the train- +ing procedure (Harlap et al., 2017; Ryabinin et al., 2021). If +new workers join during training, they get the latest model +parameters from their peers and train alongside them. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +Asynchronous training. +Another important problem is +distributed training on devices with uneven performance. +One way to solve this problem is to use asynchronous train- +ing, where nodes compute gradients at their own pace and +aggregate them using a parameter server (Recht et al., 2011; +Kijsipongse et al., 2018) or a decentralized network (Lian +et al., 2017). This idea allows full utilization of each device, +but may reduce the convergence rate due to “stale” gradi- +ents (Recht et al., 2011; Aji & Heafield, 2019). Several +studies (Li et al., 2020; Ryabinin et al., 2021; Ren et al., +2021; Diskin et al., 2021) propose hybrid techniques that +remove some synchronization points while maintaining the +per-iteration convergence. +C. Stochastic wiring details +Our approach uses stochastic wiring, a specialized routing +algorithm designed around heterogeneous unreliable devices +and high network latency. The core idea of stochastic wiring +is to route each training microbatch through random devices +from each pipeline stage, such that the workload of each +device is proportional to its performance. The performance +of the peer is measured as an exponentially weighted average +of its response time, and all peers serving a specific stage +are stored in a priority queue. We formally describe the +components of stochastic wiring in Algorithm 1. +From a system design perspective, each worker runs a sepa- +rate trainer process that forms microbatches and routes them +through pipeline stages (forward and backward pass). As +we describe earlier in Section 3.2, trainers run Interleaved +Weighted Round Robin (Katevenis et al., 1991; Tabatabaee +et al., 2020) (IWRR) scheduling to dynamically assign mi- +crobatches to peers based on each peer’s training throughput +(“samples per second”) in a balanced way. +An important observation is that stochastic wiring allows +SWARM to mitigate network latency. +Unlike existing +pipeline algorithms (Huang et al., 2019), SWARM workers +do not get blocked if their neighbors take too long to pro- +cess a minibatch. Instead, each SWARM device maintains +a queue of microbatches assigned by trainers. In case of a +latency spike, workers keep processing previously queued +microbatches, maintaining high device utilization. +D. Description and complexity of adaptive +rebalancing +Algorithm 2 contains the formal definition of the adaptive re- +balancing procedure. As described previously, each worker +of SWARM that hosts model layers continuously updates +the information about its load in parallel with processing +the incoming requests. Each T seconds, the peers measure +the total load for all stages of the pipeline, and the peer with +the lowest queue size from the stage with the minimum load +moves to the stage with the maximum load. In principle, +Algorithm 1 Pseudocode of stochastic wiring +input the number of pipeline stages N, the set of active +servers S, smoothing parameter γ, initial priority ϵ +1: ▷ Initialization +2: ema = dict() +3: queues = list() +4: for i ∈ 1, . . . , N do +5: +queues.append(PriorityQueue()) +6: end for +7: def add_server(server): +8: +ema[server] = ε +9: +for i ∈ get_blocks_served_by(server): +10: +queues[i].update(server, priority=ε) +11: def ban_server(server) : +12: +for i ∈ get_blocks_served_by(server): +13: +queues[i].update(server, priority=∞) +14: def choose_server(i): +15: +server, priority = queues[i].top() +16: +new_priority = priority + ema[server] +17: +for j ∈ get_blocks_served_by(server) : +18: +queues[j].update(server, priority=new_priority) +19: +return server +20: ▷ Forward pass with stochastic wiring +21: def forward(inputs): +22: +layer_index = 0 +23: +while layer_index < N: +24: +server = choose_server(layer_index) +25: +t = get_current_time() +26: +try: +27: +inputs = server.forward(inputs) +28: +layer_index = layer_index + 1 +29: +∆t = get_current_time() - t +30: +ema[server] = γ · ∆t + (1 − γ)· ema[server] +31: +catch (ServerFault, Timeout): +32: +ban_server(server) +33: +return inputs +the algorithm could be extended to support moving multiple +peers simultaneously; however, as we show in Appendix I, +even in the current form the algorithm bridges most of the +gap between the optimally balanced pipeline and the system +without any rebalancing. +The complexity of Algorithm 2 can be estimated as follows: +for M as the highest number of peers over all stages, we +have O(M) operations in Lines 9–11 and Lines 22–24, and +all other operations take constant time for a single stage. +These operations are nested in the loop over all stages, which +means that the total complexity of the algorithm is O(MS). +For practical numbers of both peers (e.g., < 10,000) and +stages (fewer than 100), this incurs a negligible overhead on +performance, as all communication and computation is done +in parallel with the actual forward and backward passes. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +Also, notice that only one migrating peer needs to stop +processing requests and download the weights and optimizer +statistics of the pipeline stage it starts serving: this means +that the overall network load of this procedure is relatively +small, as all DHT requests handle scalar data and do not +exceed the number of active peers for each worker. +In practice, the algorithm handles slight deviations in local +time and network/DHT latencies by allowing the peers to +wait for straggling nodes in Line 9 for a predefined time- +out. If a node does not join the rebalancing procedure by +reporting its load in time or joins the network too late, it is +omitted from the current iteration. +Algorithm 2 Adaptive rebalancing for SWARM parallelism +input peer index i, current peer stage scur, total number of +stages S, rebalancing period T +1: while active do +2: +Sleep for T seconds +3: +Measure qi as the local request queue size +4: +Write (i, qi) as the key-subkey pair to DHT[scur] +5: +Initialize minimum and maximum load stages: +smin = smax := −1, +6: +lmin := ∞, lmax := −∞ +7: +for s in 1, . . . , S do +8: +Initialize the load buffer L = 0 +9: +for (j, qj) in DHT[s] do +10: +L := L + qj +11: +end for +12: +if L > Lmax then +13: +smax := s, Lmax := L +14: +end if +15: +if L < Lmin then +16: +smin := s, Lmin := L +17: +end if +18: +end for +19: +if scur = smin then +20: +// Migrate to the maximum load stage +21: +Initialize the minimum load peer imin +:= +−1, qmin := ∞ +22: +for (j, qj) in DHT[s] do +23: +if qj < qmin then +24: +imin := j, qmin := qj +25: +end if +26: +end for +27: +if imin = i then +28: +// This peer should migrate +29: +scur := smax +30: +Download up-to-date parameters from peers in +smax +31: +end if +32: +end if +33: end while +E. Relation between SWARM and +ZeRO-Offload +In this section, we argue that depending on the use of DPU, +SWARM-parallel training is equivalent to either fully syn- +chronous training or the semi-synchronous training pro- +posed in ZeRO-Offload (Ren et al., 2021). That is, SWARM +produces exactly the same stepwise updates as conventional +distributed training algorithms and will therefore achieve a +solution in the same number of steps. +This observation is similar to how many advanced dis- +tributed training techniques (Huang et al., 2019; Rajbhan- +dari et al., 2020) are computationally equivalent to regu- +lar synchronous training on a single device. For instance, +despite using advanced distributed computation strategies, +GPipe (Huang et al., 2019) computes exactly the same math- +ematical expression to obtain gradients and applies those +gradients in the same order as any other synchronous train- +ing algorithm. On the other hand, PipeDream (Narayanan +et al., 2019) changes the order in which the updates are ap- +plied, introducing the so-called stale gradients (Recht et al., +2011). This allows PipeDream to improve device utilization +but has been shown to reduce the final model quality in +some setups (Wang et al., 2020). +Despite using randomized routing and asynchronous com- +munication between pipeline stages, SWARM still performs +optimizer steps synchronously after peers collectively reach +the required global batch size (which is a hyperparameter). +While different peers may accumulate a different number of +samples, they will all use the same gradient after averaging. +Any peer that fails or does not meet this condition is con- +sidered a straggler and must reload its state from neighbors +before it can resume training. This procedure ensures that +all surviving peers use non-stale aggregated gradients over +the specified batch size when performing the optimizer step. +The only deviation from fully synchronous training is that +SWARM uses the same approach for CPU offloading as +ZeRO-Offload, and by extension, delayed parameter up- +dates (DPU). While DPU was shown not to affect conver- +gence (Ren et al., 2021; Stich & Karimireddy, 2020; Ar- +jevani et al., 2020), one can disable this functionality and +make SWARM fully equivalent to standard training. +Naturally, these guarantees come at the cost of reduced +hardware utilization, as a small portion of devices will need +to wait after every step. However, as we show in Section 4.3, +SWARM can still train with competitive training throughput +due to the fact that large models are trained with increased +batch sizes (Brown et al., 2020). + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +F. Additional details for Section 4.1 +We benchmark four versions of the Transformer layer: +• “base”: dmodel = 768, dFFN = 3072, 12 heads; +• “xxlarge”: dmodel = 4096, dFFN = 16384, 32 heads; +• “GPT-3” (Brown et al., 2020): dmodel = 12288, +dFFN = 49152, 96 heads. +• “Ours”: dmodel = 4096, dFFN = 16384, 32 heads, 3 +layers per pipeline stage. +In Table 6, we report FLOP and parameter counts of each +version based on the expressions from (Kaplan et al., 2020). +For simplicity, we set up each experiment with 12 Trans- +former layers using 12 servers (4 for “Ours”) with a sin- +gle V100-PCIE GPU each. The servers communicate at +500Mbps under 3–6ms latency. +Due to modest communication bandwidth, smaller models +spend most of the time waiting for the network. However, +that same bandwidth allows for > 80% GPU utilization +when dealing with GPT-3-sized layers. If we co-locate 3 +GPT-3 layers per pipeline stage, the GPU utilization can +further improved to > 90%. +The time reported in Section 4.1 is the time required to +run forward and backward pass for all layers with a batch +of 1x512 tokens, not including the Adam updates. All +results are averaged over 1000 consecutive batches; the +standard deviations are below 0.1%. All four GPUs are +in the same data center but on different servers. +Each +layer is a TransformerEncoderLayer from PyTorch +1.7.0 (Paszke et al., 2019) wrapped with activation check- +pointing. We use hivemind==0.8.15 (Ryabinin & Gu- +sev, 2020) with a single synchronous trainer based on the +BERT training code from the Transformers library (Wolf +et al., 2020). However, these results are not specific to hive- +mind and are likely reproducible in FairScale (Baines et al., +2021) or PyTorch RPC. The only important detail is that the +training code should run as much communication as possi- +ble in the background while the GPUs are busy processing +batches. It is important to reuse the same connection for +multiple RPC calls so that the TCP buffer does not have +to warm up during each call. Also, our implementation +performs quantization asynchronously with communication +and other computations. +Table 6: Parameter and FLOP counts of each architecture. +Architecture +Parameters +FLOP count +“base” +7.08M +2.2 × 1010 +“xxlarge” +201M +6.2 × 1011 +“GPT-3” +1.81B +5.5 × 1012 +“Ours” +201M +1.8 × 1012 +G. Additional details for Section 4.3 +We use the standard Transformer architecture with two mod- +ifications: Rotary Positional Embeddings (Su et al., 2021) +and GeGLU activations (Shazeer, 2020). Similarly to other +models trained on Pile (Gao et al., 2020; Wang & Komat- +suzaki, 2021), we use the tokenizer of GPT-2 (Radford et al., +2019). Following (Li et al., 2021), we linearly increase +training sequence length during the initial phase. More +specifically, we begin training with sequences of up to 256 +tokens and increase them to the maximum length of 2048 +over the first 12, 000 optimizer steps. We train the model +with LAMB (You et al., 2020), following the configuration +from the original paper for a batch size of 16384. On top +of that, we set η = 10−3 and β2 = 0.95 to account for the +increased model size. +H. Additional scaling evaluation +In this experiment, we investigate the influence of the num- +ber of nodes training with SWARM parallelism on the +throughput of the pipeline. Specifically, we measure the +performance of training the same model as in Section 4.3 +in several configurations that differ in the size of the data- +parallel group at each pipeline stage, with the number of +single-GPU instances ranging from 8 to 128 (the highest +quantity of preemptible nodes that we could reliably main- +tain for a long time). To isolate the effect of worker hetero- +geneity, here we use only the T4 accelerators and measure +the average performance over 30 minutes of training. +Figure 6 shows the results of our evaluation. It can be seen +that the training performance exhibits an approximately +linear scaling pattern, which can be explained by the high +efficiency of both the stochastic wiring strategy and the +auxiliary training components such as the DHT and the +All-Reduce protocol used for gradient averaging. +8 +32 +64 +128 +Number of nodes +1 +2 +3 +4 +Throughput, samples/s +Figure 6: Scaling of SWARM parallelism throughput with +the number of nodes. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +0 +2 +4 +6 +8 +10 12 14 16 18 20 22 24 26 28 30 32 +Time, hours +13 +14 +15 +16 +17 +Samples/second +4 stages +8 stages +16 stages +32 stages +(a) Adaptive rebalancing of SWARM parallelism. +0 +2 +4 +6 +8 +10 12 14 16 18 20 22 24 26 28 30 32 +Time, hours +10 +12 +14 +16 +Samples/second +4 stages +8 stages +16 stages +32 stages +(b) No rebalancing. +Figure 7: Scaling of pipeline-parallel strategies with respect to the number of stages. +I. Adaptive rebalancing evaluation details +In this experiment, we evaluate the efficiency of adaptive +peer rebalancing between stages proposed in Section 3.2. +We use actual statistics of the number of active T4 nodes +from the 32-hour segment of the experiment described in +Section 4.3 for a representative sample of training dynamics +with unstable participation. We use these data to simulate +training dynamics by viewing it as sequence of events, each +consisting of a timestamp and a change in the number of +peers (which can be positive or negative). When a worker is +removed from the pipeline, we randomly choose the stage it +was removed from: that is, removing N peers corresponds to +N samples from the uniform distribution over four pipeline +stages. We run 10 simulations with different random seeds +and average the resulting trajectories. +The results of this evaluation are available in Figure 5; for +reference, we also provide the performance of a theoreti- +cally optimal rebalancing strategy that maintains the highest +possible throughput at every moment. It can be seen that +even with the rebalancing period T = 300, our approach +significantly improves the overall throughput of the pipeline. +When the number of peers is relatively stable, the rebal- +anced pipeline also approaches the optimal one in terms of +throughput, which shows the efficiency of rebalancing even +when moving only one node at a time. +In addition, we observed that for some brief periods, the per- +formance of the unbalanced pipeline exceeded the through- +put of the balanced one due to random choice of disconnect- +ing peers (dropping more from the “overrepresented” stages +affects the imbalanced pipeline less). However, this held +true only for ≈ 4.5% of the experiment and was quickly +mitigated by adaptive rebalancing. +As expected, decreasing T from 300 to 60 seconds improves +both the overall throughput and the speed of convergence to +optimal pipeline performance. However, the effect is not as +drastic compared to the increase in DHT data transfer. This +is also demonstrated by Table 5, which shows the relative +throughput of the three configurations compared to the opti- +mal one. Furthermore, the table displays that while initially +there is little difference between rebalancing choices, it be- +comes more pronounced later on as the imbalanced version +“drifts further” from the optimal state. +Finally, we analyze the scaling properties of rebalancing +with respect to the number of stages. To do this, we con- +duct experiments in the same setup as above (T = 300) +while changing the number of pipeline stages from 4 to +{4, 8, 16, 32}. To ensure the consistency of throughput +across all experiments, we increase the starting number of +peers accordingly while keeping the preemption rate con- +stant. As a baseline, we also evaluate the throughput of the +pipeline that has no rebalancing. +Figure 7 shows the outcome of this experiment. As dis- +played in the plots, both strategies drop in performance with +the increase in the stage count: while all stages should drop +in performance equally in expectation, in practice, the vari- +ances are too large while the number of peers is relatively +too small for the asymptotic properties to take place. This +effect results in more outliers (large drops in the number of +peers) in the preemption distribution for more stages. Still, +rebalancing allows to partially mitigate the issue: while we +observe a more consistent downward trend for the baseline +strategy, the rebalanced pipeline regains its performance +over time and achieves higher overall throughput. +J. Compression-aware architectures +Since pipeline parallelism has several distinct points of com- +munication, the network overhead can be reduced consid- +erably by reducing the size of data at these communication +points. To exploit this, we develop compression-aware ar- +chitectures that apply extreme compression at these points. +We study two distinct communication bottleneck layers: (1) +compression through a linear bottleneck layer, and (2) com- +pression through a bottleneck induced by the maxout activa- +tion function (Goodfellow et al., 2013). We also study how +compressing the activations and gradients at the communi- +cation points to 8 bits affects the predictive performance. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +J.1. Description +Fully connected layers (baseline): +Fully connected lay- +ers in models such as Transformers consist of a multilayer +perceptron with a single hidden layer and a nonlinear ac- +tivation function. Without biases and with a residual con- +nection (He et al., 2015) from the inputs to the outputs, this +can be described as MLP(x, w1, w2) = σ(xw1)w2 + x, +where x ∈ Rb×s×m, w1 ∈ Rm×h, w2 ∈ RT h×m, and σ(·) +is a nonlinear activation function such as ReLU (Krizhevsky +et al., 2012); b, s, m, and h are the batch, sequence, model, +and hidden dimensions of the neural network. To compress +the output of the MLP layer, we want to apply a compres- +sion layer between two consecutive stages. For example, +if we have 24 layers and 4 stages, we need 3 compression +layers at layers 6, 12, and 18. +Quantized activations: +A natural way to reduce the com- +munication intensity is to send activations and gradients with +respect to activations in reduced precision. However, simply +casting tensors to a lower precision may slow down conver- +gence and cause instabilities. Instead, we use dynamic 8-bit +quantization with blockwise scaling from (Dettmers et al., +2021). This technique reduces communication by ≈2x and +≈4x for half and full precision, respectively. +On the other hand, quantizing and dequantizing activations +can add compute overhead on every microbatch processed. +Our implementation circumvents that overhead by perform- +ing quantization asynchronously on the CPU. However, this +is not required, as blockwise (de)quantization takes less than +1% of total computation time: see Appendix K for details. +Bottleneck layers: +We experiment with simple bottleneck +layers that work by compressing the output features of the +MLP by linear projection: +Bottleneck(x, w1, w2, wc, wd) = += LayerNorm(LayerNorm(MLP(x, w1, w2))wc)wd, +where wc ∈ Rm×c, wd ∈ Rc×m are compression and +decompression parameters with compression dimension +c < m. We find it critical to use layer normalization (Ba +et al., 2016) to ensure training without divergence. The +parameter matrix wc resides in one stage and its outputs are +transferred to the next stage that holds the parameters wd, +which requires m/c times less communication compared +to the original model. Note that adding a bottleneck only +adds two linear layers for the forward pass and decreases the +size of MLP activations; thus, its computational overhead is +negligible (less than 1% for typical sizes, see Appendix K). +Maxout compression: +Compared to bottleneck compres- +sion, maxout compression works by using the maxout acti- +vation function (Goodfellow et al., 2013) for compression +rather than a linear projection. The maxout function of fac- +tor k takes inputs with a hidden dimension of d and reduces +this dimension by a factor of k by computing the maximum +value for each non-overlapping window of k features. We +use maxout compression as follows: +Maxout(x, w1, w2, wd) = +LayerNorm(maxoutk(LayerNorm(MLP(x, w1, w2))))wd, +where the output is reduced by a factor of k through the max- +out function in the previous stage, and then sent to the next +stage which holds the decompression matrix wd∈Rm/k×m. +J.2. Evaluating the speed-quality tradeoff +While compression techniques reduce the communication +overhead, they might also degrade the perplexity reached in +a certain time and the final perplexity after a specific number +of steps. To study these tradeoffs, we train a Transformer +language model with adaptive inputs (Baevski & Auli, 2019) +on the WikiText-103 dataset and measure how compression- +aware architecture variants affect convergence. +Our setup follows that of (Baevski & Auli, 2019) with one +difference: we use a sequence length of 2048 instead of +3072 to fit this model into our smaller GPUs. To measure +the time to solution, we look at the number of iterations +it takes to converge to the training perplexity of 22. We +evaluate the baseline model and three compression-aware +modifications from Section J.1: bottleneck, maxout, and +block-wise dynamic 8-bit quantization, each with 2 pipeline +stages and each a compression factor of 2x. +The results can be seen in Table 7. We can see that 8-bit +compression does not degrade the time to 22 perplexity and +maintains close to the final perplexity of the baseline. The +compression-aware bottleneck and maxout architectures +perform equal to each other, but degrade final perplexity +slightly and increase time to a perplexity of 22 by 26–28%. +Using these results, one can determine which method is +optimal for their hardware setup. For instance, training with +maxout with 2 pipeline stages needs 28% more steps, but +accelerates the communication phase by 2x. If communi- +cation is the limiting factor, using maxout or bottleneck +compression layers will offer improved time to perplexity +despite the performance degradation. However, the same +two techniques would result in slower training in a setup +where network bandwidth is unlimited. +In turn, 8-bit quantization reduces communication cost with- +out slowing down per-iteration convergence, making it a +“safe bet” for situations where the per-iteration convergence +must be preserved. In our large-scale experiments (Sec- +tion 4.3), we opt to using quantization since it was enough to +fully saturate the GPUs. If network bandwidth is still a lim- +iting factor, one can combine quantization with bottleneck +or maxout compression to further reduce communication. + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +Table 7: Performance of compression methods for a Transformer language model with adaptive inputs on WikiText-103. +The asterisk denotes that the difference is not statistically significant. +Method +Ppl after +286K steps +Steps to +ppl 22 +Data +transfer +Extra compute +Absolute +Relative +No compression +21.02 +1x +1x +0 +None +8-bit compression +21.13 +0.97x∗ +0.5x +1.2ms +None (overlapped) +Bottleneck +21.76 +1.26x +0.5x +1.96ms +≤ 1% +Maxout +21.83 +1.28x +0.5x +2.04ms +≤ 1% +J.3. Additional experiments +The additional experiments in this section have two pur- +poses: (1) to evaluate how compression methods vary with +the number of stages and (2) to evaluate an additional setting +that is closer to modern pretraining setups such as GPT-2/3. +While (1) has further implications for scaling, (2) is helpful +to account for confounding factors that might have been +overlooked in the main experiments on WikiText-103. The +WikiText-103 baseline uses non-BPE vocabulary, a long +sequence length, and uses adaptive inputs (Baevski & Auli, +2019), all of which are not frequently used in modern pre- +trained Transformers since GPT-2 (Radford et al., 2019). +Experimental setup: +As a baseline, we train a Trans- +former language model (Vaswani et al., 2017) on the Open- +WebText corpus (Gokaslan & Cohen, 2019). We use the +following hyperparameters: sequence size 512, 16 layers +with model dimension 1024, and hidden dimension 4096 for +a total of 253M parameters. We use byte pair encoding (Sen- +nrich et al., 2016; Radford et al., 2019) with a vocabulary +size of 50264 symbols. We do not use dropout or other +regularization, since our models underfit. We run these +experiments in Fairseq (Ott et al., 2019). +We test bottleneck and maxout compression for a compres- +sion factor of 50% and 75% compared to the original size +over two and four stages. We look at how using these +compression-aware architectures affects the performance +compared to the compression that they achieve. +Results: +The results of our compression-aware architec- +tures are shown in Table 8. We can see that while the +bottleneck architecture is competitive with maxout for a +compression factor of 2x with two stages, maxout has better +perplexities if more stages or a higher compression ratio +is used. The out-of-distribution perplexities vary consis- +tently with the in-distribution perplexity, which suggests +compression-aware architectures do not degrade the out- +of-distribution performance more than the in-distribution +performance. As such, the maxout compression is an ef- +fective technique to reduce the bandwidth requirements of +pipeline parallel training further. +While the 8-bit blockwise quantization can only compress +the activations by a factor of two (16-bit → 8-bit), it does +not affect the quality as much when compared to the base- +line. As such, the 8-bit quantization appears to be a reliable +default choice to reduce the communication overhead for +pipeline parallelism. +When considered together with the square-cube law for +distributed training and SWARM parallelism, compression- +aware architectures allow for better scaling of large neural +networks trained over preemptible low-bandwidth peers. +Thus, compression-aware architectures improve the acces- +sibility and affordability of training large models outside +HPC environments. +K. Time To Solution +In this section, we evaluate the compression-aware tech- +niques proposed in Appendix J.1 from a practitioner’s point +of view. A natural way to compare these techniques is in +terms of “the time to solution”, i.e., the wall-clock time it +takes to achieve the desired validation objective. In practice, +this time depends on three main factors: the compression +strategy, the distributed training algorithm, and the compu- +tational infrastructure. +In order to disentangle these factors, we first address the re- +lationship between the training algorithm and the infrastruc- +ture. As we discuss in Section 3.2 (and later in Appendix E), +SWARM parallelism has the same per-iteration behavior as +other synchronous methods. Theoretically, the choice of an +optimal training system should come down to whichever +algorithm has the highest training throughput. +To verify this argument in practice, we compare the per- +iteration and per-hour performance of SWARM against +fully synchronous training. For this experiment, we train +the ALBERT model (Lan et al., 2020) on the WikiText- +103 dataset (Merity et al., 2017). We use the ALBERT- +Large architecture with 4 layer groups that correspond to +4 SWARM stages without the architecture modifications +from Appendix J.1. We follow the exact hyperparameters +from the original paper: for example, we use the LAMB +optimizer (You et al., 2020) with the batch size of 4096 and + +SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient +Table 8: Results of language models trained on the OpenWebText Corpus (OWT). The baseline model has 253M parameters +and is trained for 8 GPU-days. We apply bottleneck and maxout compression to our baseline in 2 and 4 stages with a +compression factor between 2–4x. WT=WikiText, PTB=Penn Treebank, 1BW=Billion word corpus. +Validation perplexity +Model +Stages +Compression +OWT +LAMBADA +WT2 +WT103 +PTB +1BW +Baseline +– +– +19.7 +86.4 +56.2 +35.4 +133.0 +80.9 +8-bit Quantization +2 +2x +19.6 +89.1 +56.0 +35.0 +132.7 +79.8 +Bottleneck +2 +2x +19.5 +87.7 +56.5 +35.2 +129.8 +79.2 +Maxout +2 +2x +19.6 +85.4 +56.6 +35.2 +126.8 +78.8 +8-bit Quantization +4 +2x +19.7 +87.9 +56.3 +35.2 +133.9 +79.8 +Bottleneck +4 +2x +21.7 +100.0 +66.4 +40.0 +149.6 +89.5 +Maxout +4 +2x +21.4 +89.9 +63.9 +39.5 +142.1 +86.2 +Bottleneck +2 +4x +21.6 +99.8 +64.8 +39.6 +145.6 +88.3 +Maxout +2 +4x +20.5 +89.6 +60.0 +37.1 +141.7 +83.5 +Bottleneck +4 +4x +28.9 +141.6 +100.2 +58.1 +235.5 +118.3 +Maxout +4 +4x +21.3 +93.5 +63.6 +39.2 +147.7 +89.1 +0 +7.5K +15K +22.5K +30K +Steps +2 +4 +6 +8 +10 +ALBERT objective +DDP, 8xV100 +SWARM, 8xV100 +0 +24 +48 +72 +96 +120 144 168 +Hours +2 +4 +6 +8 +10 +DDP, 8xV100 +SWARM, 8xV100 +SWARM, 32xT4 +Target loss value +Figure 8: Convergence curves of ALBERT with SWARM +and standard data-parallel training. +the sequence length of 512. We train this model in three +setups: traditional distributed training with 8 V100 workers, +SWARM with 8 preemptible V100 GPUs, and SWARM +with 32 preemptible T4 workers. +To quantify the time to solution, we measure the wall time +required to achieve the ALBERT objective equal to 1.5. Ad- +ditionally, we report the per-hour cost of each experimental +setup and the total cost of achieving a loss of 1.5 using +public cloud provider pricing estimates in Table 9. +Figure 8 demonstrates that SWARM matches the per- +iteration learning curves of traditional distributed training +(PyTorch DistributedDataParallel) up to the variation com- +parable to caused by changing the random seed. However, +SWARM parallelism can achieve the loss of 1.5 more cost- +efficiently and faster by using preemptible instances. In +turn, when forced to use homogeneous and reliable GPUs, +SWARM would have slightly inferior performance com- +pared to conventional algorithms, which was first demon- +strated in Section 4.2. +Table 9: Training time and costs. +Setup +Time, hours +Cost, $ +Hourly +Total +8 × V 100, reliable +175.4 +7.834 +1374 +8 × V 100, preemptible +192.6 +5.383 +1037 +32 × T4, preemptible +140.8 +3.536 +497.8 + diff --git a/qtFKT4oBgHgl3EQfzy6t/content/tmp_files/load_file.txt b/qtFKT4oBgHgl3EQfzy6t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..de0f2746296fa96d2385334a413bbfaa2075d6c0 --- /dev/null +++ b/qtFKT4oBgHgl3EQfzy6t/content/tmp_files/load_file.txt @@ -0,0 +1,2408 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf,len=2407 +page_content='SWARM Parallelism: Training Large Models Can Be Surprisingly Communication-Efficient Max Ryabinin * 1 2 Tim Dettmers * 3 Michael Diskin 2 1 Alexander Borzunov 1 2 Abstract Many deep learning applications benefit from using large models with billions of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Training these models is notoriously expensive due to the need for specialized HPC clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' In this work, we consider alternative setups for train- ing large models: using cheap “preemptible” in- stances or pooling existing resources from multi- ple regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' We analyze the performance of exist- ing model-parallel algorithms in these conditions and find configurations where training larger models becomes less communication-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Based on these findings, we propose SWARM parallelism1, a model-parallel training algorithm designed for poorly connected, heterogeneous and unreliable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' SWARM creates tem- porary randomized pipelines between nodes that are rebalanced in case of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' We empiri- cally validate our findings and compare SWARM parallelism with existing large-scale training ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Finally, we combine our insights with compression strategies to train a large Trans- former language model with 1B shared param- eters (≈13B before sharing) on preemptible T4 GPUs with less than 200Mb/s network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Introduction For the past several years, the deep learning community has been growing more reliant on large pretrained neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' The most evident example of this trend is natural lan- guage processing, where the parameter count of models has grown from hundreds of millions (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Rad- ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=', 2019) to billions (Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Wang & Komatsuzaki, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=', 2021) to hundreds of billions (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Equal contribution 1HSE University 2Yandex 3University of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfzy6t/content/2301.11913v1.pdf'} +page_content=' Correspondence to: Max Ryabinin α, +(9) +then for any δ ∈ (0, 1), we have +P +� +L +� +Yn+1, Fˆλ(Xn+1) +� +≤ α +� +≥ 1 − δ, +where ˆλ is defined as formula (8). +Proof. The conditions of formula (9) implies that ˆλ is well de- +fined. As ˆλ is defined on the whole dataset {(Xi, Yi)}n+1 +i=1 with +the function s, one can treat {Li(ˆλ)}n+1 +i=1 as a transformation +τ applied to {(Xi, Yi)}n+1 +i=1 , i.e., +{Li(ˆλ)}n+1 +i=1 = τ +� +{(Xi, Yi)}n+1 +i=1 +� +Besides, based on Theorem 3 in [10], this transformation +preserves exchangeability, since for each permutation π1, there +exists a permutation π2 = π1, such that +π1 +� +{Li(ˆλ)}n+1 +i=1 +� += τ ◦ π2 +� +{(Xi, Yi)}n+1 +i=1 +� +, +for all possible {(Xi, Yi)}n+1 +i=1 . It follows that +P +� +Ln+1(ˆλ) ≤ Q(n+1) +1−δ (ˆλ) +� +≥ 1 − δ, +(10) +as Q(n+1) +1−δ (ˆλ) is just the corresponding 1 − δ quantile of the +exchangeable variables {Li(ˆλ)}n+1 +i=1 . (See Lemma 1 in [11]). +By definition of the function s, we have Q(n+1) +1−δ (ˆλ) ≤ α. +This combining formula (10) leads to +P +� +Ln+1(ˆλ) ≤ α +� +≥ 1 − δ, +which completes the proof. +D. Discussion +Theorem 1 shows the loss-controlling guarantee for the ideal +case where (Xn+1, Yn+1) is available, whose approach can be +approximated using Algorithm 1 in practice. The conditions of +formula (9) also implies that λ∗ is well defined, which makes +us able to obtain λ∗ based on the searching function s. Also, +by definition, one can conclude that +� +λ ∈ Λ : Q(n) +1−δ(λ) ≤ α +� +⊆ +� +λ ∈ Λ : Q(n+1) +1−δ (λ) ≤ α +� +, +which indicates that searching λ in the left set above is +reasonable. Besides, the difference between λ∗ and ˆλ can be +ignored for large n. The proof in Theorem 1 only needs s to +be a predefined function independent of {(Xi, Yi)}n+1 +i=1 . Thus, + +4 +Fig. 1. The frequencies of the prediction losses being greater than α for different δ and α on test data of HighTemp and LowTemp datasets. All bars being +near or below the preset δ confirms the controlling guarantee of LCC empirically. +Fig. 2. The distributions of the prediction losses for different δ and α on test data of HighTemp and LowTemp datasets. The losses are controlled by α and +δ properly to achieve the empirical validity in Fig. 1. +one can define s as an optimization algorithm based on another +hold-out dataset for parameter searching. +Due to the general forms of the calibrated predictors and the +loss functions, one can consider using LCC to control multiple +losses jointly. Suppose the jth loss on ith calibration sample +with λ is Lj,i(λ) and the number of losses is m. One simple +method is to search for λ∗ in the following set, +� +λ ∈ Λ : max +j +� +Q(n) +j,1−δ/m(λ) +� +≤ α +� +, +where Q(n) +j,1−δ/m(λ) is the 1−δ/m quantile of {Lj,i(λ)}n +i=1 ∪ +{B}. Therefore, to control multiple losses jointly, one may +have to calculate the 1 − δ/m quantiles, which only makes +sense when m is small. +III. EXPERIMENTS +We apply LCC to high-impact weather forecasting, which +is based on postprocessing of numerical weather prediction +(NWP) models [12] [13] [14], i.e., learning a predictor whose +inputs are forecasts made by NWP models and outputs are +corresponding high-impact weather. We use LCC to post- +process the ensemble forecasts issued by European Centre +for Medium-Range Weather Forecasts (ECMWF) [15]. The +forecasts are obtained from the THORPEX Interactive Grand +Global Ensemble (TIGGE) dataset [16]. We concentrate on 2- +m maximum temperature and minimum temperature forecasts +initialized at 0000 UTC with the forecast lead times from +12nd hour to 36th hour. The resolution of the forecast fields is +0.5◦ × 0.5◦ and the corresponding label fields with the same +resolution are calculated using the ERA5 reanalysis data [17]. +The area covers the main parts of North China, East China +and Central China, ranging from 109◦E to 122◦E in longitude +and from 29◦N to 42◦N in latitude with the grid size being +27 × 27. The HighTemp and LowTemp datasets introduced in +[5] are used for empirical studies. The inputs in HighTemp are +2-m maximum temperature forecasting fields and the corre- +sponding label fields are whether the observed 2-m maximum +temperature is above 35 °C for each grid. Similarly, the inputs +in LowTemp are 2-m minimum temperature forecasting fields +and the corresponding label fields are whether the observed 2- +m minimum temperature is below −15 °C for each grid. The +sample sizes of HighTemp and LowTemp are 1200 and 1233 +respectively. +The experimental setting is similar to that in Section IV- +B of [5]. For each dataset, all forecasts made by the NWP +model were normalized to [0, 1] by min–max normalization. +20% of the data were used for testing and 80% and 20% +of the remaining data were used for training and calibration +respectively. The normalized ensemble fields forecast by the + +Data Set = HighTemp I α = 0.3 +Data Set = HighTemp I α = 0.35 +Data Set = HighTemp I α = 0.4 +Data Set = HighTemp I α = 0.45 +Data Set = HighTemp I α = 0.5 +0.30 +0.25 +0.05 +0.00 +Model +Data Set = LowTemp I α = 0.3 +Data Set = LowTemp | α = 0.35 +Data Set = LowTemp | α = 0.45 +Data Set = LowTemp I α = 0.4 +Data Set = LowTemp I α = 0.5 + nDNN +0.30 + Unet +0.25 +0.05 +0.00 +0.1 +0.2 +0.1 +0.15 +0.2 +0.1 +0.15 +0.2 +0.1 +0.15 +0.2 +0.1 +0.2 +0.15 +6 +6 +6Data Set = HighTemp I α = 0.45 +Data Set = HighTemp I α = 0.3 +Data Set = HighTemp I α = 0.35 +Data Set = HighTemp I α = 0.4 +Data Set = HighTemp I α = 0.5 +1.0 +0.8 +0.6 +0.2 +0.0 +Model +Data Set = LowTemp I α = 0.3 +Data Set = LowTemp I α = 0.35 +Data Set = LowTemp I α = 0.4 +Data Set = LowTemp I α = 0.45 +Data Set = LowTemp I α = 0.5 +nDNN +1.0 +Unet +0.8 +0.6 +Loss +0.4 +0.2 +0.0 +0.1 +0.15 +0.2 +0.1 +0.15 +0.2 +0.1 +0.15 +0.2 +0.1 +0.15 +0.2 +0.1 +0.15 +0.2 +65 +Fig. 3. The distributions of normalized sizes for different δ and α on test data of HighTemp and LowTemp datasets. The predictions have reasonable sizes +for both U-Net and nDNN for high-impact weather forecasting. +NWP model are taken as input x and the set of grids having +high-impact weather is the corresponding label y, which can be +seen as the image segmentation problem in computer vision. +Thus, we employed two fully convolutional neural networks +[18] as our underlying algorithms. One was U-Net [19] and the +other is the naive deep neural network (nDNN), which is the +U-Net removing skip-connections. The structures of the two +networks are the same as those in [5]. To train the deep nets, +we further partitioned the data for training to validation part +(10%) and proper training part (90%), which were used for +model selection and parameter updating respectively. Adam +optimization [20] with the learning rate being 0.0001 and +the number of epochs being 1000 was employed for training, +and the model whose binary cross entropy was the lowest on +validation data was chosen as the predictive model f needing +calibration. The candidate calibrated predictor Fλ is the same +as formula (10) in [5], i.e., +Fλ(x) = {(p, q) : f(p,q)(x) ≥ 1 − λ}, +where f(p,q)(x) is the estimated probability for high-impact +weather existing at grid (p, q). The loss function is +L(y, F) = 1 − |y ∩ F| +|F| +, +which is a non-monotone loss function related to false dis- +covery introduced in [8]. The searching function s we used is +the min function. The final calibrated predictors were obtained +with the proposed LCC approach and the experimental results +are shown in Fig. 1, Fig. 2 and Fig. 3. +The frequencies of the prediction losses being more than α +are shown in Fig. 1 with bar plots for δ = 0.1, 0.15 and 0.2. +The columns represent the cases where α = 0.3, 0.35, 0.4, 0.45 +and 0.5 respectively. All bars are near or below the preset +δ, which verifies loss-controlling guarantee empirically. The +boxen plots of the losses for different δ and α are shown in +Fig. 2, which contain more information about tails by drawing +narrower boxes than box plots. It can be observed that α and +δ result in larger losses, which should be preset based on +specific applications. The informational efficiency of Fλ∗ is +measured using normalized size of the prediction set defined +as |Fλ∗(x)|/PQ in which the numbers of the vertical and the +horizontal grids of prediction fields are denoted by P and Q +respectively. The distributions of normalized sizes are shown +in Fig. 3, indicating that different α and δ cause different +normalized sizes and there should be trade-off among loss +level α, confidence level 1 − δ and informational efficiency +of the predictions. All the predictions have reasonable sizes +using LCC and there are not significant difference between +the predictions of UNet and nDNN in these cases. +We also tested other forms of searching functions such as +the max function. However, although the controlling guarantee +can be hold empirically, the constructed predictor may lose +informational efficiency for applicability, implying that the +forms of searching functions should be designed on a case- +by-case basis. +IV. CONCLUSION +This paper proposes loss-controlling calibration, which +extends conformal loss-controlling prediction to calibrating +predictive models with more general forms of calibrated +predictors and losses. The finite-sample and distribution-free +loss-controlling guarantee is proved by introducing a searching +function and the property of transformations preserving ex- +changeability in the ideal case. In addition, an approximation +approach for practical calibration is proposed, whose steps are +the same as those of conformal loss-controlling prediction, i.e., +the only difference between loss-controlling calibration and +conformal loss-controlling prediction is whether the calibrated +predictors and the loss functions satisfy specific conditions. +The method is applied to high-impact weather forecasting +problems and the loss-controlling guarantee is shown empir- +ically for the non-monotone loss related to false discovery. +Further empirical studies for a wider range of applications with +case-by-case design are needed to fully understand the power +and limitation of the proposed loss-controlling calibration. +REFERENCES +[1] V. Vovk, A. Gammerman, and G. 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Schepers et al., “The +era5 global reanalysis,” Quarterly Journal of the Royal Meteorological +Society, vol. 146, no. 730, pp. 1999–2049, 2020. +[18] Z. Li, F. Liu, W. Yang, S. Peng, and J. Zhou, “A survey of convolutional +neural networks: analysis, applications, and prospects,” IEEE Transac- +tions on Neural Networks and Learning Systems, 2021. +[19] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional net- +works for biomedical image segmentation,” in International Conference +on Medical Image Computing and Computer-Assisted Intervention. +Springer, 2015, pp. 234–241. +[20] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” +arXiv preprint arXiv:1412.6980, 2014. + diff --git a/stE3T4oBgHgl3EQfNAkn/content/tmp_files/load_file.txt b/stE3T4oBgHgl3EQfNAkn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f731c84e2b1c64d832462e7765e0ab5c9733f0fc --- /dev/null +++ b/stE3T4oBgHgl3EQfNAkn/content/tmp_files/load_file.txt @@ -0,0 +1,460 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf,len=459 +page_content='1 Loss-Controlling Calibration for Predictive Models Di Wang, Junzhi Shi, Pingping Wang, Shuo Zhuang, Hongyue Li Abstract—We propose a learning framework for calibrating predictive models to make loss-controlling prediction for ex- changeable data, which extends our recently proposed conformal loss-controlling prediction for more general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' By comparison, the predictors built by the proposed loss-controlling approach are not limited to set predictors, and the loss function can be any measurable function without the monotone assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' To control the loss values in an efficient way, we introduce transformations preserving exchangeability to prove finite-sample controlling guarantee when the test label is obtained, and then develop an approximation approach to construct predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The transformations can be built on any predefined function, which include using optimization algorithms for parameter searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' This approach is a natural extension of conformal loss-controlling prediction, since it can be reduced to the latter when the set predictors have the nesting property and the loss functions are monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Our proposed method is tested empirically for high-impact weather forecasting and the experimental results demonstrate its effectiveness for controlling the non-monotone loss related to false discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Index Terms—Loss-controlling calibration, Predictive models, Transformations preserving exchangeability, Weather forecast- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' INTRODUCTION Predictive models built on modern machine learning tech- niques have been deployed for many areas due to their expressive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' However, many of the algorithms can not provide reliable information about the difference between the prediction and the true label for a specific test object, which is important for high-risk applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' If the prediction is a set of possible labels and the difference is the miscoverage loss for set predictors, the learning framework of conformal prediction (CP) can tackle this issue with its coverage guarantee under the assumption of exchangeability of data samples [1] [2] [3] [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Furthermore, our recently proposed conformal loss-controlling prediction (CLCP) [5] extends CP to the loss satisfying monotone conditions, which ensures that the prediction loss is not greater than a preset level for high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' These This work was supported by the National Natural Science Foundation of China under Grant 62106169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (Corresponding author: Hongyue Li) Di Wang is with School of Electrical and Information Engineering, Tian- jin University, Tianjin 300072, China, and also with Tianjin Key Labo- ratory of Brain-inspired Intelligence Technology, School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (email: wangdi2015@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Junzhi Shi is with School of Electrical and Electronic Engineer- ing, Shandong University of Technology, Zibo 255000, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (email: shijz@sdut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='cn) Pingping Wang is with Qingdao Academy of Chinese Medical Science, Shandong University of Traditional Chinese Medicine, Qingdao, Shandong 266112, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (email: wangpingping@sdutcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='cn) Shuo Zhuang is with School of Computer Science and Information En- gineering, Hefei University of Technology, Hefei 230009, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (email: shuozhuang@hfut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Hongyue Li is with School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (lihongyue@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' two existing frameworks are both limited to set predictors and non-general losses, leading to this work considering general predictors and losses for loss-controlling prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' CLCP is inspired by risk-controlling prediction sets [6] and conformal risk control [7], and the purpose of CLCP is to build a set predictor Cλ∗ such that P � L � Yn+1, Cλ∗(Xn+1) � ≤ α � ≥ 1 − δ, (1) where α and δ are preset parameters for loss level and signifi- cance level respectively, and L is a monotone loss function as in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Cλ is the prediction set with the nesting property for the parameter λ ∈ Λ, where Λ is the discrete set of possible values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Cλ is usually built on some underlying predictive model learned on training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The optimal λ∗ is obtained based on n calibration data {(Xi, Yi)}n i=1, and (Xn+1, Yn+1) is the test feature-response pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The randomness of the probability in- equation above is from both {(Xi, Yi)}n i=1 and (Xn+1, Yn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Although CLCP extends CP to more general cases, the forms of the set predictor and the loss function used in CLCP are still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' To overcome this issue, one way is to use the learn then test process [8] to fuse multiple probability equations like for- mula (1) to maintain the controlling guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' However, this process can not be effectively applied to our loss-controlling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' One example is to use the Bonferroni correction to obtain the family-wise loss-controlling guarantee, where one needs to calculate the 1 − δ/|Λ| quantiles for each possible λ ∈ Λ, resulting in meaningless calculation if |Λ| is large and the number of calibration data is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' For example, if |Λ| = 1000 and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1, we need to calculate the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='9999 quantiles of losses for each possible λ, which makes sense only if the number of calibration data is more than 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Thus, the loss-controlling calibration (LCC) approach pro- posed in this paper employs predefined searching functions and the transformations preserving exchangeability to avoid the multiple hypothesis testing process used in learn then test, which is a natural extension of CLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Concretely, we aim to calibrate a predictive model f to obtained the calibrated predictor Fˆλ such that P � L � Yn+1, Fˆλ(Xn+1) � ≤ α � ≥ 1 − δ, (2) where Fλ can be a point, set or any other form of predictor built on f with the parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' L is a measurable loss function without the need of further conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The optimal ˆλ is calculated by some predefined function and all n+1 data {(Xi, Yi)}n+1 i=1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=', the controlling guarantee of formula (2) is only for the ideal case where one has the test label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' However, we can approximately obtain λ∗ ≈ ˆλ using {(Xi, Yi)}n i=1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='04378v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='LG] 11 Jan 2023 2 in practice and the controlling guarantee can still be hold empirically in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' In other words, the LCC proposed in this paper sacrifices the theoretical guarantee to efficient calibration, and the approximation is sound for large n in theory and in our empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' In the experiments, we apply LCC to high-impact weather forecasts to control the loss related to false detection, and the experimental results confirm the effectiveness of the proposed LCC approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' In summary, three contributions are made in this paper: A learning framework named loss-controlling calibration is proposed for calibrating predictive models for con- trolling general prediction losses for test objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The approach is a natural extension of CLCP and is easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' By employing transformations preserving exchangeabil- ity, the distribution-free and finite-sample controlling guarantee is proved mathematically with the exchange- ability assumption in the ideal condition where the test la- bel is obtained, and a reasonable approximation approach is proposed for practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The loss-controlling guarantee of LCC is applied to weather forecasting problems, which demonstrates the effectiveness of LCC for controlling the non-monotone loss related to false discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The remaining parts of this paper are organized as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Section II reviews conformal loss-controlling prediction and proposes the loss-controlling calibration approach with its theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Section III applies loss-controlling calibration to high-impact weather forecasting problems to empirically verify the loss-controlling guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Finally, the conclusions of this paper are drawn in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' CONFORMAL LOSS-CONTROLLING PREDICTION AND LOSS-CONTROLLING CALIBRATION This section reviews recently proposed conformal loss- controlling prediction and proposes the learning framework of loss-controlling calibration for predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Throughout this paper, let {(Xi, Yi)}n+1 i=1 be n+1 data drawn exchangeably from PXY on X ×Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (Xn+1, Yn+1) is the test object-response pair and the first n samples {(Xi, Yi)}n i=1 are calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The lower-case letter (xi, yi) represents the realization of (Xi, Yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Conformal Loss-Controlling Prediction CLCP starts with a set-valued function Cλ : X → Y′ with a parameter λ ∈ Λ, where Λ is a discrete set of possible real values of λ such as from 0 to 1 with step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Y′ denotes some space of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' For example, Y′ can be the power set of Y for single-label classification and Y′ can be equal to Y for binary image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' This set-valued function Cλ needs to satisfy the following nesting property introduced in [6]: λ1 < λ2 =⇒ Cλ1(x) ⊆ Cλ2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (3) Here we give an example of constructing the prediction set Cλ for classification problem with K classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Suppose f : X → [0, 1]K is a classifier trained on training data with the ith output being the estimation of the probability of the ith class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Then CLCP can construct the prediction set as Cλ(x) = {k : fk(x) ≥ 1 − λ}, which satisfies the nesting property of formula (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' In addition, for each realization of response y, the loss function L : Y × Y′ → R considered in CLCP should respect the following monotone property or nesting property: C1 ⊆ C2 ⊆ Y′ =⇒ L(y, C2) ≤ L(y, C1) ≤ B, (4) where B is the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' After determining Cλ and L, for preset α and δ, CLCP first calculates Li as Li(λ) = L(Yi, Cλ(Xi)) for i = 1, · · · , n, and then searches for λ∗ such that λ∗ = min � λ ∈ Λ : Q(n) 1−δ(λ) ≤ α � , where Q(n) 1−δ(λ) is the 1 − δ quantile of {Li(λ)}n i=1 ∪ {B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The finally obtained set predictor Cλ∗ satisfies the controlling guarantee of formula (1), which is proved in theory for distribution-free and finite-sample conditions, and CLCP can be seen as an extension of CP for specific forms of Cλ and L [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Loss-Controlling Calibration CLCP needs nesting properties for Cλ and L, which limits its applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Therefore, we propose loss-controlling cali- bration for general predictors and loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' We denote Fλ as a predictor built on a predictive model f learned from training data, where λ is a parameter taking values from a discrete set Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' For LCC, we emphasize that Fλ : X → Y′ can be any kind of predictor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=', Y′ does not have to be the set of label sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Besides, Λ can be any discrete set such as the sets of multi-dimensional vectors as in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Also, the loss function L : Y × Y′ → R considered for LCC can be any measurable function bounded above by B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=', for each object-response pair (x, y), L(y, Fλ(x)) ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (5) For these general conditions, one way of constructing loss- controlling guarantee is to use multiple hypothesis testing process developed in learn then test [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' However, this may lead to calculating the 1 − δ/|Λ| quantiles of losses on calibration data, which may be meaningless for our loss- controlling approach when the number of calibration data is small or moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' To overcome this issue, we propose to use a predefined function s independent of {(Xi, Yi)}n+1 i=1 to do the trick, where s stands for searching since it can be defined as an optimization algorithm for parameter searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The approach of LCC is very similar to CLCP and we first introduce it for comparison in this subsection, leaving the analysis of it to the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' After determining Fλ and L, for preset α and δ, LCC first calculates Li on calibration data as Li(λ) = L(Yi, Fλ(Xi)), (6) 3 and then search for λ such that λ∗ = s �� λ ∈ Λ : Q(n) 1−δ(λ) ≤ α �� , (7) where s : P(Λ) → Λ is the predefined searching function defined on the power set of Λ whose output is an element its input, and Q(n) 1−δ(λ) is the 1−δ quantile of {Li(λ)}n i=1 ∪{B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The final predictor built by LCC is Fλ∗, which is very similar to Fˆλ satisfying the loss-controlling guarantee as formula (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The relation between Fλ∗ and Fˆλ will be introduced in Section II-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' It can be seen that LCC is exactly CLCP if Λ ⊂ R, Fλ is a set predictor with nesting property as formula (3), L is monotone as formula (4) and s is the min function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Therefore, for LCC we also use the same notations of Li and Q(n) 1−δ(λ) as CLCP to represent similar concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Here we summarized LCC in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Algorithm 1 Loss-Controlling Calibration Input: Calibration dataset {(xi, yi)}n i=1, test input object xn+1, the predictor Fλ, the loss function L satisfies formula (7), predefined searching function s, preset α ∈ R and δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Output: Calibrated prediction for yn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 1: Based on formula (6), calculate {Li(λ)}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 2: Search for λ∗ satisfying formula (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 3: return Fλ∗(xn+1) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Theoretical Analysis of Loss-Controlling Calibration This section provides the theoretical insights of LCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Let Q(n+1) 1−δ (λ) be the 1 − δ quantile of {Li}n+1 i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Define ˆλ as ˆλ = s �� λ ∈ Λ : Q(n+1) 1−δ (λ) ≤ α �� , (8) which is very similar to λ∗ especially for large n, as Q(n+1) 1−δ (λ) and Q(n) 1−δ(λ) are nearly the same in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Here we introduce the definition of (α, δ)-loss-controlling predictors and then prove loss-controlling guarantee with ˆλ based on the theorem about transformations preserving exchangeability developed in [9] and introduced in [10] as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Given a loss function L : Y × Y′ → R and a random sample (X, Y ) ∈ X × Y, a random function F whose realization is in the space of functions X → Y′ is a (α, δ)-loss-controlling predictor if it satisfies that P � L � Y, F(X) � ≤ α � ≥ 1 − δ, where the randomness is both from F and (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Next we prove in Theorem 1 that Fˆλ is a (α, δ)-loss- controlling predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Suppose {(Xi, Yi)}n+1 i=1 are n + 1 data drawn exchangeably from PXY on X ×Y, Fλ : X → Y′ is a function with the parameter λ taking values from a discrete set Λ , L : Y × Y′ → R is a loss function satisfying formula (5) and Li(λ) is defined as formula (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Denote s : P(Λ) → Λ as any searching function defined on the power set of Λ whose output is the element of its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' For any preset α ∈ R, if L also satisfies the following conditions, min λ max i Li(λ) < α, max λ min i Li(λ) > α, (9) then for any δ ∈ (0, 1), we have P � L � Yn+1, Fˆλ(Xn+1) � ≤ α � ≥ 1 − δ, where ˆλ is defined as formula (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The conditions of formula (9) implies that ˆλ is well de- fined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' As ˆλ is defined on the whole dataset {(Xi, Yi)}n+1 i=1 with the function s, one can treat {Li(ˆλ)}n+1 i=1 as a transformation τ applied to {(Xi, Yi)}n+1 i=1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=', {Li(ˆλ)}n+1 i=1 = τ � {(Xi, Yi)}n+1 i=1 � Besides, based on Theorem 3 in [10], this transformation preserves exchangeability, since for each permutation π1, there exists a permutation π2 = π1, such that π1 � {Li(ˆλ)}n+1 i=1 � = τ ◦ π2 � {(Xi, Yi)}n+1 i=1 � , for all possible {(Xi, Yi)}n+1 i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' It follows that P � Ln+1(ˆλ) ≤ Q(n+1) 1−δ (ˆλ) � ≥ 1 − δ, (10) as Q(n+1) 1−δ (ˆλ) is just the corresponding 1 − δ quantile of the exchangeable variables {Li(ˆλ)}n+1 i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' (See Lemma 1 in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' By definition of the function s, we have Q(n+1) 1−δ (ˆλ) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' This combining formula (10) leads to P � Ln+1(ˆλ) ≤ α � ≥ 1 − δ, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Discussion Theorem 1 shows the loss-controlling guarantee for the ideal case where (Xn+1, Yn+1) is available, whose approach can be approximated using Algorithm 1 in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The conditions of formula (9) also implies that λ∗ is well defined, which makes us able to obtain λ∗ based on the searching function s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Also, by definition, one can conclude that � λ ∈ Λ : Q(n) 1−δ(λ) ≤ α � ⊆ � λ ∈ Λ : Q(n+1) 1−δ (λ) ≤ α � , which indicates that searching λ in the left set above is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Besides, the difference between λ∗ and ˆλ can be ignored for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The proof in Theorem 1 only needs s to be a predefined function independent of {(Xi, Yi)}n+1 i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Thus, 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The frequencies of the prediction losses being greater than α for different δ and α on test data of HighTemp and LowTemp datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' All bars being near or below the preset δ confirms the controlling guarantee of LCC empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The distributions of the prediction losses for different δ and α on test data of HighTemp and LowTemp datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The losses are controlled by α and δ properly to achieve the empirical validity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' one can define s as an optimization algorithm based on another hold-out dataset for parameter searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Due to the general forms of the calibrated predictors and the loss functions, one can consider using LCC to control multiple losses jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Suppose the jth loss on ith calibration sample with λ is Lj,i(λ) and the number of losses is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' One simple method is to search for λ∗ in the following set, � λ ∈ Λ : max j � Q(n) j,1−δ/m(λ) � ≤ α � , where Q(n) j,1−δ/m(λ) is the 1−δ/m quantile of {Lj,i(λ)}n i=1 ∪ {B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Therefore, to control multiple losses jointly, one may have to calculate the 1 − δ/m quantiles, which only makes sense when m is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' EXPERIMENTS We apply LCC to high-impact weather forecasting, which is based on postprocessing of numerical weather prediction (NWP) models [12] [13] [14], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=', learning a predictor whose inputs are forecasts made by NWP models and outputs are corresponding high-impact weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' We use LCC to post- process the ensemble forecasts issued by European Centre for Medium-Range Weather Forecasts (ECMWF) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The forecasts are obtained from the THORPEX Interactive Grand Global Ensemble (TIGGE) dataset [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' We concentrate on 2- m maximum temperature and minimum temperature forecasts initialized at 0000 UTC with the forecast lead times from 12nd hour to 36th hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The resolution of the forecast fields is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='5◦ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='5◦ and the corresponding label fields with the same resolution are calculated using the ERA5 reanalysis data [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The area covers the main parts of North China, East China and Central China, ranging from 109◦E to 122◦E in longitude and from 29◦N to 42◦N in latitude with the grid size being 27 × 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The HighTemp and LowTemp datasets introduced in [5] are used for empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The inputs in HighTemp are 2-m maximum temperature forecasting fields and the corre- sponding label fields are whether the observed 2-m maximum temperature is above 35 °C for each grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Similarly, the inputs in LowTemp are 2-m minimum temperature forecasting fields and the corresponding label fields are whether the observed 2- m minimum temperature is below −15 °C for each grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The sample sizes of HighTemp and LowTemp are 1200 and 1233 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The experimental setting is similar to that in Section IV- B of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' For each dataset, all forecasts made by the NWP model were normalized to [0, 1] by min–max normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 20% of the data were used for testing and 80% and 20% of the remaining data were used for training and calibration respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The normalized ensemble fields forecast by the Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='3 Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='35 Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='4 Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='45 Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='00 Model Data Set = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='3 Data Set = LowTemp | α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='35 Data Set = LowTemp | α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='45 Data Set = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='4 Data Set = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='5 nDNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='30 Unet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 6 6 6Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='45 Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='3 Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='35 Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='4 Data Set = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='0 Model Data Set = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='3 Data Set = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='35 Data Set = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='4 Data Set = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='45 Data Set = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='5 nDNN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='0 Unet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='6 Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2 65 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The distributions of normalized sizes for different δ and α on test data of HighTemp and LowTemp datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The predictions have reasonable sizes for both U-Net and nDNN for high-impact weather forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' NWP model are taken as input x and the set of grids having high-impact weather is the corresponding label y, which can be seen as the image segmentation problem in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Thus, we employed two fully convolutional neural networks [18] as our underlying algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' One was U-Net [19] and the other is the naive deep neural network (nDNN), which is the U-Net removing skip-connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The structures of the two networks are the same as those in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' To train the deep nets, we further partitioned the data for training to validation part (10%) and proper training part (90%), which were used for model selection and parameter updating respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Adam optimization [20] with the learning rate being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='0001 and the number of epochs being 1000 was employed for training, and the model whose binary cross entropy was the lowest on validation data was chosen as the predictive model f needing calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The candidate calibrated predictor Fλ is the same as formula (10) in [5], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=', Fλ(x) = {(p, q) : f(p,q)(x) ≥ 1 − λ}, where f(p,q)(x) is the estimated probability for high-impact weather existing at grid (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The loss function is L(y, F) = 1 − |y ∩ F| |F| , which is a non-monotone loss function related to false dis- covery introduced in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The searching function s we used is the min function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The final calibrated predictors were obtained with the proposed LCC approach and the experimental results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The frequencies of the prediction losses being more than α are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 1 with bar plots for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The columns represent the cases where α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='45 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' All bars are near or below the preset δ, which verifies loss-controlling guarantee empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The boxen plots of the losses for different δ and α are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 2, which contain more information about tails by drawing narrower boxes than box plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' It can be observed that α and δ result in larger losses, which should be preset based on specific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The informational efficiency of Fλ∗ is measured using normalized size of the prediction set defined as |Fλ∗(x)|/PQ in which the numbers of the vertical and the horizontal grids of prediction fields are denoted by P and Q respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The distributions of normalized sizes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' 3, indicating that different α and δ cause different normalized sizes and there should be trade-off among loss level α, confidence level 1 − δ and informational efficiency of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' All the predictions have reasonable sizes using LCC and there are not significant difference between the predictions of UNet and nDNN in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' We also tested other forms of searching functions such as the max function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' However, although the controlling guarantee can be hold empirically, the constructed predictor may lose informational efficiency for applicability, implying that the forms of searching functions should be designed on a case- by-case basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' CONCLUSION This paper proposes loss-controlling calibration, which extends conformal loss-controlling prediction to calibrating predictive models with more general forms of calibrated predictors and losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The finite-sample and distribution-free loss-controlling guarantee is proved by introducing a searching function and the property of transformations preserving ex- changeability in the ideal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' In addition, an approximation approach for practical calibration is proposed, whose steps are the same as those of conformal loss-controlling prediction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=', the only difference between loss-controlling calibration and conformal loss-controlling prediction is whether the calibrated predictors and the loss functions satisfy specific conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' The method is applied to high-impact weather forecasting problems and the loss-controlling guarantee is shown empir- ically for the non-monotone loss related to false discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Further empirical studies for a wider range of applications with case-by-case design are needed to fully understand the power and limitation of the proposed loss-controlling calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' REFERENCES [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Vovk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Gammerman, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Shafer, Algorithmic learning in a random world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQfNAkn/content/2301.04378v1.pdf'} +page_content=' Springer Science & Business Media, 2005.' 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+1University of Notre Dame, 2KAIST, 3University of Minnesota +rwan@nd.edu, jaehyungkim@kaist.ac.kr, dongyeop@umn.edu +Abstract +In NLP annotation, it is common to have multiple annotators +label the text and then obtain the ground truth labels based on +the agreement of major annotators. However, annotators are +individuals with different backgrounds, and minors’ opinions +should not be simply ignored. As annotation tasks become +subjective and topics are controversial in modern NLP tasks, +we need NLP systems that can represent people’s diverse +voices on subjective matters and predict the level of diversity. +This paper examines whether the text of the task and anno- +tators’ demographic background information can be used to +estimate the level of disagreement among annotators. Particu- +larly, we extract disagreement labels from the annotators’ vot- +ing histories in the five subjective datasets, and then fine-tune +language models to predict annotators’ disagreement. Our re- +sults show that knowing annotators’ demographic informa- +tion, like gender, ethnicity, and education level, helps predict +disagreements. In order to distinguish the disagreement from +the inherent controversy from text content and the disagree- +ment in the annotators’ different perspectives, we simulate +everyone’s voices with different combinations of annotators’ +artificial demographics and examine its variance of the fine- +tuned disagreement predictor. Our paper aims to improve the +annotation process for more efficient and inclusive NLP sys- +tems through a novel disagreement prediction mechanism. +Our code and dataset are publicly available. 1 +1 +Introduction +Supervised AI systems are trained on annotated datasets +with labels determined by consensus among multiple anno- +tators. The subjective opinions of different annotators often +bring annotation disagreement in the decision of the final +labels. Most commonly, this disagreement is addressed by +ignoring highly-disagreed cases and only including those +whose opinions were voted on by the majority as the fi- +nal label. When the labeling tasks become more subjective +and require the annotator’s own interpretation and judgment, +such as detecting offensiveness and judging social dilemmas +*This work was done while RW and JK were at the Minnesota +NLP lab. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +1https://github.com/minnesotanlp/Quantifying-Annotation- +Disagreement +" It's okay +to have +abortion." +Disagreement +Predictor +0.34 +0.57 +0.48 +Gr oup- wi se Demogr aphi cs +Per sonal Level Demogr aphi cs +Pl ai n Text wi t hout +Demogr aphi cs +Text +Disagreement +in Existed +Annotation +Pool = 0.5 +vs +Figure 1: Disagreement prediction predicts disagreement +only from input text or input text with (group-wise or in- +dividual against the majority) annotators’ demographic in- +formation. +(Reidsma and op den Akker 2008), this majority-based ag- +gregation often fails to learn the true distribution of anno- +tators’ voices. The increasing subjectivity of NLP problems +in modern NLP will cause the annotators’ disagreement not +only due to the potential random error in the process but +also because annotators may interpret the text with different +views and make judgments based on their own connotations. +Different demographics, cultural backgrounds, and liv- +ing experiences influence how people receive and interpret +information. This difference is more visible in subjective +tasks. For example, Sap et al. found more consecutive an- +notators who have higher scores on racist beliefs are more +likely to label African American English as toxic rather than +label anti-Black language as toxic (Sap et al. 2022). In these +cases, the aggregated singular labels can bring bias by using +less inclusive and societal-representative labels to accom- +modate everyone’s voices in subjective studies. +This paper assumes that annotators’ disagreement poten- +tially comes from the limited representations of the annota- +tor group assigned or controversy of the text in nature. This +study focuses on exploring the relationship between the an- +notator group and natural controversy in text by developing +a disagreement predictor with and without the information +about the annotator group, as depicted in Figure 1. In partic- +ular, we analyze annotators’ disagreement from five subjec- +tive task datasets to answer the following research questions: +arXiv:2301.05036v1 [cs.CL] 12 Jan 2023 + +Q1: Is it possible to predict the level of annotators’ disagree- +ment with text using language models? Does knowing +annotators’ identities, like demographic information in +addition to text, help predict annotation disagreement? +Q2: Is the disagreement caused by the natural controversy of +the text or by the biased distribution of the assigned an- +notators? +Our research demonstrates that disagreement is pre- +dictable in subjective annotation tasks by using Roberta +model (Liu et al. 2019) to predict both hard disagreement +(binary label) and soft disagreement (continuous label). We +further design two demographic augmentation experiments +and find that bringing the annotator-level demographic in- +formation can significantly improve disagreement prediction +performance. Finally, based on the findings, we simulated +artificial annotators’ backgrounds to predict disagreement to +check whether the disagreement score will be changed in +a wider annotator population. In short, we propose a dis- +agreement measurement that can efficiently suggest the op- +timal number of annotators and assign an appropriate demo- +graphic group of annotators per text, possibly helping im- +prove the fairness and quality of subjective annotation. +2 +Related Works +Tasks like toxicity detection (Sap et al. 2020; Yu 2022), sen- +timent analysis (Potts et al. 2021), and social, ethical label- +ing (Forbes et al. 2020; Hendrycks et al. 2021) are highly +subjective and controversial. One can think one post is of- +fensive, but others may consider it acceptable. There is no +one objective ground truth. R¨ottger et al. summarized three +key challenges of descriptive annotation in subjective NLP +tasks: interpretation of disagreement, label aggregation, and +representativeness of annotators. +Due to the absence of absolute ground truth, the interpre- +tation of disagreement becomes complicated (Alm 2011). +For instance, the disagreement may result in considerably +different reliabilities: whether the annotators disagree on the +most critical or least crucial instances (Foley 2018). In addi- +tion, researchers commonly use some notion of the agree- +ment to measure the task’s subjectiveness, such as using +inter-annotator agreement metrics Cohen’s Kappa (Cohen +1960) or Fleiss’ Kappa (Fleiss 1971) to measure annota- +tions’ reliability. But when presenting the final results in +the downstream task, people usually use the aggregated la- +bels that can conceal informative disagreement and evalu- +ation metrics that are unaware of the task’s subjectiveness +(R¨ottger et al. 2022). +Rather than ’correct’ or ’wrong,’ Alm pointed out the +concept of acceptability. There might be multiple accept- +able answers in subjective tasks. However, aggregating la- +bels through major voting will increase the risk of discarding +minority voices. To address the problem, Davani, D´ıaz, and +Prabhakaran treats predicting each annotator’s judgments as +separate subtasks, which achieved the same or better per- +formance than aggregating labels in the data before train- +ing. They also further evaluate the model uncertainty using +the variance of the predicted annotation label. However, this +still concerns that recognizing the aggregated major votes as +the final targets does not always represent all acceptable an- +swers. On the other hand, Uma, Almanea, and Poesio used +posterior calibration with a soft-loss approach to learning +from data containing disagreement. They noticed that tem- +perature scaling only functions with data where disagree- +ments are caused by label overlap and not with data where +disagreements are caused by annotator subjective judgment +or language ambiguity. This aligns with Foley ’s finding for +tasks with subjective labels: without collecting additional +labels, models reach the ceiling of performance given the +small dataset size and the inherent disagreement between +annotators on which documents are controversial. +In previous research, annotators’ demographics have +shown importance in improving the annotation quality in +subjective tasks. For example, Prabhakaran, Davani, and +D’iaz demonstrated that the agreement scores could be very +significant among different socio-demographic groups anno- +tators identified when certain individual annotators disagree +with the majority labels. Further, Gordon et al. proposed jury +learning, a recommender system approach defining which +people or groups, in what proportion, determine the classi- +fier’s prediction. For instance, a jury learning model would +recommend women and black jurors for online hate speech +detection, who are mainly targeted in online harassment. +However, many public datasets didn’t collect annotators’ +demographics with their annotations. Further, the datasets +that reported the annotator’s demographics also have imbal- +anced representative concerns. For example, the race is often +skewed, and dominant with the white race (Sap et al. 2020; +Forbes et al. 2020; Hendrycks et al. 2021; Sap et al. 2022). +As shown above, researchers have implicitly resolved the +label disagreement using majority votes, annotator selec- +tion, and the soft-loss approach(Uma et al. 2021). Different +from the interrater disagreement resolution, which defines +disagreement as a sign of poor quality or mistakes to be +resolved(Oortwijn, Ossenkoppele, and Betti 2021), our re- +search explicitly quantifies disagreement as our task target +and further distinguishes the nuance among various socio- +demographic groups. +3 +Methods +This section presents our method for quantifying subjective +annotation disagreements. Our main idea is modeling the +annotation disagreement using demographic information of +each annotator as additional inputs, with the pre-trained lan- +guage model, e.g., RoBERTa (Liu et al. 2019). In Section +3.1, we first introduce the mathematical notations. Then, we +elaborate on the details of the proposed method in Section +3.2. Finally, we provide a way to simulate the annotators’ +demographic information in Section 3.3. +3.1 +Preliminaries +We first describe the problem setup of our interest under a +text classification scenario with K classes. Let D = (X, Y) +denote the given annotated dataset where X is a set of +texts and Y is the annotation matrix of X. Specifically, +each entry of Y, yi(x) ∈ {1, . . . , K}, represents ith an- + +Binaray +Continuous +? ? ? ? ? +0 +0 +? ? × ? ? +? × ? × ? +1 +1/5 = 0.2 +1 +2/5 = 0.4 +RoBERTa-based +Disagreement +Predictor +"It's okay to have +abortion." +The annotators are a [21-29] year's old [high +school] [white] [woman] +Age: [21-29], Education: [high school], +Race: [white], Gender: [woman] +Demogr aphi c I nf or mat i on +( Sent ence f or mat vs Templ at ed- col on f or mat ) +Text +I nput +Gr ound- t r ut h di sagr eement f r om annot at or s' vot i ng r ecor ds +Di sagr eement +Label s +Figure 2: Our proposed disagreement predictor that takes the task sentence and/or (group or person) demographic information +as input and ground-truth disagreement among annotators as labels. The demographic information is concatenated to the task +sentence either in sentence format or templated-colon format. The ground-truth labels are aggregated from the annotators’ +voting records as binary labels with a threshold (i.e., 3/5) or continuous labels as they are. +notation assigned to text x ∈ X.2 We assume that there +are N different annotations for each text x and y(x) = +[y1(x), . . . , yN(x)] denotes all annotations assigned to x. +Then, rk(x) = �N +i=1 1[yi(x) = k]/N denotes the agree- +ment rate of x to the label k where �K +k=1 rk(x) = 1. In +addition, we assume that T different demographic informa- +tion of all N annotators is available such as gender, age +and race3, and denote it as d(t)(x) = [d(t) +1 (x), . . . , d(t) +N (x)] +where t = 1, . . . , T. Remarkably, majority voting, which +is a popular common practice of assigning the label from +the multiple annotations y(x) to the maximally agreed la- +bel, can be represented as ymaj(x) := arg maxk rk(x). +Binary vs Continuous disagreement labels. From the +agreement rate rk(x), we first compute a binary disagree- +ment label ¯rb(x) = 1[rymaj(x) ̸= 1], which indicates if +there are different opinions among the annotators for this in- +stance x. We further define a continuous disagreement label +¯rc(x) = 1−rymaj(x), that has the scale of 0 (everyone agrees +with the same annotation result) to 1 (a significant number +of people holding different opinions on the annotation re- +sults). Namely, the binary label ¯rb indicates the existence of +at least some different opinions, and the continuous label ¯rc +measures the degree of disagreement among the annotators. +Without loss of generality, we refer both types of disagree- +ment as ¯r. The text with highest disagreement means anno- +tators hold different opinions, and this text content is very +controversial. +3.2 +Disagreement Prediction with Demographic +Information +Our goal is to predict the disagreement ¯r(x) of given text +x because it provides an effective way to understand which +2We clarify that the ith annotation could be labeled by different +annotators between different texts in X. +3This assumption will be relaxed in Section 4.4 +content is controversial or not. To this end, our first idea is to +utilize the pre-trained language model, e.g., RoBERTa (Liu +et al. 2019), for training a predictor fθ of the disagreement +of given text. Specifically, we train the model by minimizing +a mean square error (MSE) loss as follow: +minimize +θ +LMSE(fθ(x), ¯r(x)) +(1) +However, the annotators’ disagreement is not only from +the controversy of the text in nature but also from the lim- +ited representations of the assigned annotator group. Hence, +more than just using text as input is needed to capture the +disagreement fully. +Incorporation of demographics: Group vs Personal. +To this end, our key idea is incorporating the demographic +information of annotators {d(t)(x)}T +t=1 to train the model +fθ. Intuitively, it is expected to encode the valuable informa- +tion of the disagreement of the text x, especially related to +limited representations of the annotator group assigned. To +be specific, we propose two different ways to incorporate the +demographic information: (1) Text with group demographic +information and (2) Text with personal demographic infor- +mation. +Text with group demographic information �xgroup is con- +structed by listing all N annotators’ information d(t)(x) in +one string and then concatenating with the targeted text x: +�xgroup = Combine[d(1)(x), . . . , d(T )(x), x] +(2) +Therefore, the group demographics supplemented text also +has the same number of instances as the original dataset. +On the other hand, text with personal demographic infor- +mation �xperson is constructed by concatenating only one an- +notator’s demographic with text: +�xperson = Combine[d(1) +j (x), . . . , d(T ) +j +(x), x] +(3) + +Table 1: Examples from the five disagreement datasets used in this paper. A stands for annotator. +Datasets +Text +Annotation Distribution +Disagreement Label +SBIC +“Abortion destruction of the nuclear family +contraceptives feminism convincing women to wait for +children damaging economy so youth cannot leave +the nest ramping up tensions between sexes all +serves one primary goal to lower the population.” +A1 (age: 32, politics: liberal, race: white, gender: woman) +votes for inoffensive +A2 (age: 34, politics: liberal, race: white, gender: woman) +votes for inoffensive +A3 (age: 29, politics: mod-liberal, race: hispanic, gender: woman) +votes for offensive +−→ Aggregated Label: inoffensive +Binary: 1 +Continuous: 1/3 +SChem101 +“It’s okay to have abortion.” +A1 (age: 30-39, education: high school, race: white, gender: woman) +votes for people ocassional think this +A2 (age: 40-49, education: grad, race: white, gender: man) +votes for controversial +A3 (age: 30-39, education: bachelor, race: white, gender: man) +votes for common belief +A4 (age: 21-29, education: high school, race: white, gender: woman) +votes for controversial +A5 (age: 30-39 , education: bachelor, race: hispanic, gender: woman) +votes for controversial +−→ Aggregated Label: controversial +Binary: 1 +Continuous: 2/5 +Dilemmas +1st action: “refusing to do a survey on the credit card +reader while paying with cash at the Office Max.” +2nd action: “saying my bf has no right to dictate +who I tell about my abortion.” +1 annotator votes for the first action is less ethical +while 4 others vote the second action is less ethical +−→ Aggregated Label: 2nd action is less ethical +Binary: 1 +Continuous: 1/5 +Dynasent +“Had to remind him to toast the sandwich.” +4 annotators believe it’s negative while one think it is neutral +−→ Aggregated Label: negative +Binary: 1 +Continuous: 1/5 +Politeness +“Where did you learn English? +How come you’re taking on a third language?” +5 annotators politeness scores are 5, 13, 9, 11, 11 +with the maximum of 25. +−→ Aggregated Label: impolite +Binary: 0 +Continuous: 0 +where j = 1, . . . , N and hence it results in N times larger +dataset with N different annotators. +Format: Templated vs Sentence. For combining the de- +mographic information and text, we further propose two +different ways with specific templates: (1) Templated for- +mat and (2) Sentence format. Templated format represents +the category and value of each demographic information in +a separate sentence, then concatenate all of them with the +given text. For example, if one annotator is 36 years white +woman, this demographic information is converted to ”Age: +36, Color: white, Gender: women”, then concatenated with +the original sentence in case of the text with person demo- +graphic. On the other hand, sentence format represents the +demographic information with a natural sentence, e.g., the +annotator is a 36 years old white woman., then concatenate +it with the original sentence. +With these demographics supplemented text �x (�xgroup or +�xperson), we train our model similar to the case with the orig- +inal sentence x in Equation (1): +minimize +θ +LMSE(fθ(�x), ¯r(x)) +(4) +An illustration of the proposed demographic-based dis- +agreement predictor is presented in Figure 2. +3.3 +Simulation of Demographic Information +In addition, we propose a simulation of demographic infor- +mation, which is a novel approach to analyze how the dif- +ferent annotator groups impact disagreement prediction. It +is expected to separately reveal the inherent disagreement of +annotators from the controversy of the text in nature. Specif- +ically, instead of ground-truth {d(t)(x)}T +t=1, we combine the +artificial demographic information {¯d(t)(x)}T +t=1 with the +given text x and annotations y(x), to simulate the scenario +with different annotators. Such as, the gender demographic +type has four possible options: woman, man, transgender, +non-binary; and ethnicity with seven options: white, black +or African American, American Indian or Alaska Native, +Asian, Native Hawaiian or other pacific islanders, Hispanic, +or some other race. Overall, we have a total 28 = 4×7 differ- +ent combinations of the annotator’s demographic informa- +tion for the simulation, while the ground-truth demographic +information is one of them; hence, it offers an opportunity to +explore the more extensive range of demographic informa- +tion with the increased number of instances. Then, we obtain +a predicted disagreement using fθ, which is trained with x +and {d(t)(x)}T +t=1 as introduced in Section 3.2. +Then, we evaluate whether the predicted disagreement is +easily or hard to be changed among the simulated demo- +graphic profiles so that we can distinguish whether the dis- +agreement comes from the controversy of text or uncertainty +from annotators for the disagreement label. For example, if +the variation of predicted disagreement among the simulated +combinations is high and the average change of the predicted +disagreement between the simulated combinations and real +disagreement is large, it might reveal that disagreement is +highly related to the uncertainty of annotators. In contrast, +the lower variation and smaller change between real dis- +agreements indicate the disagreement is based on the con- +troversy in the text, which is stable disagreement among var- +ious kinds of people. + +4 +Experiments +4.1 +Benchmark Datasets +To obtain the annotators’ disagreement, we choose the fol- +lowing five datasets of subjective tasks that include annota- +tors’ voting records in the raw format.4 +Social Bias Inference Corpus (SBIC) (Sap et al. 2020) +contains 150k structured annotations of social media posts. +Each post has three different annotators. Annotators indi- +cated whether the post could be considered “offensive to +anyone.” The offensiveness is a categorical variable with +three possible answers (yes, maybe, no). +Social Chemistry 101 (SChem101) (Forbes et al. 2020) +is a corpus of cultural norms via free-text rules-of-thumb +created by crowd workers. A rule-of-thumb is a judgment of +action which is further broken down into 12 theoretically- +motivated dimensions of people’s judgments. Our study +focuses on the anticipated agreement category. It reflects +workers’ opinion on what portion of people probably agree +with the judgment given the action. The category has five +possible answers: almost no one believes, people occasion- +ally think this, controversial, common belief, universally +true. Each rule of thumb is annotated by five workers. +Scruples-dilemmas (Lourie, Bras, and Choi 2021) is a +resource for normative ranking actions. Each instance pairs +two unrelated actions and identifies which action crowd +workers found less ethical. Each instance is annotated by +five different annotators. +Dyna-Sentiment (Potts et al. 2021) is an English lan- +guage benchmark task for ternary sentiment analysis. Each +Yelp review is validated by five crowd workers into three +possible sentiment results: positive, negative, and neutral. +Wikipedia Politeness (Danescu-Niculescu-Mizil et al. +2013) is a collection of requests from Wikipedia Talk pages, +annotated with politeness. Each Wikipedia request is anno- +tated by five annotators on a 1 to 25 scale. As Danescu- +Niculescu-Mizil et al. ignored neutral cases for politeness +prediction, we extracted the disagreement between the bi- +nary classes of request, i.e., polite and impolite. +Disagreement Label Distributions The Figure 3 shows +the distributions of disagreement scores among five datasets. +For dynasent dataset, since the majority of the dataset has +disagreement between 0.3 to 0.6. The prediction concentrate +around 0.4 to 0.5. The comparison among multiple datasets +reflects that the subject topics influence the crowd annota- +tors’ disagreement. For example, most texts regarding offen- +siveness had consensus opinions from the annotators, while +most annotators disagreed regarding sentiment. +4.2 +Experimental Details +All the experiments are conducted by fine-tuning RoBERTa- +base (Liu et al. 2019) using Adam optimizer (Kingma and +Ba 2015) with a fixed learning rate 1e-5 and the default hy- +perparameters of Adam. For the text classification tasks, the +model is fine-tuned with batch size 8 for 15 epochs. +4Note that only the SBIC and SChem101 datasets report anno- +tators’ demographic information, so we used these two datasets to +evaluate the effect of including demographic information in dis- +agreement prediction. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Disagreement Label +0 +1 +2 +3 +4 +5 +Density +polite +dilemmas +dynasent +SBIC +SChem +Figure 3: Disagreement distributions for five datasets +Table 2: Evaluation results of vanilla (RoBERTa) classifiers +only with text input on the five datasets with disagreement. +Binary Label +Continuous Label +Datasets +F1 (↑) +MSE (↓) +F1 (↑) +MSE (↓) +SBIC +61.5 +0.309 +66.0 +0.086 +SChem101 +0.0 +0.905 +52.3 +0.056 +Dilemmas +0.0 +0.330 +34.2 +0.165 +DynaSent +74.9 +0.361 +11.8 +0.114 +Politeness +55.9 +0.490 +56.8 +0.110 +To the best of our knowledge, we couldn’t find any ex- +isting disagreement predictors to be used as baselines. As a +result, we compare our predictors with different input types +and disagreement labeling setups. Different versions of pre- +trained language models were tested, but RoBERTa always +performed better. For the evaluation of the performance of +the trained disagreement predictor, we use both 1) hard score +F1 and 2) soft score Mean Square Error (MSE), and com- +pare the measurement effect of binary disagreement label +and continuous disagreement rate. +4.3 +Main Results +Disagreement prediction only with text +From Table 2, +we notice that continuous disagreement achieves better pre- +diction than binary disagreement for most of the datasets. +Among the datasets, the disagreement prediction models +work the best in the SBIC dataset. The binary label predic- +tion are close to continuous prediction for SBIC and Polite- +ness datasets. But SChem and Dilemmas have 0 F1 scores +which only give 0 outputs. That means the binary label is +not reliable for the two datasets. +For Dynasent, the binary label has an inconsistent perfor- +mance based on hard score F1 and soft score MSE. We think +one potential reason is that the binary disagreement is highly +unbalanced while converting a continuous prediction to cate- +gorical labels like 0, 0.33, 0.67, and 1 is easy to accidentally +assign an intermediate value to a wrong group. Therefore, +even though we used both F1 and MSE metrics, they are +used to have a parallel comparison between the binary label + +Table 3: Evaluation results of vanilla (RoBERTa) classifiers with text and demographics inputs on the SBIC and Social +Chemistry datasets. Note that only these two datasets among the five include the demographics of the annotators. +Group of demographics +Personal level demographics +Datasets +Input Setup +Label Type +F1 (↑) +MSE (↓) +F1 (↑) +MSE (↓) +SBIC +Demographics(sentence) ; Text +Continuous +65.4 +0.086 +85.6 +0.033 +Demographics(templated) ; Text +Continuous +64.6 +0.087 +85.6 +0.033 +SChem101 +Demographics(sentence) ; Text +Continuous +14.6 +0.065 +48.8 +0.009 +Demographics(templated) ; Text +Continuous +16.9 +0.073 +50.7 +0.009 +Table 4: Examples of four types of predictions: (1) the predictions are the same as the collected disagreement, (2) both predic- +tions are the same but differ from the collected disagreement. When the prediction given only text differ from the given text +with demographics, (3) the former is close to collected disagreement, and (4) the latter is close to collected disagreement. +Type Input +Predicted +Disagreement +Predicted Disagreement +with Demographics +Collected +Disagreement +(1) age: 22, politics: liberal, race: white, gender: woman . [SEP] what is the +most +common name for a black +dude +king kong. +0.003 +0.015 +0 +(2) age: 25, politics: liberal, race: asian, gender: woman. [SEP] swear to god +they +are +the same +people +that +objectify +women +and +reify rape +culturemisogyny . More importantly y do they feel so threatened. +1.034 +0.971 +1 +(3) age: 18 , politics: other, race: asian, gender: man . [SEP] two strong +men needed to lift some furniture . +0.223 +0.096 +0 +(4) age: 29, politics: liberal, race: white, gender: woman. [SEP] 13 reasons why +is making me +wish i went to a real +fucking +highschool . +0.332 +0.01 +0.5 +and continuous label setup. Among the binary classification, +we consider F1 as the metric of model goodness, on the op- +posite, we use MSE to evaluate the regression fitness. +Disagreement prediction with text and demographic in- +formation +Further, by comparing different experiment se- +tups for disagreement with demographic information in Ta- +ble 3, we focus on the different effects of a group of de- +mographics or the personal level of demographics. The re- +sults show that personal-level demographics improve the +disagreement prediction more than group-level demograph- +ics. One potential reason is that the annotator’s level of de- +mographics may imitate the annotation process that each an- +notator labels the text without knowing each other. And also +because concatenating personal level demographics can be +considered as oversampling that group-level setup can not. +Qualitative Results Analysis +Lastly, we categorize pre- +diction into four types and provide an example per each +in Table 4. Using Local Interpretable Model-Agnostic Ex- +planations (LIME) (Ribeiro, Singh, and Guestrin 2016), we +found demographics have been used in prediction with the +text. Blue text is important feature to predict agreement +while orange text is used to predict disagreement. +4.4 +Simulation of Everyone’s Voices with +Artificial Demographics +One remaining question is how to reflect everyone’s diverse +opinions on such subjective and socially sensitive annotation +tasks. To explore this aspect, we run additional experiments +with the simulated demographics introduced in Section 3.3. +Namely, we simulate a different combination of all possi- +ble artificial demographic groups, rather than using the real +annotators’ demographics used in model training (Section +4.3). Then, the disagreement of the simulated demographic +information and the text is predicted using the fine-tuned dis- +agreement predictor introduced in Section 3.2. +Our study is motivated by the Intersectionality theory +(Crenshaw 1990), assuming that people’s perspectives are +shaped by the intersection of all available demographic cat- +egories. We set four gender types, seven ethnicity types, and +five age ranges (see A.1 for details), and thus we have 140 +(4×7×5) artificial annotators’ unique demographic charac- +teristics. Since we only trained our disagreement predic- +tor with demographic information on SBIC and SChem101 +datasets, the simulation experiments are also applied to these +two datasets. We randomly sampled 600s text instances in +each dataset and concatenated them with 140 artificial an- +notators’ demographic information in the colon template to +predict continuous disagreement. +To visualize the simulation result of 140 artificial annota- +tors per text, we made a scatter plot based on the mean and + +Need many +annotators +(a) SBIC Dataset +(b) SChem 101 Dataset +Need less +annotators +"It is understandable to want to +spend holidays with family." +Figure 4: Disagreement prediction with simulated demographic information on (a) SBIC and (b) Schem101 datasets, respec- +tively. Different shapes and colors indicate the different disagreement labels as denoted in the legend. Best viewed in color. +variance of 140 disagreement prediction as shown in Fig- +ure 4. The color and shape denoted at the legend shows the +text’s disagreement label in the original dataset. The higher +points in the plot means higher predicted disagreement rate. +The more rightward point implies a greater variance in the +disagreement prediction among the 140 artificial annotators. +The difficulty of disagreement prediction is related to the +dataset’s topic, quality etc. SBIC is collected from social me- +dia data while SChem is created by crowdsourcing, which +might explain why the clusters are more clear in the Figure +4(b) than in the Figure 4(a). From 4(b), most text are pre- +dicted into corresponding disagreement clusters. But some +outliers are predicted to be more controversial or agreeable. +For example, the circled outlier has an original 0.5 disagree- +ment label but ends up with a 0.04 disagreement prediction +among 140 artificial annotators. The text is ”It is understand- +able to want to spend holidays with family.” Those outliers in +the simulation experiment show the disagreement rate would +change if the annotator change. Other than the outliers, the +disagreement clusters shows they are less influenced by an- +notator change. With this simulation, we can distinguish dis- +agreements caused by the natural controversy of the text or +by the biased distribution of the assigned annotators. +5 +Discussion and Future Work +We could think of potential applications in NLP data anno- +tation pipeline using our disagreement prediction model: +Annotator number estimation. We could potentially use +the predicted disagreement score in order to decide the ap- +propriate number of annotators in a cost-efficient manner, +e.g., we may not need three or five annotators for the text +being predicted zero disagreements. For instance, we may +need one or two annotators if a text is predicted to have +lower disagreement scores. Other than that, we can assign +five or even more annotators to those texts being predicted +as highly disagreeable. +Annotator group assignment. Additionally, we suggest +considering the annotation disagreement as a critical factor +in finding the optimal group of annotator pools. This can +be used as a novel annotator assignment supporting sys- +tem for the data annotation pipeline. In the current annota- +tor recruiting process, there is usually some uncontrollable +randomness from annotators, either from skewed represen- +tatives or individual variations. We present a low-cost ap- +proach to simulate as diverse as possible artificial annota- +tion pools to identify the controversial samples that maxi- +mize the disagreement. Thus, we avoid ignoring human bias +and listening to opinions from a more diverse group of peo- +ple to avoid polarized analysis. We hope our study can evoke +others’ attention in designing a more fair and representative +annotation pipeline. +Potential risk of using demographic information. Last +but not least, though our research shows that annotators’ +demographics help disagreement prediction, we should be +careful about collecting private and personal information. +Also, we admit that NLP or AI systems trained on demo- +graphic information might make another bias toward certain +demographic groups. +6 +Conclusion +Overall, we propose a disagreement prediction framework +that measures annotators’ disagreement in subjective tasks, +predicts disagreement with/without demographic informa- +tion and simulates 140 artificial annotators to build a rela- +tively fair annotation pool. Our results show that the annota- +tors’ disagreement could be fairly predictable from the text +and even better performs when we know the demographic +information of the annotators. With our disagreement pre- +dictor, we believe we could shed light on various applica- +tions of data annotation in a more effective and inclusive +manner. +Acknowledgments +We thank Dr. Maxwell Forbes for sharing the demographic +information data for Social Chemistry 101 dataset. We also + +0.00 +1.0 +0.25 +0.50 +0.8 +0.75 +1.00 +0.6 +0.4 +0.2 +8 +0.0 +0.00 +0.01 +0.02 +0.03 +Disagreement Variance0.00 +1.0 +0.50 +1.00 +0.8 +0.6 +0.4 +0.2 +0.0 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +Disagreement Variancethank the anonymous reviewers and Minnesota NLP mem- +bers for their insightful comments and suggestions. +References +Alm, C. O. 2011. Subjective Natural Language Problems: +Motivations, Applications, Characterizations, and Implica- +tions. In ACL. +Cohen, J. 1960. +A coefficient of agreement for nominal +scales. Educational and psychological measurement, 20(1): +37–46. +Crenshaw, K. 1990. Mapping the margins: Intersectionality, +identity politics, and violence against women of color. 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Scaling and +Disagreements: Bias, Noise, and Ambiguity. Frontiers in +Artificial Intelligence, 5. +Uma, A.; Fornaciari, T.; Dumitrache, A.; Miller, T.; Cham- +berlain, J. P.; Plank, B.; Simpson, E.; and Poesio, M. 2021. +SemEval-2021 Task 12: Learning with Disagreements. In +SEMEVAL. +Yang, Z.; Dai, Z.; Yang, Y.; Carbonell, J.; Salakhutdinov, +R. R.; and Le, Q. V. 2019. Xlnet: Generalized autoregressive +pretraining for language understanding. Advances in neural +information processing systems, 32. +Yu, X. 2022. +Hate Speech and Counter Speech De- +tection: Conversational Context Does Matter. +ArXiv, +abs/2206.06423. + +A +Appendices +A.1 +Simulation Setup of Artificial Annotators +We set gender with four possible options: woman, man, +transgender, non-binary; ethnicity with seven options: white, +black or African American, American Indian or Alaska Na- +tive, Asian, Native Hawaiian or other pacific islanders, His- +panic, or some other race. Also, we set the age with five +ranges: 18 to 29, 30 to 39, 40 - 49, 50-59, and 60 to elder. +A.2 +Annotators Distributions +Our analysis finds that the annotators’ pool in the SBIC +dataset was relatively gender-balanced and age-balanced +(55% women, 42% men, 1% non-binary; 36±10 years old), +but racially skewed (82% White, 4% Asian, 4% Hispanic, +4% Black). And it was also politically skewed (63% liberal, +20% conservative). Overall, workers agreed on a post be- +ing offensive at a rate of 76%. Later, Sap et al. showed that +annotator identity and beliefs are highly related to their toxi- +city ratings in their annotators with attitudes paper (Sap et al. +2022). Similar to the demographic distribution in the SBIC +dataset, the crowd worker pool in SChem101 is also gender- +balanced and race-skewed: 55% were women and 45% men. +89% of workers identified as white, 7% as Black. 39% were +in the 30-39 age range, 27% in the 21-29, and 19% in the +40-49 age range. Regarding education, 44% had a bache- +lor’s degree, and 36% had some college experience or an +associate’s degree. However, even though some people con- +sider one rule as a common belief, other people may think +no one believes it. +A.3 +Group v.s. Personal Demographics Setup +Table 5 shows one example of text with individual annota- +tors’ demographics or with the group of annotators’ demo- +graphics. +A.4 +Disagreement Prediction Given Only Or +Partial Demographics +To further evaluate how annotators’ demographics influence +disagreement prediction, we also tested the inputs of only +demographics, which performed much worse than the inputs +including text. Notably, this experimental input setup might +mislead, assuming people from certain social groups always +have the kind of opinion regardless of the text context. +Based on our above study, we controlled the demograph- +ics in the templated format of individual annotators and the +label in the continuous format, which is the optimal setup. +And we evaluated the input of text with partial demographic +information as shown in Table 6. It shows that the predic- +tions are given input of text with a single demographic fac- +tor, or only demographics perform worse than predicting +with text and intersectional demographic information. We +also tried using random forests given only demographic fea- +tures to predict annotation disagreement. The age feature +was the most important. +Table 5: Setups between group and personal demographics +Input +Disagreement Label +age: 22, politics: liberal, +race: white, gender: woman. +what is the most common +name for a black dude king kong. +0 +age: 34, politics: liberal, +race: white, gender: woman. +what is the most common +name for a black dude king kong +0 +age: 57, politics: mod-liberal, +race: white, gender: woman. +what is the most common +name for a black dude king kong +0 +age: 22, politics: liberal, +race: white, gender: woman. +age: 34, politics: liberal, +race: white, gender: woman. +age: 57, politics: mod-liberal, +race: white, gender: woman. +what is the most common +name for a black dude king kong +0 +Table 6: Evaluation results of vanilla (RoBERTa) classifiers +with only demographics or text and partial demographics. +Personal level demographics +Datasets +Input Setup +F1 (↑) +MSE (↓) +SBIC +Only demographics +32.29 +0.172 +SChem +Only demographics +14.0 +0.134 +Age; text +49.39 +0.008 +Education; text +47.04 +0.009 +Race; text +47.04 +0.009 +Gender; text +47.04 +0.008 +Table 7: Evaluation results of different classifiers with text +input on SChem Dataset with disagreement. +Text with Continuous Label +Model +F1 (↑) +MSE (↓) +BERT +25.18 +0.061 +XLNet +20.22 +0.068 +AlBERTa +20.81 +0.066 +A.5 +Results of Other Language Models on +Disagreement Prediction +We only reported Roberta in our main paper, which showed +the best performance. But we have also conducted experi- +ments with other language models like BERT(Devlin et al. +2018), XLNet(Yang et al. 2019), and AlBERTa(Lan et al. +2019). Table 7 shows the other language models’ prediction +results on SChem as an example. + diff --git a/vNE4T4oBgHgl3EQfWwyu/content/tmp_files/load_file.txt b/vNE4T4oBgHgl3EQfWwyu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4492cbcf428637c873bb1b6497e95320989bc778 --- /dev/null +++ b/vNE4T4oBgHgl3EQfWwyu/content/tmp_files/load_file.txt @@ -0,0 +1,818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf,len=817 +page_content='Everyone’s Voice Matters: Quantifying Annotation Disagreement Using Demographic Information Ruyuan Wan1*, Jaehyung Kim2∗, Dongyeop Kang3 1University of Notre Dame, 2KAIST, 3University of Minnesota rwan@nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='edu, jaehyungkim@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='kr, dongyeop@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='edu Abstract In NLP annotation, it is common to have multiple annotators label the text and then obtain the ground truth labels based on the agreement of major annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' However, annotators are individuals with different backgrounds, and minors’ opinions should not be simply ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' As annotation tasks become subjective and topics are controversial in modern NLP tasks, we need NLP systems that can represent people’s diverse voices on subjective matters and predict the level of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' This paper examines whether the text of the task and anno- tators’ demographic background information can be used to estimate the level of disagreement among annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Particu- larly, we extract disagreement labels from the annotators’ vot- ing histories in the five subjective datasets, and then fine-tune language models to predict annotators’ disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Our re- sults show that knowing annotators’ demographic informa- tion, like gender, ethnicity, and education level, helps predict disagreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In order to distinguish the disagreement from the inherent controversy from text content and the disagree- ment in the annotators’ different perspectives, we simulate everyone’s voices with different combinations of annotators’ artificial demographics and examine its variance of the fine- tuned disagreement predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Our paper aims to improve the annotation process for more efficient and inclusive NLP sys- tems through a novel disagreement prediction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Our code and dataset are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 1 1 Introduction Supervised AI systems are trained on annotated datasets with labels determined by consensus among multiple anno- tators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The subjective opinions of different annotators often bring annotation disagreement in the decision of the final labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Most commonly, this disagreement is addressed by ignoring highly-disagreed cases and only including those whose opinions were voted on by the majority as the fi- nal label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' When the labeling tasks become more subjective and require the annotator’s own interpretation and judgment, such as detecting offensiveness and judging social dilemmas This work was done while RW and JK were at the Minnesota NLP lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='com/minnesotanlp/Quantifying-Annotation- Disagreement " It\'s okay to have abortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='" Disagreement Predictor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='48 Gr oup- wi se Demogr aphi cs Per sonal Level Demogr aphi cs Pl ai n Text wi t hout Demogr aphi cs Text Disagreement in Existed Annotation Pool = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='5 vs Figure 1: Disagreement prediction predicts disagreement only from input text or input text with (group-wise or in- dividual against the majority) annotators’ demographic in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' (Reidsma and op den Akker 2008), this majority-based ag- gregation often fails to learn the true distribution of anno- tators’ voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The increasing subjectivity of NLP problems in modern NLP will cause the annotators’ disagreement not only due to the potential random error in the process but also because annotators may interpret the text with different views and make judgments based on their own connotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Different demographics, cultural backgrounds, and liv- ing experiences influence how people receive and interpret information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' This difference is more visible in subjective tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For example, Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' found more consecutive an- notators who have higher scores on racist beliefs are more likely to label African American English as toxic rather than label anti-Black language as toxic (Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In these cases, the aggregated singular labels can bring bias by using less inclusive and societal-representative labels to accom- modate everyone’s voices in subjective studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' This paper assumes that annotators’ disagreement poten- tially comes from the limited representations of the annota- tor group assigned or controversy of the text in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' This study focuses on exploring the relationship between the an- notator group and natural controversy in text by developing a disagreement predictor with and without the information about the annotator group, as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In partic- ular, we analyze annotators’ disagreement from five subjec- tive task datasets to answer the following research questions: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='05036v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='CL] 12 Jan 2023 Q1: Is it possible to predict the level of annotators’ disagree- ment with text using language models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Does knowing annotators’ identities, like demographic information in addition to text, help predict annotation disagreement?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Q2: Is the disagreement caused by the natural controversy of the text or by the biased distribution of the assigned an- notators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Our research demonstrates that disagreement is pre- dictable in subjective annotation tasks by using Roberta model (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2019) to predict both hard disagreement (binary label) and soft disagreement (continuous label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We further design two demographic augmentation experiments and find that bringing the annotator-level demographic in- formation can significantly improve disagreement prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Finally, based on the findings, we simulated artificial annotators’ backgrounds to predict disagreement to check whether the disagreement score will be changed in a wider annotator population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In short, we propose a dis- agreement measurement that can efficiently suggest the op- timal number of annotators and assign an appropriate demo- graphic group of annotators per text, possibly helping im- prove the fairness and quality of subjective annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2 Related Works Tasks like toxicity detection (Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Yu 2022), sen- timent analysis (Potts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2021), and social, ethical label- ing (Forbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2021) are highly subjective and controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' One can think one post is of- fensive, but others may consider it acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' There is no one objective ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' R¨ottger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' summarized three key challenges of descriptive annotation in subjective NLP tasks: interpretation of disagreement, label aggregation, and representativeness of annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Due to the absence of absolute ground truth, the interpre- tation of disagreement becomes complicated (Alm 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For instance, the disagreement may result in considerably different reliabilities: whether the annotators disagree on the most critical or least crucial instances (Foley 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In addi- tion, researchers commonly use some notion of the agree- ment to measure the task’s subjectiveness, such as using inter-annotator agreement metrics Cohen’s Kappa (Cohen 1960) or Fleiss’ Kappa (Fleiss 1971) to measure annota- tions’ reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' But when presenting the final results in the downstream task, people usually use the aggregated la- bels that can conceal informative disagreement and evalu- ation metrics that are unaware of the task’s subjectiveness (R¨ottger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Rather than ’correct’ or ’wrong,’ Alm pointed out the concept of acceptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' There might be multiple accept- able answers in subjective tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' However, aggregating la- bels through major voting will increase the risk of discarding minority voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' To address the problem, Davani, D´ıaz, and Prabhakaran treats predicting each annotator’s judgments as separate subtasks, which achieved the same or better per- formance than aggregating labels in the data before train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' They also further evaluate the model uncertainty using the variance of the predicted annotation label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' However, this still concerns that recognizing the aggregated major votes as the final targets does not always represent all acceptable an- swers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' On the other hand, Uma, Almanea, and Poesio used posterior calibration with a soft-loss approach to learning from data containing disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' They noticed that tem- perature scaling only functions with data where disagree- ments are caused by label overlap and not with data where disagreements are caused by annotator subjective judgment or language ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' This aligns with Foley ’s finding for tasks with subjective labels: without collecting additional labels, models reach the ceiling of performance given the small dataset size and the inherent disagreement between annotators on which documents are controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In previous research, annotators’ demographics have shown importance in improving the annotation quality in subjective tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For example, Prabhakaran, Davani, and D’iaz demonstrated that the agreement scores could be very significant among different socio-demographic groups anno- tators identified when certain individual annotators disagree with the majority labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Further, Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' proposed jury learning, a recommender system approach defining which people or groups, in what proportion, determine the classi- fier’s prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For instance, a jury learning model would recommend women and black jurors for online hate speech detection, who are mainly targeted in online harassment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' However, many public datasets didn’t collect annotators’ demographics with their annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Further, the datasets that reported the annotator’s demographics also have imbal- anced representative concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For example, the race is often skewed, and dominant with the white race (Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Forbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' As shown above, researchers have implicitly resolved the label disagreement using majority votes, annotator selec- tion, and the soft-loss approach(Uma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Different from the interrater disagreement resolution, which defines disagreement as a sign of poor quality or mistakes to be resolved(Oortwijn, Ossenkoppele, and Betti 2021), our re- search explicitly quantifies disagreement as our task target and further distinguishes the nuance among various socio- demographic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 3 Methods This section presents our method for quantifying subjective annotation disagreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Our main idea is modeling the annotation disagreement using demographic information of each annotator as additional inputs, with the pre-trained lan- guage model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=', RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='1, we first introduce the mathematical notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Then, we elaborate on the details of the proposed method in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Finally, we provide a way to simulate the annotators’ demographic information in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='1 Preliminaries We first describe the problem setup of our interest under a text classification scenario with K classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Let D = (X, Y) denote the given annotated dataset where X is a set of texts and Y is the annotation matrix of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Specifically, each entry of Y, yi(x) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' , K}, represents ith an- Binaray Continuous ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 0 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' × ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' × ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' × ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 1 1/5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2 1 2/5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 RoBERTa-based Disagreement Predictor "It\'s okay to have abortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='" The annotators are a [21-29] year\'s old [high school] [white] [woman] Age: [21-29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Education: [high school],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Race: [white],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=" Gender: [woman] Demogr aphi c I nf or mat i on ( Sent ence f or mat vs Templ at ed- col on f or mat ) Text I nput Gr ound- t r ut h di sagr eement f r om annot at or s' vot i ng r ecor ds Di sagr eement Label s Figure 2: Our proposed disagreement predictor that takes the task sentence and/or (group or person) demographic information as input and ground-truth disagreement among annotators as labels." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The demographic information is concatenated to the task sentence either in sentence format or templated-colon format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The ground-truth labels are aggregated from the annotators’ voting records as binary labels with a threshold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=', 3/5) or continuous labels as they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' notation assigned to text x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2 We assume that there are N different annotations for each text x and y(x) = [y1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' , yN(x)] denotes all annotations assigned to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Then, rk(x) = �N i=1 1[yi(x) = k]/N denotes the agree- ment rate of x to the label k where �K k=1 rk(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In addition, we assume that T different demographic informa- tion of all N annotators is available such as gender, age and race3, and denote it as d(t)(x) = [d(t) 1 (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' , d(t) N (x)] where t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Remarkably, majority voting, which is a popular common practice of assigning the label from the multiple annotations y(x) to the maximally agreed la- bel, can be represented as ymaj(x) := arg maxk rk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Binary vs Continuous disagreement labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' From the agreement rate rk(x), we first compute a binary disagree- ment label ¯rb(x) = 1[rymaj(x) ̸= 1], which indicates if there are different opinions among the annotators for this in- stance x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We further define a continuous disagreement label ¯rc(x) = 1−rymaj(x), that has the scale of 0 (everyone agrees with the same annotation result) to 1 (a significant number of people holding different opinions on the annotation re- sults).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Namely, the binary label ¯rb indicates the existence of at least some different opinions, and the continuous label ¯rc measures the degree of disagreement among the annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Without loss of generality, we refer both types of disagree- ment as ¯r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The text with highest disagreement means anno- tators hold different opinions, and this text content is very controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2 Disagreement Prediction with Demographic Information Our goal is to predict the disagreement ¯r(x) of given text x because it provides an effective way to understand which 2We clarify that the ith annotation could be labeled by different annotators between different texts in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 3This assumption will be relaxed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 content is controversial or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' To this end, our first idea is to utilize the pre-trained language model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=', RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2019), for training a predictor fθ of the disagreement of given text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Specifically, we train the model by minimizing a mean square error (MSE) loss as follow: minimize θ LMSE(fθ(x), ¯r(x)) (1) However, the annotators’ disagreement is not only from the controversy of the text in nature but also from the lim- ited representations of the assigned annotator group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Hence, more than just using text as input is needed to capture the disagreement fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Incorporation of demographics: Group vs Personal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' To this end, our key idea is incorporating the demographic information of annotators {d(t)(x)}T t=1 to train the model fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Intuitively, it is expected to encode the valuable informa- tion of the disagreement of the text x, especially related to limited representations of the annotator group assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' To be specific, we propose two different ways to incorporate the demographic information: (1) Text with group demographic information and (2) Text with personal demographic infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Text with group demographic information �xgroup is con- structed by listing all N annotators’ information d(t)(x) in one string and then concatenating with the targeted text x: �xgroup = Combine[d(1)(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' , d(T )(x), x] (2) Therefore, the group demographics supplemented text also has the same number of instances as the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' On the other hand, text with personal demographic infor- mation �xperson is constructed by concatenating only one an- notator’s demographic with text: �xperson = Combine[d(1) j (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' , d(T ) j (x), x] (3) Table 1: Examples from the five disagreement datasets used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' A stands for annotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Datasets Text Annotation Distribution Disagreement Label SBIC “Abortion destruction of the nuclear family contraceptives feminism convincing women to wait for children damaging economy so youth cannot leave the nest ramping up tensions between sexes all serves one primary goal to lower the population.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' A1 (age: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' politics: liberal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' race: white,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' gender: woman) votes for inoffensive A2 (age: 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' politics: liberal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' race: white,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' gender: woman) votes for inoffensive A3 (age: 29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' politics: mod-liberal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' race: hispanic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' gender: woman) votes for offensive −→ Aggregated Label: inoffensive Binary: 1 Continuous: 1/3 SChem101 “It’s okay to have abortion.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' A1 (age: 30-39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' education: high school,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' race: white,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' gender: woman) votes for people ocassional think this A2 (age: 40-49,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' education: grad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' race: white,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' gender: man) votes for controversial A3 (age: 30-39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' education: bachelor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' race: white,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' gender: man) votes for common belief A4 (age: 21-29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' education: high school,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' race: white,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' gender: woman) votes for controversial A5 (age: 30-39 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' education: bachelor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' race: hispanic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' gender: woman) votes for controversial −→ Aggregated Label: controversial Binary: 1 Continuous: 2/5 Dilemmas 1st action: “refusing to do a survey on the credit card reader while paying with cash at the Office Max.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2nd action: “saying my bf has no right to dictate who I tell about my abortion.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 1 annotator votes for the first action is less ethical while 4 others vote the second action is less ethical −→ Aggregated Label: 2nd action is less ethical Binary: 1 Continuous: 1/5 Dynasent “Had to remind him to toast the sandwich.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 4 annotators believe it’s negative while one think it is neutral −→ Aggregated Label: negative Binary: 1 Continuous: 1/5 Politeness “Where did you learn English?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' How come you’re taking on a third language?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 5 annotators politeness scores are 5, 13, 9, 11, 11 with the maximum of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' −→ Aggregated Label: impolite Binary: 0 Continuous: 0 where j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' , N and hence it results in N times larger dataset with N different annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Format: Templated vs Sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For combining the de- mographic information and text, we further propose two different ways with specific templates: (1) Templated for- mat and (2) Sentence format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Templated format represents the category and value of each demographic information in a separate sentence, then concatenate all of them with the given text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For example, if one annotator is 36 years white woman, this demographic information is converted to ”Age: 36, Color: white, Gender: women”, then concatenated with the original sentence in case of the text with person demo- graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' On the other hand, sentence format represents the demographic information with a natural sentence, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=', the annotator is a 36 years old white woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=', then concatenate it with the original sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' With these demographics supplemented text �x (�xgroup or �xperson), we train our model similar to the case with the orig- inal sentence x in Equation (1): minimize θ LMSE(fθ(�x), ¯r(x)) (4) An illustration of the proposed demographic-based dis- agreement predictor is presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='3 Simulation of Demographic Information In addition, we propose a simulation of demographic infor- mation, which is a novel approach to analyze how the dif- ferent annotator groups impact disagreement prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' It is expected to separately reveal the inherent disagreement of annotators from the controversy of the text in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Specif- ically, instead of ground-truth {d(t)(x)}T t=1, we combine the artificial demographic information {¯d(t)(x)}T t=1 with the given text x and annotations y(x), to simulate the scenario with different annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Such as, the gender demographic type has four possible options: woman, man, transgender, non-binary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' and ethnicity with seven options: white, black or African American, American Indian or Alaska Native, Asian, Native Hawaiian or other pacific islanders, Hispanic, or some other race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Overall, we have a total 28 = 4×7 differ- ent combinations of the annotator’s demographic informa- tion for the simulation, while the ground-truth demographic information is one of them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' hence, it offers an opportunity to explore the more extensive range of demographic informa- tion with the increased number of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Then, we obtain a predicted disagreement using fθ, which is trained with x and {d(t)(x)}T t=1 as introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Then, we evaluate whether the predicted disagreement is easily or hard to be changed among the simulated demo- graphic profiles so that we can distinguish whether the dis- agreement comes from the controversy of text or uncertainty from annotators for the disagreement label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For example, if the variation of predicted disagreement among the simulated combinations is high and the average change of the predicted disagreement between the simulated combinations and real disagreement is large, it might reveal that disagreement is highly related to the uncertainty of annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In contrast, the lower variation and smaller change between real dis- agreements indicate the disagreement is based on the con- troversy in the text, which is stable disagreement among var- ious kinds of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='1 Benchmark Datasets To obtain the annotators’ disagreement, we choose the fol- lowing five datasets of subjective tasks that include annota- tors’ voting records in the raw format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 Social Bias Inference Corpus (SBIC) (Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2020) contains 150k structured annotations of social media posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Each post has three different annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Annotators indi- cated whether the post could be considered “offensive to anyone.” The offensiveness is a categorical variable with three possible answers (yes, maybe, no).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Social Chemistry 101 (SChem101) (Forbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2020) is a corpus of cultural norms via free-text rules-of-thumb created by crowd workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' A rule-of-thumb is a judgment of action which is further broken down into 12 theoretically- motivated dimensions of people’s judgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Our study focuses on the anticipated agreement category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' It reflects workers’ opinion on what portion of people probably agree with the judgment given the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The category has five possible answers: almost no one believes, people occasion- ally think this, controversial, common belief, universally true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Each rule of thumb is annotated by five workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Scruples-dilemmas (Lourie, Bras, and Choi 2021) is a resource for normative ranking actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Each instance pairs two unrelated actions and identifies which action crowd workers found less ethical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Each instance is annotated by five different annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Dyna-Sentiment (Potts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2021) is an English lan- guage benchmark task for ternary sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Each Yelp review is validated by five crowd workers into three possible sentiment results: positive, negative, and neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Wikipedia Politeness (Danescu-Niculescu-Mizil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2013) is a collection of requests from Wikipedia Talk pages, annotated with politeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Each Wikipedia request is anno- tated by five annotators on a 1 to 25 scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' As Danescu- Niculescu-Mizil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ignored neutral cases for politeness prediction, we extracted the disagreement between the bi- nary classes of request, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=', polite and impolite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Disagreement Label Distributions The Figure 3 shows the distributions of disagreement scores among five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For dynasent dataset, since the majority of the dataset has disagreement between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The prediction concentrate around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The comparison among multiple datasets reflects that the subject topics influence the crowd annota- tors’ disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For example, most texts regarding offen- siveness had consensus opinions from the annotators, while most annotators disagreed regarding sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2 Experimental Details All the experiments are conducted by fine-tuning RoBERTa- base (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2019) using Adam optimizer (Kingma and Ba 2015) with a fixed learning rate 1e-5 and the default hy- perparameters of Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For the text classification tasks, the model is fine-tuned with batch size 8 for 15 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 4Note that only the SBIC and SChem101 datasets report anno- tators’ demographic information, so we used these two datasets to evaluate the effect of including demographic information in dis- agreement prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 Disagreement Label 0 1 2 3 4 5 Density polite dilemmas dynasent SBIC SChem Figure 3: Disagreement distributions for five datasets Table 2: Evaluation results of vanilla (RoBERTa) classifiers only with text input on the five datasets with disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Binary Label Continuous Label Datasets F1 (↑) MSE (↓) F1 (↑) MSE (↓) SBIC 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='309 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='086 SChem101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='905 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='056 Dilemmas 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='330 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='165 DynaSent 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='361 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='114 Politeness 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='490 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='110 To the best of our knowledge, we couldn’t find any ex- isting disagreement predictors to be used as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' As a result, we compare our predictors with different input types and disagreement labeling setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Different versions of pre- trained language models were tested, but RoBERTa always performed better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For the evaluation of the performance of the trained disagreement predictor, we use both 1) hard score F1 and 2) soft score Mean Square Error (MSE), and com- pare the measurement effect of binary disagreement label and continuous disagreement rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='3 Main Results Disagreement prediction only with text From Table 2, we notice that continuous disagreement achieves better pre- diction than binary disagreement for most of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Among the datasets, the disagreement prediction models work the best in the SBIC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The binary label predic- tion are close to continuous prediction for SBIC and Polite- ness datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' But SChem and Dilemmas have 0 F1 scores which only give 0 outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' That means the binary label is not reliable for the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For Dynasent, the binary label has an inconsistent perfor- mance based on hard score F1 and soft score MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We think one potential reason is that the binary disagreement is highly unbalanced while converting a continuous prediction to cate- gorical labels like 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='67, and 1 is easy to accidentally assign an intermediate value to a wrong group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Therefore, even though we used both F1 and MSE metrics, they are used to have a parallel comparison between the binary label Table 3: Evaluation results of vanilla (RoBERTa) classifiers with text and demographics inputs on the SBIC and Social Chemistry datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Note that only these two datasets among the five include the demographics of the annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Group of demographics Personal level demographics Datasets Input Setup Label Type F1 (↑) MSE (↓) F1 (↑) MSE (↓) SBIC Demographics(sentence) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Text Continuous 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='086 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='033 Demographics(templated) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Text Continuous 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='087 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='033 SChem101 Demographics(sentence) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Text Continuous 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='065 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='009 Demographics(templated) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Text Continuous 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='073 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='009 Table 4: Examples of four types of predictions: (1) the predictions are the same as the collected disagreement, (2) both predic- tions are the same but differ from the collected disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' When the prediction given only text differ from the given text with demographics, (3) the former is close to collected disagreement, and (4) the latter is close to collected disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Type Input Predicted Disagreement Predicted Disagreement with Demographics Collected Disagreement (1) age: 22, politics: liberal, race: white, gender: woman .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' [SEP] what is the most common name for a black dude king kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='015 0 (2) age: 25, politics: liberal, race: asian, gender: woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' [SEP] swear to god they are the same people that objectify women and reify rape culturemisogyny .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' More importantly y do they feel so threatened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='971 1 (3) age: 18 , politics: other, race: asian, gender: man .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' [SEP] two strong men needed to lift some furniture .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='096 0 (4) age: 29, politics: liberal, race: white, gender: woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' [SEP] 13 reasons why is making me wish i went to a real fucking highschool .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='5 and continuous label setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Among the binary classification, we consider F1 as the metric of model goodness, on the op- posite, we use MSE to evaluate the regression fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Disagreement prediction with text and demographic in- formation Further, by comparing different experiment se- tups for disagreement with demographic information in Ta- ble 3, we focus on the different effects of a group of de- mographics or the personal level of demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The re- sults show that personal-level demographics improve the disagreement prediction more than group-level demograph- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' One potential reason is that the annotator’s level of de- mographics may imitate the annotation process that each an- notator labels the text without knowing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' And also because concatenating personal level demographics can be considered as oversampling that group-level setup can not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Qualitative Results Analysis Lastly, we categorize pre- diction into four types and provide an example per each in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Using Local Interpretable Model-Agnostic Ex- planations (LIME) (Ribeiro, Singh, and Guestrin 2016), we found demographics have been used in prediction with the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Blue text is important feature to predict agreement while orange text is used to predict disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 Simulation of Everyone’s Voices with Artificial Demographics One remaining question is how to reflect everyone’s diverse opinions on such subjective and socially sensitive annotation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' To explore this aspect, we run additional experiments with the simulated demographics introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Namely, we simulate a different combination of all possi- ble artificial demographic groups, rather than using the real annotators’ demographics used in model training (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Then, the disagreement of the simulated demographic information and the text is predicted using the fine-tuned dis- agreement predictor introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Our study is motivated by the Intersectionality theory (Crenshaw 1990), assuming that people’s perspectives are shaped by the intersection of all available demographic cat- egories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We set four gender types, seven ethnicity types, and five age ranges (see A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='1 for details), and thus we have 140 (4×7×5) artificial annotators’ unique demographic charac- teristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Since we only trained our disagreement predic- tor with demographic information on SBIC and SChem101 datasets, the simulation experiments are also applied to these two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We randomly sampled 600s text instances in each dataset and concatenated them with 140 artificial an- notators’ demographic information in the colon template to predict continuous disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' To visualize the simulation result of 140 artificial annota- tors per text, we made a scatter plot based on the mean and Need many annotators (a) SBIC Dataset (b) SChem 101 Dataset Need less annotators "It is understandable to want to spend holidays with family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='" Figure 4: Disagreement prediction with simulated demographic information on (a) SBIC and (b) Schem101 datasets, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Different shapes and colors indicate the different disagreement labels as denoted in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' variance of 140 disagreement prediction as shown in Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The color and shape denoted at the legend shows the text’s disagreement label in the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The higher points in the plot means higher predicted disagreement rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The more rightward point implies a greater variance in the disagreement prediction among the 140 artificial annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The difficulty of disagreement prediction is related to the dataset’s topic, quality etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' SBIC is collected from social me- dia data while SChem is created by crowdsourcing, which might explain why the clusters are more clear in the Figure 4(b) than in the Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' From 4(b), most text are pre- dicted into corresponding disagreement clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' But some outliers are predicted to be more controversial or agreeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For example, the circled outlier has an original 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='5 disagree- ment label but ends up with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='04 disagreement prediction among 140 artificial annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The text is ”It is understand- able to want to spend holidays with family.” Those outliers in the simulation experiment show the disagreement rate would change if the annotator change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Other than the outliers, the disagreement clusters shows they are less influenced by an- notator change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' With this simulation, we can distinguish dis- agreements caused by the natural controversy of the text or by the biased distribution of the assigned annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 5 Discussion and Future Work We could think of potential applications in NLP data anno- tation pipeline using our disagreement prediction model: Annotator number estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We could potentially use the predicted disagreement score in order to decide the ap- propriate number of annotators in a cost-efficient manner, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=', we may not need three or five annotators for the text being predicted zero disagreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' For instance, we may need one or two annotators if a text is predicted to have lower disagreement scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Other than that, we can assign five or even more annotators to those texts being predicted as highly disagreeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Annotator group assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Additionally, we suggest considering the annotation disagreement as a critical factor in finding the optimal group of annotator pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' This can be used as a novel annotator assignment supporting sys- tem for the data annotation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In the current annota- tor recruiting process, there is usually some uncontrollable randomness from annotators, either from skewed represen- tatives or individual variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We present a low-cost ap- proach to simulate as diverse as possible artificial annota- tion pools to identify the controversial samples that maxi- mize the disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Thus, we avoid ignoring human bias and listening to opinions from a more diverse group of peo- ple to avoid polarized analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We hope our study can evoke others’ attention in designing a more fair and representative annotation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Potential risk of using demographic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Last but not least, though our research shows that annotators’ demographics help disagreement prediction, we should be careful about collecting private and personal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Also, we admit that NLP or AI systems trained on demo- graphic information might make another bias toward certain demographic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 6 Conclusion Overall, we propose a disagreement prediction framework that measures annotators’ disagreement in subjective tasks, predicts disagreement with/without demographic informa- tion and simulates 140 artificial annotators to build a rela- tively fair annotation pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Our results show that the annota- tors’ disagreement could be fairly predictable from the text and even better performs when we know the demographic information of the annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' With our disagreement pre- dictor, we believe we could shed light on various applica- tions of data annotation in a more effective and inclusive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Acknowledgments We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Maxwell Forbes for sharing the demographic information data for Social Chemistry 101 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We also 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='03 Disagreement Variance0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='05 Disagreement Variancethank the anonymous reviewers and Minnesota NLP mem- bers for their insightful comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' References Alm, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Seattle, United States: Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Uma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Almanea, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' and Poesio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Scaling and Disagreements: Bias, Noise, and Ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Frontiers in Artificial Intelligence, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Uma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Fornaciari, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Dumitrache, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Miller, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Cham- berlain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Plank, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Simpson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' and Poesio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' SemEval-2021 Task 12: Learning with Disagreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' In SEMEVAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Carbonell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Salakhutdinov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' and Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Xlnet: Generalized autoregressive pretraining for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Advances in neural information processing systems, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Hate Speech and Counter Speech De- tection: Conversational Context Does Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ArXiv, abs/2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='06423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' A Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='1 Simulation Setup of Artificial Annotators We set gender with four possible options: woman, man, transgender, non-binary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' ethnicity with seven options: white, black or African American, American Indian or Alaska Na- tive, Asian, Native Hawaiian or other pacific islanders, His- panic, or some other race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Also, we set the age with five ranges: 18 to 29, 30 to 39, 40 - 49, 50-59, and 60 to elder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='2 Annotators Distributions Our analysis finds that the annotators’ pool in the SBIC dataset was relatively gender-balanced and age-balanced (55% women, 42% men, 1% non-binary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 36±10 years old), but racially skewed (82% White, 4% Asian, 4% Hispanic, 4% Black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' And it was also politically skewed (63% liberal, 20% conservative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Overall, workers agreed on a post be- ing offensive at a rate of 76%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Later, Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' showed that annotator identity and beliefs are highly related to their toxi- city ratings in their annotators with attitudes paper (Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Similar to the demographic distribution in the SBIC dataset, the crowd worker pool in SChem101 is also gender- balanced and race-skewed: 55% were women and 45% men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 89% of workers identified as white, 7% as Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 39% were in the 30-39 age range, 27% in the 21-29, and 19% in the 40-49 age range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Regarding education, 44% had a bache- lor’s degree, and 36% had some college experience or an associate’s degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' However, even though some people con- sider one rule as a common belief, other people may think no one believes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='3 Group v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Personal Demographics Setup Table 5 shows one example of text with individual annota- tors’ demographics or with the group of annotators’ demo- graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='4 Disagreement Prediction Given Only Or Partial Demographics To further evaluate how annotators’ demographics influence disagreement prediction, we also tested the inputs of only demographics, which performed much worse than the inputs including text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Notably, this experimental input setup might mislead, assuming people from certain social groups always have the kind of opinion regardless of the text context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Based on our above study, we controlled the demograph- ics in the templated format of individual annotators and the label in the continuous format, which is the optimal setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' And we evaluated the input of text with partial demographic information as shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' It shows that the predic- tions are given input of text with a single demographic fac- tor, or only demographics perform worse than predicting with text and intersectional demographic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' We also tried using random forests given only demographic fea- tures to predict annotation disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' The age feature was the most important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Table 5: Setups between group and personal demographics Input Disagreement Label age: 22, politics: liberal, race: white, gender: woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' what is the most common name for a black dude king kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 0 age: 34, politics: liberal, race: white, gender: woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' what is the most common name for a black dude king kong 0 age: 57, politics: mod-liberal, race: white, gender: woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' what is the most common name for a black dude king kong 0 age: 22, politics: liberal, race: white, gender: woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' age: 34, politics: liberal, race: white, gender: woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' age: 57, politics: mod-liberal, race: white, gender: woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' what is the most common name for a black dude king kong 0 Table 6: Evaluation results of vanilla (RoBERTa) classifiers with only demographics or text and partial demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Personal level demographics Datasets Input Setup F1 (↑) MSE (↓) SBIC Only demographics 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='172 SChem Only demographics 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='134 Age;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' text 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='008 Education;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' text 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='009 Race;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' text 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='009 Gender;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' text 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='008 Table 7: Evaluation results of different classifiers with text input on SChem Dataset with disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Text with Continuous Label Model F1 (↑) MSE (↓) BERT 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='061 XLNet 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='068 AlBERTa 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='066 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content='5 Results of Other Language Models on Disagreement Prediction We only reported Roberta in our main paper, which showed the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' But we have also conducted experi- ments with other language models like BERT(Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2018), XLNet(Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2019), and AlBERTa(Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} +page_content=' Table 7 shows the other language models’ prediction results on SChem as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE4T4oBgHgl3EQfWwyu/content/2301.05036v1.pdf'} diff --git a/vtE0T4oBgHgl3EQfsgGN/content/tmp_files/2301.02580v1.pdf.txt b/vtE0T4oBgHgl3EQfsgGN/content/tmp_files/2301.02580v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0118414bac0549af169d854437321bda6b3fb5c --- /dev/null +++ b/vtE0T4oBgHgl3EQfsgGN/content/tmp_files/2301.02580v1.pdf.txt @@ -0,0 +1,1252 @@ +Neuro-DynaStress: Predicting Dynamic +Stress Distributions in Structural +Components +Hamed Bolandi1,2*, Gautam Sreekumar2, Xuyang Li1,2, Nizar +Lajnef1 and Vishnu Naresh Boddeti2 +1*Civil and Environmental Engineering, Michigan State +University, Shaw Lane, East Lansing, 48824, MI, USA. +2Computer Science and Engineering, Michigan State University, +Shaw Lane, East Lansing, 48824, MI, USA. +*Corresponding author(s). E-mail(s): bolandih@msu.edu; +Contributing authors: sreekum1@msu.edu; lixuyan1@msu.edu; +lajnefni@msu.edu; vishnu@msu.edu; +Abstract +Structural components are typically exposed to dynamic loading, such +as earthquakes, wind, and explosions. Structural engineers should be +able to conduct real-time analysis in the aftermath or during extreme +disaster events requiring immediate corrections to avoid fatal fail- +ures. As a result, it is crucial to predict dynamic stress distribu- +tions during highly disruptive events in real time. Currently avail- +able high-fidelity methods, such as Finite Element Models (FEMs), +suffer from their inherent high complexity and are computationally +prohibitive. Therefore, to reduce computational cost while preserving +accuracy, a deep learning model, Neuro-DynaStress, is proposed to +predict the entire sequence of stress distribution based on finite ele- +ment simulations using a partial differential equation (PDE) solver. +The model was designed and trained to use the geometry, boundary +conditions and sequence of loads as input and predict the sequences +of high-resolution stress contours. The proposed framework’s perfor- +mance is compared to finite element simulations using a PDE solver. +Keywords: Deep Learning; Finite Element Analysis, Dynamic Stress +Distribution, Structural Engineering +1 +arXiv:2301.02580v1 [physics.geo-ph] 19 Dec 2022 + +2 +Neuro-DynaStress: Predicting Dynamic Stress Distributions +Gusset plate +Load +Neuro-DynaStress +Fig. 1: Overview: Unlike FEM, our proposed Neuro-DynaStress is compu- +tationally efficient and facilitates real-time analysis. The existing workflow for +FEM applications includes: (i) modeling the geometry and its components, +(ii) specifying material properties, boundary conditions, meshing, and loading, +(iii) dynamic analysis, which may be time-consuming based on the complexity +of the model. Our Neuro-DynaStress takes geometry, boundary condition, and +load as input and predicts the dynamic stress distribution at all time steps in +one shot. +1 Introduction +Numerical analysis methods, such as Finite Element Analysis (FEA), are +typically used to conduct stress analysis of various structures and systems +for which it is impractical or hard to determine an analytical solution. +Researchers commonly use FEA methods to evaluate the design, safety and +maintenance of different structures in various fields, including aerospace, +automotive, architecture and civil structural systems. The current workflow +for FEA applications includes: (i) modeling the geometry and its components, +(ii) specifying material properties, boundary conditions, meshing, and loading, +(iii) dynamic analysis, which may be time-consuming based on the complexity +of the model. The time requirement constraint and the complexity of the +current FEA workflow make it impractical for real-time or near real-time +applications, such as in the aftermath of a disaster or during extreme disrup- +tive events that require immediate corrections to avoid catastrophic failures. +Based on the steps of FEA described above, performing a complete stress +analysis with conventional FEA has a high computational cost. In order + +Neuro-DynaStress: Predicting Dynamic Stress Distributions +3 +to overcome this problem, some recent works have proposed deep neural +network (DNN)-based methods to predict stress distributions in both intact +and damaged structural components [1, 2], bypassing the need for static +finite element analysis. But these works are not suitable for dynamic finite +element analysis. We propose an architecture that can act as a surrogate for +FEA solvers for dynamic FEA while avoiding the computational bottlenecks +involved. To demonstrate its utility, we model the stress distribution in gusset +plates under dynamic loading. Bridges and buildings rely heavily on gusset +plates as one of their most critical components. Gusset plates are designed +to withstand lateral loads such as earthquakes and winds, which makes fast +dynamic models valuable in avoiding catastrophic failures. +The main idea here is to train a model that can later be used when real- +time estimations are needed, such as in the aftermath of extreme disruptive +events. For example, focusing on critical structural components, there is a need +for immediate assessment following a disaster or during extremely disruptive +events to guide corrective actions. Engineers could rely on the proposed compu- +tationally efficient algorithms to determine stress distributions over damaged +gusset plates and apply the proper rehabilitation actions. They need to be +able to analyze gusset plates quickly and accurately, which is what our model +can provide. To our knowledge, this work is the first to predict dynamic stress +distribution in the specific domain of steel plates. +2 Related Work +The most recent works in data-driven applications of scientific machine +learning have included design and topology optimization [3, 4], data-driven +approaches in fluid dynamics [5, 6], molecular dynamics simulation [7, 8], and +material properties prediction [9–12]. Atalla et al. [13] and Levin et al. [14] have +used neural regression for FEA model updating. More recently, DL has shown +promise in solving traditional mechanics problems. Some researchers used +DL for structural damage detection, a promising alternative to conventional +structural health monitoring methods [15, 16]. +Javadi et al. [17] used a typical neural network in FEA as a surrogate for +the traditional constitutive material model. They simplified the geometry into +a feature vector which approaches hard to generalize complicated cases. The +numerical quadrature of the element stiffness matrix in the FEA on a per- +element basis was optimized by Oishi et al. [18] using deep learning. Their +approach helps to accelerate the calculation of the element stiffness matrix. +Convolutional Neural Networks (CNN) are commonly used in tasks involving +2D information due to the design of their architecture. Recently, Madani et +al. [19] developed a CNN architecture for stress prediction of arterial walls in +atherosclerosis. Also, Liang et al. [20] proposed a CNN model for aortic wall +stress prediction. Their method is expected to allow real-time stress analysis +of human organs for a wide range of clinical applications. + +4 +Neuro-DynaStress: Predicting Dynamic Stress Distributions +Gulgec et al. [21] proposed a CNN architecture to classify simulated dam- +aged and intact samples and localize the damage in steel gusset plates. Modares +et al. [22] conducted a study on composite materials to identify the presence +and type of structural damage using CNNs. Also, in order to detect concrete +cracks without calculating the defect features, Cha et al. [23] proposed a vision- +based method based on convolutional neural networks (CNNs). Do et al. [24] +proposed a method for forecasting the crack propagation in risk assessment of +engineering structures based on “long short-term memory” and “multi-layer +neural network”. An approach for predicting stress distribution on all layers of +non-uniform 3D parts was presented by Khadilkar et al. [25]. More recently, Nie +et al. [26] developed a CNN-based method to predict the low-resolution stress +field in a 2D linear cantilever beam. Jiang et al. [27] developed a conditional +generative adversarial network for low-resolution von Mises stress distribution +prediction in solid structures. +Some studies have been conducted to develop methods of predicting struc- +tural response using ML models. Dong et al. [28] proposed a support vector +machine approach to predict nonlinear structural responses. Wu et al. [29] Uti- +lized deep convolutional neural networks to estimate the structural dynamic +responses. Long short-term memory (LSTM) [30] was used by Zhang et +al. [31] to predict nonlinear structural response under earthquake loading. +Fang et al. [32] proposed a deep-learning-based structural health monitoring +(SHM) framework capable of predicting a dam’s structural dynamic responses +once explosions are experienced using LSTM. Kohar et al. [33] used 3D- +CNN-autoencoder and LSTM to predict the force-displacement response and +deformation of the mesh in vehicle crash-worthiness. Schwarzer et al. [34] +construct a neural network architecture that combines a graph convolutional +neural network (GCN) with a recurrent neural network (RNN) to predict frac- +ture propagation in brittle materials. Lazzara et al. [35] proposed a dual-phase +LSTM Auto-encoder-based surrogate model to predict aircraft dynamic land- +ing response over time. Jahanbakht et al. [36] presented an FEA-inspired DNN +using an attention transformer to predict the sediment distribution in the wide +coral reef. +The few models that studied stress predictions suffer from the problem of +low-resolution predictions, making them unsuitable for decision-making after +a catastrophic failure. To the best of our knowledge, this is the first work to +predict dynamic stress distribution in the specific domain of steel plates with +high accuracy and low latency. The algorithm takes the geometry, boundary +conditions, and time histories as input and renders the dynamic von Mises +stress distribution as an output. We modeled the steel plates as gusset plates +with dynamic loading applied at different edges, different boundary conditions, +and varying complex geometries. + +Neuro-DynaStress: Predicting Dynamic Stress Distributions +5 +3 Methods +3.1 Data Generation +Two-dimensional steel plate structures with five edges, E1 to E5 denoting edges +1 to 5, as shown in Fig. 2, are considered homogeneous and isotropic linear +elastic materials. Various geometries are generated by changing the position of +each node in horizontal and vertical directions, as shown in Fig. 2, which led to +1024 unique pentagons. The material properties remain unchanged, isotropic +for all samples. The 2D steel plates approach the geometry of gusset plates. +Gusset plates connect beams and columns to braces in steel structures. The +behavior and analysis of these components are critical since various reports +have observed failures of gusset plates subject to lateral loads [37–40]. The +boundary conditions and time-history load cases are considered to simulate +similar conditions in common gusset plate structures under external loading. +Some of the most common gusset plates configurations in practice are shown +in Fig. 3. +30 cm +15 cm +15 cm +5 cm +15 cm +5 cm +E1 +E3 +E4 +E5 +E2 +15 cm +Fig. 2: Basic schematic topology for initializing the steel plate geometries. +Column +Gusset plate +Brace +Beam +(a) +(b) +(c) +(d) +Fig. 3: Some of the most common gusset plates in practice. +A total of 57,344 unique samples were created by combining 14 random +time-history load cases and four most common boundary conditions in gusset +plates. Boundary conditions are shown in Fig. 4, mimicking the real gusset +plates’ boundary conditions. All the translation and rotational displacements + +6 +Neuro-DynaStress: Predicting Dynamic Stress Distributions +were fixed at the boundary conditions. The range for width and height of the +plates is from 30 cm to 60 cm. Each time history consists of 100 time steps +generated with random sine and cosine frequencies. The frequencies range +between 1 and 3 HZ, with amplitudes ranging from 2 to 10 kN at intervals +of 2 kN. All time histories in horizontal and vertical directions are shown in +Fig. 5. Considering 100 time steps, each interval is 0.01 seconds, making the +total time equal to 1 second. All the details for the input variables used to +initialize the population are shown in Table 1. +E1 +E3 +E4 +E5 +E2 +E1 +E3 +E4 +E5 +E2 +E1 +E3 +E4 +E5 +E2 +(a) +(b) +(c) +(d) +E1 +E3 +E4 +E5 +E2 +Fig. 4: Different types of boundary conditions for initializing population. +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +−15000 +−10000 +−5000 +0 +5000 +10000 +15000 +Time (s) +Load (N) +(a) +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +0 +20 +40 +60 +80 +100 +−10000 +−5000 +0 +5000 +10000 +Time (s) +Load (N) +(b) +Fig. 5: Time histories (a) Horizontal direction (b) Vertical direction +3.2 Input Data +The geometry is encoded as a 200 × 200 matrix and, incidentally, a binary +image. 0 (black) and 1 (white) denote outside and inside of the geometry, as + +Neuro-DynaStress: Predicting Dynamic Stress Distributions +7 +Table 1: Input variable +Geometry +Boundary +conditions +Load +position +Frequencies +(HZ) +Load +(kN) +Time +steps +Total +time (s) +pentagon +E2 +E4E5 +1,1.5,2,2.5,3 +2,4,6,8,10 +100 +1 +pentagon +E2E3 +E5 +1,1.5,2,2.5,3 +2,4,6,8,10 +100 +1 +pentagon +E1E2 +E4 +1,1.5,2,2.5,3 +2,4,6,8,10 +100 +1 +pentagon +E3 +E2E5 +1,1.5,2,2.5,3 +2,4,6,8,10 +100 +1 +shown in Fig. 6(a). The boundary condition is also represented by another +200 × 200 pixel binary image, where the constrained edges are defined by +1 (white) as shown in Fig. 6(b). Moreover, each time step of time histories +for horizontal and vertical components is encoded in the load position of the +corresponding frame. Load positions in each time step have values between 0 +and 1, corresponding to each time step of time histories, and all remaining +elements are zero. All the load frames of each sample in horizontal and vertical +directions are saved as tensors of dimension 100 × 200 × 200. Figs. 6(c) and +6(d) show loads in the horizontal and vertical directions. The colored load +positions in Figs. 6(c) and 6(d) are used only for visualization. Each row of +Fig. 6 represents one of the simulated samples. Details of boundary conditions +and their load positions are described in Table 1. +Fig. 6: Input and output representation for stress distribution prediction: (a) +geometry, (b) boundary condition, (c) horizontal load, (d) vertical load, (e) +output +3.3 Output Data +FEA was performed using the Partial Differential Equation (PDE) solver +in the MATLAB toolbox to obtain the stress distributions of each sample. +We used transient-planestress function of MATLAB PDE solver to generate + +(b) +(d) +(e) +(c) +(a)8 +Neuro-DynaStress: Predicting Dynamic Stress Distributions +dynamic stress contours as the ground truth of our model. We defined the +geometry, boundary condition, material properties and time histories as input +and PDE solver returns the sequence of stress distributions corresponding +to the inputs. The MATLAB PDE toolbox mesh generator only generates +unstructured triangulated meshes incompatible with CNN. The minimum and +maximum triangulated mesh sizes are 5 and 10mm, respectively. Since each +element should be represented by one pixel in an image, we develop a 200×200 +grid surface equal to the dimensions of the largest possible geometry. Figs. 7(a) +and 7(b) show the unstructured mesh and the 200 × 200 grid surface on top +of a random sample. The stress values are then interpolated between the tri- +angular elements and grids to determine a stress distribution compatible with +our CNN network. The stress values of all the elements outside the material +geometry are assigned to zero, as shown in Fig. 6(e). +Fig. 7: A sample of mesh generation: (a) unstructured triangular mesh, (b) +structured gird surface +The dimension of the largest sample is 600 × 600 mm, and the smallest +is 300 × 300 mm. Using a mesh grid of 200 × 200 on top of samples made +each element 3 × 3 mm, which means that each frame of output has 40000 +pixels. This high-resolution dataset led to achieving significant accuracy. The +maximum and minimum von Mises stress values for elements among the entire +dataset are 279,370 and -980 MPa, respectively. We normalized all the output +data between 0 and 1 to ensure faster convergence and encoded it to 200×200 +for each frame. +3.4 Stress Calculation +The steps for linear finite element analysis’ stress calculation, which is part +of phase (iii) of FEA’s workflow elaborated in the introduction section, are as +follows: + +(a) +(b)Neuro-DynaStress: Predicting Dynamic Stress Distributions +9 +KQ = F +(1) +where K denotes a global stiffness matrix, F is the load vector applied at +each node, and Q denotes the displacement. A stiffness matrix K consists of +elemental stiffness matrices Ke: +Ke = AeBT DB +(2) +where B represents strain-displacement matrix; D represents stress-strain +matrix; and Ae represents area of element. Mesh geometry and material prop- +erties determine B and D. This will be followed by adding the local stiffness +matrix ke to the global stiffness matrix. The displacement boundary conditions +are encoded using the corresponding rows and columns in the global stiffness +matrix K. Solving Q can be achieved using direct factorization or iterative +methods. +As a result of calculating the global displacement using equation 1, we can +calculate the nodal displacements q then we can calculate the stress tensors of +each element as follows: +σ = DBq +(3) +where σ specifies the tensor of an element. The 2-D von Mises Stress +criterion is then used to calculate each element’s von Mises Stress: +σvm = +� +σ2x + σ2y − σxσy + 3τ 2xy +(4) +where σvm denotes von Mises Stress, σx, σy are the normal stress compo- +nents and τxy is the shear stress component. +4 Proposed Methodology +We use convolutional layers to encode the spatial information from the input. +Our hypothesis is that these layers will combine the information in geome- +try, boundary conditions, and load. A key characteristic of dynamic structural +systems is the temporal dependence of their states. LSTM is a suitable architec- +ture for modeling temporal information in sequence and hence is a good choice +to model structural dynamic systems in our experiments. For high-quality +2D reconstructions, we use transposed convolutional layers in our decoder. +For further improving training and performance, we use modules from the +recently proposed feature-aligned pyramid networks (FaPN) [41]. FaPN allows +the decoder to access information from the encoder directly. Overall, our net- +work architecture consists of four modules: encoder consisting of convolutional +layers, temporal module made using LSTM modules, decoder consisting of +transposed convolutional layers, and alignment modules acting as connections +between encoder and decoder. The number of layers in each module and the +number of layers in LSTM modules were chosen based on their performance. + +10 +Neuro-DynaStress: Predicting Dynamic Stress Distributions +The architecture is illustrated schematically in Fig. 8. The size of layers and +hyper-parameters used in the network are summarized in Table 2. +LSTM +Feature-aligned Pyramid Network (FaPN) +Elementwise Addition +Conv Layer +Transposed Conv Layer +Fig. 8: Architecture for the proposed Neuro-DynaStress. The convolutional +encoder maps the raw input data to a latent space. LSTM layers processes the +information across different time frames. The final output is obtained from the +resulting latent representation using transposed convolutional layers. +Table 2: Network layers and hyper-parameters +Type of layers +Number of layers +First layer (H×W×C) +Last layer (H×W×C) +Conv +6 +200×200×16 +7×7×512 +LSTM +4 +1×1×512 +1×1×512 +ConvT +5 +13×13×256 +200×200×16 +FaPN +4 +13×13×256 +100×100×32 +Batch size +Learning rate +Weight decay +Loss function +8 +10−4 +10−5 +MAE +5 Loss Function and Performance Metrics +We use Mean Absolute Error (MAE), defined in Eq. 5 as the primary training +loss and metric. To ensure that we do not overfit to a single metric, we also +use Mean Relative Percentage Error (MRPE) to evaluate the overall quality +of predicted stress distribution. +MAE = +1 +NT +n,t +� +N,T +|S(n, t) − ˆS(n, t)| +(5) +MRPE = +MAE +max |S(n, t), ˆS(n, t)| +× 100 +(6) + +Neuro-DynaStress: Predicting Dynamic Stress Distributions +11 +where S(n, t) is the true stress value at a node n at time step t, as computed by +FEA, and ˆS(n, t) is the corresponding stress value predicted by our model, N +is the total number of mesh nodes in each frame of a sample, and T is a total +number of time steps in each sample. As mentioned earlier, we set T = 100 in +our experiments. +6 Implementation and Computational +Performance +We implemented our model using PyTorch [42] and PyTorch Lightning. +AdamW optimizer [43] was used as the optimizer with a learning rate of 10−4. +We found that a batch size of 8 gave the best results. The computational per- +formance of the model was evaluated on an AMD EPYC 7313 16-core processor +and two NVIDIA A6000 48G GPUs. The time required during the training +phase for a single sample with 100 frames and a batch size of 8 was 10 sec- +onds. In the training phase, one forward and backward pass was considered. +The inference time for one sample was less than 5 ms which can be consid- +ered a real-time requirement. The most powerful FE solvers take between 10 +minutes to an hour to solve the same. Therefore, Neuro-DynaStress is about +72 × 104 times faster than conventional FE solvers. We consider the minimum +time for all processes of modeling geometry, meshing, and analysis of one sam- +ple in FE solver to be about 10 minutes. MATLAB PDE solver does not use +GPU acceleration. This demonstrates that our proposed approach can achieve +the real-time requirement during the validating phase. +7 Results and Discussions +7.1 Quantitative Evaluation +Our model is trained on the training dataset for 45 epochs and evaluated on +the validation dataset using separate metrics. The training dataset consisted +of 48,755, while the validating dataset contained 8,589 samples, together form- +ing the 80%-20% split of the whole dataset. The model predicts five frames +of output from a sequence of five previous inputs until all 100 frames are pre- +dicted. The best validation performance was obtained when we sequenced five +frames during validation. The best checkpoint during validation, at epoch 40, +is the basis for all error metrics. MRPE for the validating dataset is just 2.3%. +7.2 Qualitative Evaluation +The prediction results for a few randomly selected samples from the vali- +dation dataset are visualized in Figs. 9a and 9b. The first row represents 5 +frames out of 100 frames of one reference sample. The second row illustrates +the prediction corresponding to the frames in the first row, and the last row +represents the error in the corresponding predictions. The columns represent +the time steps 1, 25, 50, 75 and 100 seconds. We visualized frames at intervals + +12 +Neuro-DynaStress: Predicting Dynamic Stress Distributions +of 25 seconds to evaluate different ranges of dynamic stress prediction. +Reference +t=1 +t=25 +t=50 +t=75 +t=100 +Predict +Error +0 +2000 +4000 +6000 +8000 +10000 +12000 +0 +2000 +4000 +6000 +(a) +Reference +t=1 +t=25 +t=50 +t=75 +t=100 +Predict +Error +0 +5000 +10000 +15000 +20000 +25000 +5000 +10000 +15000 +(b) +Fig. 9: Successful predicted dynamic stress distribution and their correspond- +ing errors in different time sequences for two samples. The top row corresponds +to reference frames and the middle row shows the predictions. The bottom row +shows the absolute error between corresponding frames (Unit = MPa) + +Neuro-DynaStress: Predicting Dynamic Stress Distributions +13 +For visualization purposes, the references and predictions in Figs. 9a and +9b are scaled to the same range using the maximum and the minimum of each +sample. The errors are scaled independently. As it can be seen in Fig. 9a, the +predicted frames are quite similar to their corresponding references. Although +the geometry contains sharp corners and edges, which are areas that are hard +for CNN to reconstruct, our model is able to predict it. The errors, except for +a small part of the first frame, are in an acceptable range which shows the pre- +diction accuracy of our model. Fig. 9b shows another successful reconstruction. +Comparing references with their corresponding predicted frames demonstrates +that our Neuro-DynaStress model can capture both load variations and max- +imum stress values at the same time. Furthermore, these results demonstrate +that our model is able to predict a dynamic stress distribution with a high +variation of distributed stress. +Fig.10 shows a random failure sample. In spite of the model’s success in +predicting most parts of the frames, it is not able to reconstruct high-stress +concentrations at angles of 90 degrees. Since CNNs typically struggle in +handling sharp edges, smoothening the sharp corners using Gaussian filters +during data preprocessing may help the network to train better. Furthermore, +as the loads in frames t = 25 and t = 75 are lower than in other frames, the +prediction in those frames is acceptable. +Reference +t=1 +t=25 +t=50 +t=75 +t=100 +Predict +Error +0 +10000 +20000 +30000 +40000 +50000 +60000 +5000 +10000 +15000 +20000 +25000 +Fig. 10: Failed predicted dynamic stress distribution and their corresponding +errors in different time sequences. (Unit = MPa) +It is also important that the predictions are temporally consistent. In +order to qualitatively demonstrate the temporal consistency of the proposed +method, Fig. 11a shows a comparison of stress values across 100 frames for + +14 +Neuro-DynaStress: Predicting Dynamic Stress Distributions +0 +20 +40 +60 +80 +100 +0 +5000 +10000 +15000 +20000 +0 +20 +40 +60 +80 +100 +0 +2500 +5000 +7500 +10000 +12500 +15000 +17500 +0 +20 +40 +60 +80 +100 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Time (s) +Stress (MPa) +Reference +Predict +Error +(a) +0 +20 +40 +60 +80 +100 +0 +2000 +4000 +6000 +8000 +0 +20 +40 +60 +80 +100 +0 +5000 +10000 +15000 +20000 +0 +20 +40 +60 +80 +100 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Time (s) +Stress (MPa) +Reference +Predict +Error +(b) +Fig. 11: Comparison of stress values across 100 frames for predictions, ref- +erences, and errors in a randomly selected element. (a) Successful predictions +(b) Unsuccessful predictions (Units = MPa-T). +successful predictions in a randomly selected element. As can be seen, the +references and the predicted distributions are almost identical in most time +sequences, with errors close to zero, despite the stress varying widely with +time. Fig. 11a illustrates how prediction fits with reference more closely when +there is more temporal smoothness at peak points. For instance, a good +match between prediction and reference can be seen in the rightmost graph +in Fig. 11a, where the stress variation follows a smooth Gaussian distribution +in the last peak. However, in the remaining graphs, the prediction has good +correlation with the reference despite a lack of smoothness in most peak stress +values. Moreover, based on the graphs in Fig. 11a, we can conclude that the +model is better at predicting stress in valleys compared to peaks. +We have also illustrated some of the unsuccessful predictions in Fig. 11b to +identify the limitations of our proposed model. It can be seen that in all graphs +with non-Gaussian stress distributions, the model finds it difficult to capture +the peak stress values accurately. However, in the first two graphs from the + +Neuro-DynaStress: Predicting Dynamic Stress Distributions +15 +(a) +(b) +(c) +(d) +(a) +(b) +(c) +(d) +(a) +(b) +Fig. 12: Relative errors across 100 frames in the randomly selected sample. +Graphs in the center represent the MRPE per frame. (a) and (c) in each figure +represent the reference; (b) and (d) refer to their corresponding predictions. +Arrows refer to the MRPE of the presented frame. (Units = MPa-T). +left in Fig. 11b, the predictions perfectly fit later peaks of the reference since +the stress values in the reference have Gaussian distributions at these points. +Figs. 12a and 12b depict the MRPE of randomly selected samples across 100 +frames and frames corresponding to the minimum and maximum MRPE. As +can be seen for both samples, the minimum errors are around zero, with only +a few frames exceeding the error by more than 2%. +7.3 Ablation Study +The efficiency of architecture can be attributed to several design choices we +have made. Our architecture models the temporal dependency between time +frames and the relationship between different elements in an input. Even + +16 +Neuro-DynaStress: Predicting Dynamic Stress Distributions +though self-attention has shown state-of-the-art performance in sequence mod- +eling, they are not suitable for tasks without large amounts of data. Hence, we +use LSTMs for sequence modeling. To demonstrate our claim, we compare our +architecture against other baseline architectures. We compare against three +architectures as shown in Table 3. The model with multi head self-attention +is very similar to our architecture, except the LSTM modules in our model +are replaced with self-attention modules. The details of the other models are +represented in Table 3. We will refer to our architecture as Neuro-DynaStress. +The results are shown in Table 3, and the best results are highlighted in bold. +Table 3: Architecture comparison +Architecture for modeling temporal information +Multi-headed self-attention +LSTM +LSTM +LSTM +FaPN +✓ +✓ +✓ +× +Skip connection +✓ +✓ +× +× +MRPE(%) +4.5 +2.3 +6.6 +9.7 +8 Conclusion +We propose Neuro-DynaStress model equipped with Convolutional Neural +Network (CNN) and Long Short Term Memory (LSTM) to predict the entire +sequence of dynamic stress distribution. The model was designed and trained to +use the geometry, boundary conditions and the sequence of loads as input and +predicts the sequence of high-resolution dynamic stress contours. The convolu- +tional components are used to extract spatial features and the LSTM captures +the temporal dependence between the frames. Feature alignment modules are +used to improve the training and performance of our model. The model is +trained using synthetic data generated using the PDE toolbox in MATLAB. +Neuro-DynaStress can predict dynamic stress distribution with a mean rela- +tive percentage error of 2.3%, which is considered an acceptable error rate in +engineering communities. +Declarations +• This research was funded in part by the National Science Foundation grant +CNS 1645783. +• There is no conflict of interest among the authors of this paper +• The datasets generated during and/or analyzed during the current study +are available from the corresponding author upon reasonable request. + +Neuro-DynaStress: Predicting Dynamic Stress Distributions +17 +References +[1] Bolandi, H., Li, X., Salem, T., Boddeti, V., Lajnef, N.: Bridging finite +element and deep learning: High-resolution stress distribution prediction +in structural components. 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Advances in neural +information processing systems 32 (2019) +[43] Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. arXiv +preprint arXiv:1711.05101 (2017) + diff --git a/vtE0T4oBgHgl3EQfsgGN/content/tmp_files/load_file.txt b/vtE0T4oBgHgl3EQfsgGN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f49bf3539ebf313841ed3f60743d088ad9ba7d7a --- /dev/null +++ b/vtE0T4oBgHgl3EQfsgGN/content/tmp_files/load_file.txt @@ -0,0 +1,1000 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf,len=999 +page_content='Neuro-DynaStress: Predicting Dynamic Stress Distributions in Structural Components Hamed Bolandi1,2*, Gautam Sreekumar2, Xuyang Li1,2, Nizar Lajnef1 and Vishnu Naresh Boddeti2 1*Civil and Environmental Engineering, Michigan State University, Shaw Lane, East Lansing, 48824, MI, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 2Computer Science and Engineering, Michigan State University, Shaw Lane, East Lansing, 48824, MI, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' E-mail(s): bolandih@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Contributing authors: sreekum1@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' lixuyan1@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' lajnefni@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' vishnu@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Abstract Structural components are typically exposed to dynamic loading, such as earthquakes, wind, and explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Structural engineers should be able to conduct real-time analysis in the aftermath or during extreme disaster events requiring immediate corrections to avoid fatal fail- ures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' As a result, it is crucial to predict dynamic stress distribu- tions during highly disruptive events in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Currently avail- able high-fidelity methods, such as Finite Element Models (FEMs), suffer from their inherent high complexity and are computationally prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Therefore, to reduce computational cost while preserving accuracy, a deep learning model, Neuro-DynaStress, is proposed to predict the entire sequence of stress distribution based on finite ele- ment simulations using a partial differential equation (PDE) solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The model was designed and trained to use the geometry, boundary conditions and sequence of loads as input and predict the sequences of high-resolution stress contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The proposed framework’s perfor- mance is compared to finite element simulations using a PDE solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Keywords: Deep Learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Finite Element Analysis, Dynamic Stress Distribution, Structural Engineering 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='02580v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='geo-ph] 19 Dec 2022 2 Neuro-DynaStress: Predicting Dynamic Stress Distributions Gusset plate Load Neuro-DynaStress Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 1: Overview: Unlike FEM, our proposed Neuro-DynaStress is compu- tationally efficient and facilitates real-time analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The existing workflow for FEM applications includes: (i) modeling the geometry and its components, (ii) specifying material properties, boundary conditions, meshing, and loading, (iii) dynamic analysis, which may be time-consuming based on the complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Our Neuro-DynaStress takes geometry, boundary condition, and load as input and predicts the dynamic stress distribution at all time steps in one shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 1 Introduction Numerical analysis methods, such as Finite Element Analysis (FEA), are typically used to conduct stress analysis of various structures and systems for which it is impractical or hard to determine an analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Researchers commonly use FEA methods to evaluate the design, safety and maintenance of different structures in various fields, including aerospace, automotive, architecture and civil structural systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The current workflow for FEA applications includes: (i) modeling the geometry and its components, (ii) specifying material properties, boundary conditions, meshing, and loading, (iii) dynamic analysis, which may be time-consuming based on the complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The time requirement constraint and the complexity of the current FEA workflow make it impractical for real-time or near real-time applications, such as in the aftermath of a disaster or during extreme disrup- tive events that require immediate corrections to avoid catastrophic failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Based on the steps of FEA described above, performing a complete stress analysis with conventional FEA has a high computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' In order Neuro-DynaStress: Predicting Dynamic Stress Distributions 3 to overcome this problem, some recent works have proposed deep neural network (DNN)-based methods to predict stress distributions in both intact and damaged structural components [1, 2], bypassing the need for static finite element analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' But these works are not suitable for dynamic finite element analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We propose an architecture that can act as a surrogate for FEA solvers for dynamic FEA while avoiding the computational bottlenecks involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' To demonstrate its utility, we model the stress distribution in gusset plates under dynamic loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Bridges and buildings rely heavily on gusset plates as one of their most critical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Gusset plates are designed to withstand lateral loads such as earthquakes and winds, which makes fast dynamic models valuable in avoiding catastrophic failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The main idea here is to train a model that can later be used when real- time estimations are needed, such as in the aftermath of extreme disruptive events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' For example, focusing on critical structural components, there is a need for immediate assessment following a disaster or during extremely disruptive events to guide corrective actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Engineers could rely on the proposed compu- tationally efficient algorithms to determine stress distributions over damaged gusset plates and apply the proper rehabilitation actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' They need to be able to analyze gusset plates quickly and accurately, which is what our model can provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' To our knowledge, this work is the first to predict dynamic stress distribution in the specific domain of steel plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 2 Related Work The most recent works in data-driven applications of scientific machine learning have included design and topology optimization [3, 4], data-driven approaches in fluid dynamics [5, 6], molecular dynamics simulation [7, 8], and material properties prediction [9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Atalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [13] and Levin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [14] have used neural regression for FEA model updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' More recently, DL has shown promise in solving traditional mechanics problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Some researchers used DL for structural damage detection, a promising alternative to conventional structural health monitoring methods [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Javadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [17] used a typical neural network in FEA as a surrogate for the traditional constitutive material model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' They simplified the geometry into a feature vector which approaches hard to generalize complicated cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The numerical quadrature of the element stiffness matrix in the FEA on a per- element basis was optimized by Oishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [18] using deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Their approach helps to accelerate the calculation of the element stiffness matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Convolutional Neural Networks (CNN) are commonly used in tasks involving 2D information due to the design of their architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Recently, Madani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [19] developed a CNN architecture for stress prediction of arterial walls in atherosclerosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Also, Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [20] proposed a CNN model for aortic wall stress prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Their method is expected to allow real-time stress analysis of human organs for a wide range of clinical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 4 Neuro-DynaStress: Predicting Dynamic Stress Distributions Gulgec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [21] proposed a CNN architecture to classify simulated dam- aged and intact samples and localize the damage in steel gusset plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Modares et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [22] conducted a study on composite materials to identify the presence and type of structural damage using CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Also, in order to detect concrete cracks without calculating the defect features, Cha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [23] proposed a vision- based method based on convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Do et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [24] proposed a method for forecasting the crack propagation in risk assessment of engineering structures based on “long short-term memory” and “multi-layer neural network”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' An approach for predicting stress distribution on all layers of non-uniform 3D parts was presented by Khadilkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' More recently, Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [26] developed a CNN-based method to predict the low-resolution stress field in a 2D linear cantilever beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [27] developed a conditional generative adversarial network for low-resolution von Mises stress distribution prediction in solid structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Some studies have been conducted to develop methods of predicting struc- tural response using ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [28] proposed a support vector machine approach to predict nonlinear structural responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [29] Uti- lized deep convolutional neural networks to estimate the structural dynamic responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Long short-term memory (LSTM) [30] was used by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [31] to predict nonlinear structural response under earthquake loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [32] proposed a deep-learning-based structural health monitoring (SHM) framework capable of predicting a dam’s structural dynamic responses once explosions are experienced using LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Kohar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [33] used 3D- CNN-autoencoder and LSTM to predict the force-displacement response and deformation of the mesh in vehicle crash-worthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Schwarzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [34] construct a neural network architecture that combines a graph convolutional neural network (GCN) with a recurrent neural network (RNN) to predict frac- ture propagation in brittle materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Lazzara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [35] proposed a dual-phase LSTM Auto-encoder-based surrogate model to predict aircraft dynamic land- ing response over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Jahanbakht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' [36] presented an FEA-inspired DNN using an attention transformer to predict the sediment distribution in the wide coral reef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The few models that studied stress predictions suffer from the problem of low-resolution predictions, making them unsuitable for decision-making after a catastrophic failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' To the best of our knowledge, this is the first work to predict dynamic stress distribution in the specific domain of steel plates with high accuracy and low latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The algorithm takes the geometry, boundary conditions, and time histories as input and renders the dynamic von Mises stress distribution as an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We modeled the steel plates as gusset plates with dynamic loading applied at different edges, different boundary conditions, and varying complex geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Neuro-DynaStress: Predicting Dynamic Stress Distributions 5 3 Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='1 Data Generation Two-dimensional steel plate structures with five edges, E1 to E5 denoting edges 1 to 5, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 2, are considered homogeneous and isotropic linear elastic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Various geometries are generated by changing the position of each node in horizontal and vertical directions, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 2, which led to 1024 unique pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The material properties remain unchanged, isotropic for all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The 2D steel plates approach the geometry of gusset plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Gusset plates connect beams and columns to braces in steel structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The behavior and analysis of these components are critical since various reports have observed failures of gusset plates subject to lateral loads [37–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The boundary conditions and time-history load cases are considered to simulate similar conditions in common gusset plate structures under external loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Some of the most common gusset plates configurations in practice are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 30 cm 15 cm 15 cm 5 cm 15 cm 5 cm E1 E3 E4 E5 E2 15 cm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 2: Basic schematic topology for initializing the steel plate geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Column Gusset plate Brace Beam (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 3: Some of the most common gusset plates in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' A total of 57,344 unique samples were created by combining 14 random time-history load cases and four most common boundary conditions in gusset plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Boundary conditions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 4, mimicking the real gusset plates’ boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' All the translation and rotational displacements 6 Neuro-DynaStress: Predicting Dynamic Stress Distributions were fixed at the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The range for width and height of the plates is from 30 cm to 60 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Each time history consists of 100 time steps generated with random sine and cosine frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The frequencies range between 1 and 3 HZ, with amplitudes ranging from 2 to 10 kN at intervals of 2 kN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' All time histories in horizontal and vertical directions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Considering 100 time steps, each interval is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='01 seconds, making the total time equal to 1 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' All the details for the input variables used to initialize the population are shown in Table 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Time (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Load (N) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 5: Time histories (a) Horizontal direction (b) Vertical direction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='2 Input Data The geometry is encoded as a 200 × 200 matrix and, incidentally, a binary image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 0 (black) and 1 (white) denote outside and inside of the geometry, as Neuro-DynaStress: Predicting Dynamic Stress Distributions 7 Table 1: Input variable Geometry Boundary conditions Load position Frequencies (HZ) Load (kN) Time steps Total time (s) pentagon E2 E4E5 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5,2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5,3 2,4,6,8,10 100 1 pentagon E2E3 E5 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5,2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5,3 2,4,6,8,10 100 1 pentagon E1E2 E4 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5,2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5,3 2,4,6,8,10 100 1 pentagon E3 E2E5 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5,2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5,3 2,4,6,8,10 100 1 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The boundary condition is also represented by another 200 × 200 pixel binary image, where the constrained edges are defined by 1 (white) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Moreover, each time step of time histories for horizontal and vertical components is encoded in the load position of the corresponding frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Load positions in each time step have values between 0 and 1, corresponding to each time step of time histories, and all remaining elements are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' All the load frames of each sample in horizontal and vertical directions are saved as tensors of dimension 100 × 200 × 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 6(c) and 6(d) show loads in the horizontal and vertical directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The colored load positions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 6(c) and 6(d) are used only for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Each row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 6 represents one of the simulated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Details of boundary conditions and their load positions are described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 6: Input and output representation for stress distribution prediction: (a) geometry, (b) boundary condition, (c) horizontal load, (d) vertical load, (e) output 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='3 Output Data FEA was performed using the Partial Differential Equation (PDE) solver in the MATLAB toolbox to obtain the stress distributions of each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We used transient-planestress function of MATLAB PDE solver to generate (b) (d) (e) (c) (a)8 Neuro-DynaStress: Predicting Dynamic Stress Distributions dynamic stress contours as the ground truth of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We defined the geometry, boundary condition, material properties and time histories as input and PDE solver returns the sequence of stress distributions corresponding to the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The MATLAB PDE toolbox mesh generator only generates unstructured triangulated meshes incompatible with CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The minimum and maximum triangulated mesh sizes are 5 and 10mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Since each element should be represented by one pixel in an image, we develop a 200×200 grid surface equal to the dimensions of the largest possible geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 7(a) and 7(b) show the unstructured mesh and the 200 × 200 grid surface on top of a random sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The stress values are then interpolated between the tri- angular elements and grids to determine a stress distribution compatible with our CNN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The stress values of all the elements outside the material geometry are assigned to zero, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 6(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 7: A sample of mesh generation: (a) unstructured triangular mesh, (b) structured gird surface The dimension of the largest sample is 600 × 600 mm, and the smallest is 300 × 300 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Using a mesh grid of 200 × 200 on top of samples made each element 3 × 3 mm, which means that each frame of output has 40000 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' This high-resolution dataset led to achieving significant accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The maximum and minimum von Mises stress values for elements among the entire dataset are 279,370 and -980 MPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We normalized all the output data between 0 and 1 to ensure faster convergence and encoded it to 200×200 for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='4 Stress Calculation The steps for linear finite element analysis’ stress calculation, which is part of phase (iii) of FEA’s workflow elaborated in the introduction section, are as follows: (a) (b)Neuro-DynaStress: Predicting Dynamic Stress Distributions 9 KQ = F (1) where K denotes a global stiffness matrix, F is the load vector applied at each node, and Q denotes the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' A stiffness matrix K consists of elemental stiffness matrices Ke: Ke = AeBT DB (2) where B represents strain-displacement matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' D represents stress-strain matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' and Ae represents area of element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Mesh geometry and material prop- erties determine B and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' This will be followed by adding the local stiffness matrix ke to the global stiffness matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The displacement boundary conditions are encoded using the corresponding rows and columns in the global stiffness matrix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Solving Q can be achieved using direct factorization or iterative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' As a result of calculating the global displacement using equation 1, we can calculate the nodal displacements q then we can calculate the stress tensors of each element as follows: σ = DBq (3) where σ specifies the tensor of an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The 2-D von Mises Stress criterion is then used to calculate each element’s von Mises Stress: σvm = � σ2x + σ2y − σxσy + 3τ 2xy (4) where σvm denotes von Mises Stress, σx, σy are the normal stress compo- nents and τxy is the shear stress component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 4 Proposed Methodology We use convolutional layers to encode the spatial information from the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Our hypothesis is that these layers will combine the information in geome- try, boundary conditions, and load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' A key characteristic of dynamic structural systems is the temporal dependence of their states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' LSTM is a suitable architec- ture for modeling temporal information in sequence and hence is a good choice to model structural dynamic systems in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' For high-quality 2D reconstructions, we use transposed convolutional layers in our decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' For further improving training and performance, we use modules from the recently proposed feature-aligned pyramid networks (FaPN) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' FaPN allows the decoder to access information from the encoder directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Overall, our net- work architecture consists of four modules: encoder consisting of convolutional layers, temporal module made using LSTM modules, decoder consisting of transposed convolutional layers, and alignment modules acting as connections between encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The number of layers in each module and the number of layers in LSTM modules were chosen based on their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 10 Neuro-DynaStress: Predicting Dynamic Stress Distributions The architecture is illustrated schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The size of layers and hyper-parameters used in the network are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' LSTM Feature-aligned Pyramid Network (FaPN) Elementwise Addition Conv Layer Transposed Conv Layer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 8: Architecture for the proposed Neuro-DynaStress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The convolutional encoder maps the raw input data to a latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' LSTM layers processes the information across different time frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The final output is obtained from the resulting latent representation using transposed convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Table 2: Network layers and hyper-parameters Type of layers Number of layers First layer (H×W×C) Last layer (H×W×C) Conv 6 200×200×16 7×7×512 LSTM 4 1×1×512 1×1×512 ConvT 5 13×13×256 200×200×16 FaPN 4 13×13×256 100×100×32 Batch size Learning rate Weight decay Loss function 8 10−4 10−5 MAE 5 Loss Function and Performance Metrics We use Mean Absolute Error (MAE), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 5 as the primary training loss and metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' To ensure that we do not overfit to a single metric, we also use Mean Relative Percentage Error (MRPE) to evaluate the overall quality of predicted stress distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' MAE = 1 NT n,t � N,T |S(n, t) − ˆS(n, t)| (5) MRPE = MAE max |S(n, t), ˆS(n, t)| × 100 (6) Neuro-DynaStress: Predicting Dynamic Stress Distributions 11 where S(n, t) is the true stress value at a node n at time step t, as computed by FEA, and ˆS(n, t) is the corresponding stress value predicted by our model, N is the total number of mesh nodes in each frame of a sample, and T is a total number of time steps in each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' As mentioned earlier, we set T = 100 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 6 Implementation and Computational Performance We implemented our model using PyTorch [42] and PyTorch Lightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' AdamW optimizer [43] was used as the optimizer with a learning rate of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We found that a batch size of 8 gave the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The computational per- formance of the model was evaluated on an AMD EPYC 7313 16-core processor and two NVIDIA A6000 48G GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The time required during the training phase for a single sample with 100 frames and a batch size of 8 was 10 sec- onds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' In the training phase, one forward and backward pass was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The inference time for one sample was less than 5 ms which can be consid- ered a real-time requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The most powerful FE solvers take between 10 minutes to an hour to solve the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Therefore, Neuro-DynaStress is about 72 × 104 times faster than conventional FE solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We consider the minimum time for all processes of modeling geometry, meshing, and analysis of one sam- ple in FE solver to be about 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' MATLAB PDE solver does not use GPU acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' This demonstrates that our proposed approach can achieve the real-time requirement during the validating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 7 Results and Discussions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='1 Quantitative Evaluation Our model is trained on the training dataset for 45 epochs and evaluated on the validation dataset using separate metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The training dataset consisted of 48,755, while the validating dataset contained 8,589 samples, together form- ing the 80%-20% split of the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The model predicts five frames of output from a sequence of five previous inputs until all 100 frames are pre- dicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The best validation performance was obtained when we sequenced five frames during validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The best checkpoint during validation, at epoch 40, is the basis for all error metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' MRPE for the validating dataset is just 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='2 Qualitative Evaluation The prediction results for a few randomly selected samples from the vali- dation dataset are visualized in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 9a and 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The first row represents 5 frames out of 100 frames of one reference sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The second row illustrates the prediction corresponding to the frames in the first row, and the last row represents the error in the corresponding predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The columns represent the time steps 1, 25, 50, 75 and 100 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We visualized frames at intervals 12 Neuro-DynaStress: Predicting Dynamic Stress Distributions of 25 seconds to evaluate different ranges of dynamic stress prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Reference t=1 t=25 t=50 t=75 t=100 Predict Error 0 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 (a) Reference t=1 t=25 t=50 t=75 t=100 Predict Error 0 5000 10000 15000 20000 25000 5000 10000 15000 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 9: Successful predicted dynamic stress distribution and their correspond- ing errors in different time sequences for two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The top row corresponds to reference frames and the middle row shows the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The bottom row shows the absolute error between corresponding frames (Unit = MPa) Neuro-DynaStress: Predicting Dynamic Stress Distributions 13 For visualization purposes, the references and predictions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 9a and 9b are scaled to the same range using the maximum and the minimum of each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The errors are scaled independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' As it can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 9a, the predicted frames are quite similar to their corresponding references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Although the geometry contains sharp corners and edges, which are areas that are hard for CNN to reconstruct, our model is able to predict it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The errors, except for a small part of the first frame, are in an acceptable range which shows the pre- diction accuracy of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 9b shows another successful reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Comparing references with their corresponding predicted frames demonstrates that our Neuro-DynaStress model can capture both load variations and max- imum stress values at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Furthermore, these results demonstrate that our model is able to predict a dynamic stress distribution with a high variation of distributed stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='10 shows a random failure sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' In spite of the model’s success in predicting most parts of the frames, it is not able to reconstruct high-stress concentrations at angles of 90 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Since CNNs typically struggle in handling sharp edges, smoothening the sharp corners using Gaussian filters during data preprocessing may help the network to train better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Furthermore, as the loads in frames t = 25 and t = 75 are lower than in other frames, the prediction in those frames is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Reference t=1 t=25 t=50 t=75 t=100 Predict Error 0 10000 20000 30000 40000 50000 60000 5000 10000 15000 20000 25000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 10: Failed predicted dynamic stress distribution and their corresponding errors in different time sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' (Unit = MPa) It is also important that the predictions are temporally consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' In order to qualitatively demonstrate the temporal consistency of the proposed method, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 11a shows a comparison of stress values across 100 frames for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='14 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Time (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Stress (MPa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Reference ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Time (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Stress (MPa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Reference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Predict ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 11: Comparison of stress values across 100 frames for predictions, ref- erences, and errors in a randomly selected element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' (a) Successful predictions (b) Unsuccessful predictions (Units = MPa-T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' successful predictions in a randomly selected element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' As can be seen, the references and the predicted distributions are almost identical in most time sequences, with errors close to zero, despite the stress varying widely with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 11a illustrates how prediction fits with reference more closely when there is more temporal smoothness at peak points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' For instance, a good match between prediction and reference can be seen in the rightmost graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 11a, where the stress variation follows a smooth Gaussian distribution in the last peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' However, in the remaining graphs, the prediction has good correlation with the reference despite a lack of smoothness in most peak stress values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Moreover, based on the graphs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 11a, we can conclude that the model is better at predicting stress in valleys compared to peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We have also illustrated some of the unsuccessful predictions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 11b to identify the limitations of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' It can be seen that in all graphs with non-Gaussian stress distributions, the model finds it difficult to capture the peak stress values accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' However, in the first two graphs from the Neuro-DynaStress: Predicting Dynamic Stress Distributions 15 (a) (b) (c) (d) (a) (b) (c) (d) (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 12: Relative errors across 100 frames in the randomly selected sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Graphs in the center represent the MRPE per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' (a) and (c) in each figure represent the reference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' (b) and (d) refer to their corresponding predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Arrows refer to the MRPE of the presented frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' (Units = MPa-T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' left in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 11b, the predictions perfectly fit later peaks of the reference since the stress values in the reference have Gaussian distributions at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 12a and 12b depict the MRPE of randomly selected samples across 100 frames and frames corresponding to the minimum and maximum MRPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' As can be seen for both samples, the minimum errors are around zero, with only a few frames exceeding the error by more than 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='3 Ablation Study The efficiency of architecture can be attributed to several design choices we have made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Our architecture models the temporal dependency between time frames and the relationship between different elements in an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Even 16 Neuro-DynaStress: Predicting Dynamic Stress Distributions though self-attention has shown state-of-the-art performance in sequence mod- eling, they are not suitable for tasks without large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Hence, we use LSTMs for sequence modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' To demonstrate our claim, we compare our architecture against other baseline architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We compare against three architectures as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The model with multi head self-attention is very similar to our architecture, except the LSTM modules in our model are replaced with self-attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The details of the other models are represented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' We will refer to our architecture as Neuro-DynaStress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The results are shown in Table 3, and the best results are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Table 3: Architecture comparison Architecture for modeling temporal information Multi-headed self-attention LSTM LSTM LSTM FaPN ✓ ✓ ✓ × Skip connection ✓ ✓ × × MRPE(%) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='7 8 Conclusion We propose Neuro-DynaStress model equipped with Convolutional Neural Network (CNN) and Long Short Term Memory (LSTM) to predict the entire sequence of dynamic stress distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The model was designed and trained to use the geometry, boundary conditions and the sequence of loads as input and predicts the sequence of high-resolution dynamic stress contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The convolu- tional components are used to extract spatial features and the LSTM captures the temporal dependence between the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Feature alignment modules are used to improve the training and performance of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' The model is trained using synthetic data generated using the PDE toolbox in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Neuro-DynaStress can predict dynamic stress distribution with a mean rela- tive percentage error of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content='3%, which is considered an acceptable error rate in engineering communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Declarations This research was funded in part by the National Science Foundation grant CNS 1645783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' There is no conflict of interest among the authors of this paper The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=' Neuro-DynaStress: Predicting Dynamic Stress Distributions 17 References [1] Bolandi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=', Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=', Salem, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=', Boddeti, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=', Lajnef, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} +page_content=': Bridging finite element and deep learning: High-resolution stress distribution prediction in structural components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfsgGN/content/2301.02580v1.pdf'} 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Error-bounded lossy compression has been considered one of the +most effective solutions to the above problem. However, little work has been done to improve error-bounded lossy compression for +Adaptive Mesh Refinement (AMR) simulation data. Unlike the previous work that only leverages 1D compression, in this work, we +propose an approach (TAC) to leverage high-dimensional SZ compression for each refinement level of AMR data. To remove the data +redundancy across different levels, we propose several pre-process strategies and adaptively use them based on the data +characteristics. We further optimize TAC to TAC+ by improving the lossless encoding stage of SZ compression to efficiently handle +many small AMR data blocks after the pre-processing. Experiments on 8 AMR datasets from a real-world large-scale AMR simulation +demonstrate that TAC+ can improve the compression ratio by up to 4.9× under the same data distortion, compared to the +state-of-the-art method. In addition, we leverage the flexibility of our approach to tune the error bound for each level, which achieves +much lower data distortion on two application-specific metrics. +! +1 +INTRODUCTION +T +He increase in supercomputer performance over the +past decades has been insufficient to solve many chal- +lenging modeling and simulation problems. For example, +the complexity of solving evolutionary partial differential +equations scales as Ω(n4), where n is the number of mesh +points per dimension. Thus, the performance improvement +of about three orders of magnitudes over the past 30 years +has meant just a 5.6× gain in spatio-temporal resolution [1]. +To address this issue, many high-performance computing +(HPC) simulation packages [2] (such as AMReX [3] and +Athena++ [4]) use Adaptive Mesh Refinement (AMR)— +which applies computation to selective regions of most +interest—to increase resolution. Compared to the method +where a high resolution is applied everywhere, the AMR +method greatly reduces the computational complexity and +storage overhead; thus, it is one of the most widely used +frameworks for many HPC applications [5], [6], [7], [8]. +Although AMR can save storage space to some extent, +AMR applications running on supercomputers still generate +large amounts of data, bringing challenges to data transmis- +sion and storage. For example, one Nyx simulation [9] with a +resolution of 40963 (i.e., 0.5×20483 mesh points in the coarse +level and 0.5×40963 in the fine level ) can generate up to 1.8 +TB of data for a single snapshot; a total of 1.8 PB of disk stor- +age is needed assuming running the simulation 5 times with +200 snapshots dumped per simulation. Therefore, reducing +data size is necessary to lower the storage overhead and I/O +cost and improve the overall application performance for +running large-scale AMR simulations on supercomputers. +• +Daoce Wang, Sian Jin, Jiannan Tian, and Dingwen Tao (corresponding +author) are with Indiana University, Bloomington, IN 47405, USA. +• +Jesus Pulido, Pascal Grosset, and James Arhes are with Los Alamos Na- +tional Laboratory, Los Alamos, NM 87545, USA. +• +Kai Zhao is with University of Alabama at Birmingham, AL 35294, USA. +A straightforward way to address this issue is to use +data compression. However, traditional lossless compres- +sion techniques such as GZIP [10] and Zstandard [11] can +only provide a compression ratio by up to 2× for scientific +data [12]. On the other hand, a new generation of lossy +compressors that can provide strict error control (called +“error-bounded” lossy compression) has been developed, +such as SZ [13], [14], [15], ZFP [16], MGARD [17], and +TTHRESH [18]. Using those error-bounded lossy compres- +sors, scientists can achieve relatively high compression ratios +while minimizing the quality loss of reconstructed data and +post-analysis, as demonstrated in [19], [20], [21], [22], [23], +[24], [25], [26]. +Only a few existing contributions have investigated +error-bounded lossy compression for AMR applications +and datasets. A common approach is to generate uniform- +resolution data by up-sampling the coarse-level data and +merging them with the finest-level data and then performing +compression on the merged data. However, this approach +introduces redundant information to the data, which signif- +icantly degrades the compression ratio, especially when the +up-sampling rate is high or there are multiple coarse levels +to up-sample. +Recently, Luo et al. introduced zMesh [27], a technique +that groups data points that are mapped to the same or adja- +cent geometric coordinates such that the dataset is smoother +and more compressible. However, since zMesh maps data +points from different AMR levels to adjacent geometric co- +ordinates and generates a 1D array, it cannot adopt 3D com- +pression which most HPC simulations use. Moreover, zMesh +is designed for patch-based AMR data1 with redundancy +across different AMR levels to improve the compression +1. The patch-based AMR data redundantly saves the data block to +be refined at the next finer level in the current coarse level (will be +introduced in detail in Section 2.3). +arXiv:2301.01901v1 [cs.DC] 5 Jan 2023 + +ratio. However, these coarser levels of redundant data are of- +ten not used for post-analysis or visualization and hence can +be directly discarded to improve the compression ratio. For +this case, the reorganization approach proposed by zMesh +cannot improve the data smoothness appropriately (will be +illustrated in Section 4). +To solve these issues, we propose TAC that removes the +redundant data in coarser level(s) and employs 3D lossy +compression for each level. We note that each level may +contain many empty/zero regions, where data points are +saved in other levels, which may significantly decrease the +data smoothness and hence reduce the compression ratio. +To this end, TAC either removes these empty regions us- +ing adaptive partition strategy or partially pads them with +appropriate values, based on the density of empty regions. +TAC also has an optimization to reduce the time complexity +of removing empty regions. +Although our partition strategies can remove empty re- +gions, there are still several challenges. Specifically, the par- +tition strategies can generate many (e.g., 3,000+) small data +blocks with totally different shapes (e.g., 10 different shapes), +whereas the SZ compressor performs poorly on relatively +small data sets. This is because the SZ compressor uses thou- +sands of Huffman trees to encode these small blocks sepa- +rately, leading to a low encoding efficiency. A naive solution +to the heavy Huffman encoding cost is to linearize/merge +the thousands of small blocks into fewer larger blocks and +then pass these larger blocks to SZ. This approach can reduce +the overhead of the Huffman trees for encoding and hence +increase the amount of data encoded together. +However, TAC still has two limitations: (1) Most of the +merged small blocks are not adjacent in the original dataset, +leading to rapid changes in the data values at the boundaries +of these non-neighboring blocks, which impacts the accuracy +of SZ’s predictor negatively. (2) Although we can compress +data blocks with the same shape together, the SZ compressor +must be called multiple times for each shape, resulting in +the inevitable low performance of Huffman encoding. To +address these limitations, we further optimize TAC to TAC+ +by designing a Shared Huffman Encoding (SHE) approach +for the SZ compressor. This approach allows individual pre- +dictions for each small block while being encoded using a +single shared Huffman tree, which can improve the predic- +tion accuracy and compression ratio accordingly. +The main contributions are summarized as follows. +• We propose to leverage 3D SZ compression to compress +each level of an AMR dataset separately. We propose +a hybrid compression approach based on the following +three pre-process strategies and data characteristics. +• We propose an optimized sparse tensor representation +to efficiently partition data and remove empty regions +for sparse AMR data. +• We propose an enhanced k-d tree approach to reduce +the time overhead of removing empty regions. +• We propose a padding approach to improve the smooth- +ness and compressibility of dense AMR data. +• We employ the SHE approach in the SZ compressor to +reduce the high time and storage costs of compressing +multiple small blocks after the partition. +• We tune the error bound for each AMR level to fur- +ther improve the compression quality in terms of two +application-specific post-analysis metrics. +• Experiments show that, compared to the state-of-the-art +approach zMesh, our proposed AMR compression can +improve the compression ratio by up to 4.9× under the +same data distortion on the tested datasets. +We evaluate our proposed compression method on eight +datasets from three real-world AMR simulation runs. The +AMR simulations are well-known, open-source cosmology +simulations—Nyx [9]. We compare our method with four +baselines including zMesh using generic metrics such as +compression ratio and peak signal-to-noise ratio (PSNR) and +application-specific metrics such as power spectrum and +halo finder. Our code is available at https://github.com/ +FabioGrosso/3dAMRcomp. +In Section 2, we present background information about +error-bounded lossy compression, AMR method, k-d tree, +and related work on AMR data compression. In Section 3, +we describe our proposed pre-process strategies, SHE ap- +proach, and hybrid compression. In Section 4, we show the +experimental results on different AMR datasets. In Section 5, +we conclude our work and discuss the future work. +2 +BACKGROUND AND RELATED WORK +2.1 +Lossy Compression for Scientific Data +There are two main categories for data compression: lossless +and lossy compression. Compared to lossless compression, +lossy compression can offer a much higher compression ra- +tio by trading a little bit of accuracy. There are some well- +developed lossy compressors for images and videos such +as JPEG [28] and MPEG [29], but they do not have a good +performance on the scientific data because they are mainly +designed for integers rather than floating points. +In recent years there is a new generation of lossy com- +pressors that are designed for scientific data, such as SZ [13], +[14], [15], ZFP [16], MGARD [17], and TTHRESH [18]. These +lossy compressors provide parameters that allow users to +finely control the information loss introduced by lossy +compression. Unlike traditional lossy compressors such as +JPEG [28] for images (in integers), SZ, ZFP, MGARD, and +TTHRESH are designed to compress floating-point data and +can provide a strict error-controlling scheme based on the +user’s requirements. Generally, lossy compressors provide +multiple compression modes, such as error-bounding mode +and fixed-rate mode. Error-bounding mode requires users +to set an error type, such as the point-wise absolute error +bound and point-wise relative error bound, and an error +bound level (e.g., 10−3). The compressor ensures that the +differences between the original data and the reconstructed +data do not exceed the user-set error bound level. +In this work, we focus on the SZ lossy compression +(2021 R&D 100 Award Winner [30]) because SZ typically +provides a higher compression ratio than ZFP [22], [31] +and higher (de)compression speeds than MGARD [31], [32] +and TTHRESH [18]. SZ is a prediction-based error-bounded +lossy compressor for scientific data. It has three main steps: +(1) predict each data point’s value based on its neighboring +points by using an adaptive, best-fit prediction method; (2) +quantize the difference between the real value and predicted +value based on the user-set error bound; and (3) apply a +customized Huffman coding and lossless compression. +2 + +Fig. 1: Visualization (one zoom-in 2D slice) of three key timesteps gen- +erated from an AMR-based cosmology simulation. The grid structure +changes with the universe’s evolution. The red boxes indicate different +resolutions within one AMR level. +2.2 +AMR Method and AMR data +AMR is a method of adapting the accuracy of a solution +by using a non-uniform grid to increase computational and +storage savings while still achieving the desired accuracy. +AMR applications change the mesh or spatial resolution +based on the level of refinement needed by the simulation +and use finer mesh in the regions with more importance/interest +and coarser mesh in the regions with less importance/interest. +Figure 1 shows that during an AMR run, the mesh will be +refined when the value meets the refinement criteria, e.g., +refining a block when its norm of the gradients or maximum +value is larger than a threshold. +Fig. 2: A typical example of AMR data storage and usage. +Clearly, the data generated by an AMR application are +hierarchical data with different resolutions. The data of each +AMR level are usually stored separately (e.g., in a 1D array). +For example, Figure 2 (left) shows a simple example of two- +level AMR data; “0” means high resolution (the fine level) +and “1” for low resolution (the coarse level). When the AMR +data are needed for post analysis or visualization, users will +typically convert the data from different levels to a uniform +resolution. In the previous example, we will up-sample the +data in the coarse level and combine it with the data in the +fine level, as shown in Figure 2 (right). +2.3 +Tree-based and Patch-based AMR Data +There are two types of techniques to represent AMR data: +patch-based AMR and tree-based AMR [33]. The main dif- +ference between them is that the patch-based AMR tech- +nique generates AMR data with redundancy across different +levels. In other words, the patch-based AMR data structure +redundantly saves data blocks to be refined at the next level +in the current level, simplifying the computation in the re- +finement process. By comparison, the tree-based AMR tech- +nique organizes the grids on the tree leaves, so there is no +redundant data across different levels. But tree-based AMR +data is more complex for post analysis and visualization +compared to patch-based AMR data [34]. +In this work, we focus on a state-of-the-art patch-based +AMR framework AMReX. Note that since the redundant +coarser-level data in the patch-based AMR will not often be +used in post-analysis, we discard them during compression +to improve the compression ratio. +2.4 +Existing AMR Data Compression +1D AMR Compression: The main challenge for AMR data +compression is that the AMR data is comprehensive and +hierarchical with different resolutions. A naive approach is +to compress the 1D data of each AMR level separately. How- +ever, this approach loses most of the topological/spatial in- +formation, which is critical for data compression. zMesh [27] +is a state-of-the-art AMR data compression based on the 1D +approach. Different from the naive 1D approach, zMesh re- +organizes the 1D data based on each point’s coordinate in +the 2D layout; in other words, zMesh puts the points neigh- +bored in the 2D layout closer in the 1D array. It can increase +the data smoothness/compressibility to benefit the follow- +ing 1D compression such as SZ on patch-based AMR data +with redundancy across different AMR levels. However, +zMesh does not leverage high-dimensional compression, +while many previous studies [14], [35] proved that lever- +aging more dimensional information (e.g., spatial/temporal +information) can significantly improve the compression per- +formance. Moreover, it only focuses on 2D AMR data. Our +work aims to leverage high-dimensional data compression +and supports 3D AMR data. +High-dimensional AMR Compression: Similar to the idea +described in Section 2.2, a straightforward way to leverage +3D compression on 3D AMR data is to compress different +levels together by up-sampling coarse levels. However, this +approach must handle extra redundant data generated by +the up-sampling process. As shown in Figure 2, 1A, 1B, and +1C are redundant points in the compression. Note that the +storage overhead of these redundant points will be higher +when more data are in the coarse levels or the up-sampling +rate is higher, especially for 3D AMR data. This is because we +only need to duplicate one point from the coarse level 4 times +for 2D AMR data but 8 times for 3D AMR data, with an up- +sampling rate of 2. Another limitation of this approach is that +it cannot apply different compression configurations (e.g., +error bound) to different AMR levels. This is because after +up-sampling all data points will have the same importance. +However, the purpose of using the AMR method is to set +different interests to different AMR levels, so the error bound +for each AMR level can be chosen adaptively. +2.5 +k-D Tree for Particle Data Compression +k-d tree [36] is a binary tree in which every node represents a +certain space. Without loss of generality, for the 3D case, ev- +ery non-leaf node in a k-d tree splits the space into two parts +by a 2D plane associated with one of the three dimensions. +The left subspace is associated with the left child of the node, +while the right subspace is associated with the right child. k- +d tree is commonly used in particle data compression [37], +[38], [39] to locate each particle and remove empty regions. +Specifically, a k-d tree keeps dividing the space in between +3 + +口1B +1B +Ivl 0.bin (0A,0B,0C,0D) +A +1A +1B +1B +1B +Ivl 1.bin(1A,1B,1C) +1C +0A +0B +1C +1C +OA +0B +OC +OD +1C +1C +oC +OD(a) z10 fine level +(b) z10 coarse level +Fig. 3: Visualization of data distributions of an example AMR data +“z10”, where z = redshift. Non-empty regions are shown in red. +along one dimension until the space is empty or contains +only one particle. We will optimize the classic k-d tree and +use it to remove empty regions and increase the compress- +ibility for each AMR level (to be detailed in Section 3.3). +3 +OUR PROPOSED DESIGN +In this section, we propose a pre-processing approach for +AMR data to leverage high-dimensional SZ lossy compres- +sion at each AMR level. Specifically, we propose three pre- +process strategies to mitigate the issue of irregular data +distribution. We further propose Shared Huffman Encoding +(SHE) and integrate it into the SZ compressor to further +improve the compression performance for AMR data. We +also propose an adaptive approach to select the best-fit pre- +processing strategy based on the data density of each level. +3.1 +Ghost-Shell Padding for High-density Data +To compress the AMR data in 3D, besides the aforemen- +tioned 3D baseline, we can also compress each level sepa- +rately in 3D. In that way, however, the data will be split into +multiple levels, and each level will have many empty regions +and an irregular data distribution, as shown in Figure 3. +A naive solution to handle the irregular 3D data is to fill +the empty regions with zeros and pass a large 3D block to +the compressor. Although the padded zeros will increase +the size of data for compression, for high-density data such +as z10’s coarse level shown in Figure 3b (i.e., about 77% +density), the size overhead will be small. +However, these padded zeros can also greatly reduce +the performance of compression, especially for prediction- +based lossy compression such as SZ, because these zeros can +significantly affect the prediction accuracy of SZ, resulting +in high compression errors on the boundaries, as shown in +Figure 5a. More specifically, as mentioned in Section 3.2, SZ +uses each point’s neighboring points’ values to predict its +value. Thus, for those boundary points which are adjacent +to padded zeros, SZ will involve zero(s) in the prediction, +while the actual values of these empty regions are typically +non-zeros (saved in other AMR levels), which will seriously +mislead the prediction. +To eliminate the above issue of padding zeroes, we pro- +pose to use a ghost-shell padding strategy (GSP) to diffuse +neighboring values to a padding layer. Figure 4 illustrates +the high-level idea, and the detailed algorithm is described +in Algorithm 1. Specifically, we first partition the data into +Fig. 4: A 2D example of GSP approach. Non-empty blocks are in navy +blue; padded blocks are in light blue/red; padded blocks based on more +than one non-empty neighbor are in red. +Algorithm 1: Proposed Ghost Shell Padding Method +Input: Data, x, y +Output: Data after padding +1 for each unit block bi do +2 +if bi is empty and bi has non-empty neighbor then +3 +for each non-empty neighbor nj do +4 +pad slice = avg (first y slices of nj next +to bi); +5 +if overlap edge then +6 +pad = pad/2; +7 +else if overlap corner then +8 +pad = pad/3; +9 +else +10 +continue; +11 +end +12 +add an x-layers pad slice to bi next to +nj; +13 +end +14 +end +15 end +16 return padded Data +unit blocks and then pad each empty unit block by using the +average of its non-empty neighbors’ boundary data values. +Note that some empty unit blocks have more than one +non-empty neighbor such as the red box shown in Figure 4. +For these blocks, we will use the average value of all its +neighbors for padding. Correspondingly, we will remove +these padded values in the decompression based on the +saved padding information. Note that since the padding pro- +cess is only for non-empty blocks, this metadata overhead is +almost negligible for high-density data (e.g., 0.1%). +(a) ZF (CR=156.7, PSNR=32.8dB) +(b) GSP (CR=161.3, PSNR=33.5dB) +Fig. 5: Visual comparison (one slice) of compression errors of two ap- +proaches using SZ based on Nyx’s “baryon density” field (i.e., z10’s +coarse level, 77% density). Brighter means higher compression error. +The error bound is the relative error bound of 6.7 × 10−3. +4 + +口After padding, each boundary point will be predicted us- +ing the average of all the boundary data in the unit block(s) +to which it belongs or is neighbored. As shown in Figure 5, +compared to the zero-filling (ZF) approach, GSP can signif- +icantly reduce the overall compression error, especially for +the boundary data. Moreover, the GSP approach can provide +a similar compression ratio to the ZF approach on this high- +density data and hence a better rate-distortion. A detailed +evaluation will be presented in Section 4. +3.2 +Optimized Sparse Tensor Representation for Low- +density Data +When most of the regions in the data are empty (e.g., about +77% of the data is empty in Figure 3a), the large amount +of padded data would greatly increase the size of data for +compression, resulting in a low compression ratio. +To solve this issue, we propose to use a naive sparse- +tensor-based approach (called NaST) to remove the empty +regions, as shown in Figure 6. NaST includes four main +steps in the compression process: (1) partition the 3D data +into multiple unit blocks, (2) remove the empty blocks, (3) +linearize the remaining 3D blocks into a 4D array, and (4) +pass the 4D array to the compressor. Note that in the de- +compression process, we will put the unit blocks from the +decompressed 4D array back into the original data. +Fig. 6: Workflow of the naive sparse tensor (NaST) method (empty +regions marked in pink and non-empty regions marked in blue). +However, in order to completely remove the empty re- +gions to form a sparse representation, the unit block size +needs to be relatively small compared to the input data size +(e.g., 163 vs. 5123), resulting in a high proportion of data on +the boundary. While the linearized unit blocks are usually +not adjacent in the original data, so the boundary data be- +tween them are not smooth. Thus, it is harder for prediction- +based compressors such as SZ to predict the boundary data +values. As a result, the NaST method without optimizing the +boundary data would have low compression performance. +To address the above problem, we propose an optimized +sparse tensor representation (called OpST) to effectively re- +move the empty regions as well as maintain a relatively +large unit block size so as to reduce the portion of boundary +data. A detailed description of our algorithm can be found +in Algorithm 2. We use a 2D example to demonstrate our +approach, as illustrated in Figure 7. Specifically, (1) we par- +tition the data into many small unit blocks. (2) For each unit +block, we use the dynamic programming method to initiate +an array BS to save the dimension/size of the maximum +square whose bottom-right corner is that unit block (line +6, which will be discussed in the next paragraph). (3) We +extract the sub-blocks (composed of multiple unit blocks) +from the original data according to the sizes saved in BS +(lines 13). (4) Since the original data will be changed after the +Fig. 7: A 2D example of our proposed OpST approach. The sub-blocks +are extracted according to our optimized sizes saved in BS. E.g., a 2-by- +2 sub-block B0 is extracted according to BS1[2][1]. +extraction, we need to partially update BS based on maxSide +(lines 14, will be discussed later). We loop (3) and (4) from the +bottom-right corner to the top-left corner until the original +data is empty. (5) After extracting all the sub-blocks, we put +them into multiple 3D arrays (to be compressed) based on +their sizes. Note that the sub-blocks with the same size will +be merged into the same array for easy compression. +Algorithm 2: Proposed Optimized Sparse Tensor Method +Input: Sparse 3D data S +Output: multiple 4D array Dn +1 for each unit block b(x, y, z) do +2 +if b(x, y, z) is non-empty then +3 +if x is 0 or y is 0 or z is 0 then +4 +BS(x, y, z) = 1 +5 +else +6 +BS(x, y, z) = min(BS(x − 1, y, z), BS(x, y − +1, z), BS(x, y, z − 1), BS(x − 1, y − +1, z), BS(x, y − 1, z − 1), BS(x − 1, y, z − +1), BS(x − 1, y − 1, z − 1)) + 1 ; +/* BS(x,y,z) is the dimension size of the +maximum cube whose bottom right rear +corner is the unit block with index +(x,y,z) in the original data */ +7 +maxSide = max(maxSide, BS(x, y, z)) +8 +end +9 +end +10 end +11 for each unit block b(x, y, z) do +12 +if BS(x, y, z) ≥ 1 then +13 +size = BS(x, y, z) +Dsize ← S((x − size : x) ∗ blkSize, (y − size : +y) ∗ blkSize, (z − size : z) ∗ blkSize) ; +/* put +the sub-block to the according to 4D array */ +14 +b(x − size : x, y − size : y, z − size : z) ← empty +BS(x − size : x, y − size : y, z − size : z) = 0 +BS = updateBs(BS, x, y, z, maxSide) +15 +end +16 end +17 return Dn +When initializing the BS in step (2), we start with the +b′[i][j] with i = 0 or j = 0 (i.e., on the top-left edge), +where b′[·][·] are the unit blocks: if b′[i][j] is empty, we will +set BS[i][j] to 0 otherwise 1. For the remaining unit blocks, if +it is empty, BS[i][j] will be 0; otherwise, BS[i][j] will be set +to 1 plus the minimum value among its three neighboring +blocks (i.e., upper block, left block, and upper-left block). In +other words, we have BS[i][j] = 1+min(BS[i][j−1], BS[i− +1][j], BS[i−1][j−1]) for the 2D case. For example, BS1[2][1] +is 2 because all its upper-left neighbors are 1 (as shown in +Figure 7). However, both BS1[1][1] and BS2[1][2] can only +reach 1 because one of their neighbors is set to 0, having no +chance to form a sub-block with the size of 2. +Moreover, as mentioned in step (3), we need to update +5 + +Bo +0 +1 +1 +0 +1 +0 +0 +1 +1 +1 +2 +0 +0 +1 +0 +0 +0 +1 +2 +0 +0 +0 +0 +0 +0 +0 +BS2 +BS3 +BS.(a) NaST(CR=245, PSNR=77.5dB) +(b) OpST(CR=248, PSNR=78.0dB) +Fig. 8: Visual comparison (one slice) of compression errors of two ap- +proaches using SZ based on Nyx’s “baryon density” field (i.e., z10’s fine +level, 23% density). Brighter means higher compression error. The error +bound is the relative error bound of 7.2 × 10−4. +BS after each extraction. Specifically, for each sub-block we +extract, we have to set its corresponding values in BS to +zeros. For instance, as shown in Figure 7, after we extract +a 2-by-2 sub-block B0 at BS1[2][1], we need to set BS2[1][0], +BS2[1][1], BS2[2][0], and BS2[2][1] to zeros. In addition, we +also need to recalculate a part of BS (line 17 in Algorithm 2) +because the extraction could influence other BS values. For +example, we need to recalculate BS2[1][2] (marked in bold +orange) after extracting B0. Note that this update is a partial +update as the BS values to be updated will be bounded by +maxSide which is the dimension size of the largest cube in the +dataset (line 7). +Similar to NaST, in decompression, we will put the sub- +blocks back to reconstruct the data based on the saved co- +ordinates. Note that after our optimization, each sub-block +size will be relatively large (e.g., 963 vs the original data size +of 5123), the overhead of saving the coordinates of all the +sub-blocks will be negligible (e.g., 0.1%). +Finally, we show a visual comparison of the compression +quality between NaST and OpST in Figure 8. Note that both +use SZ with the same error bound. Brighter means more +errors. We can observe that compared to the NaST method, +OpST can significantly reduce the overall compression error, +especially for the data points on the boundary. It is worth +noting that even with a lower error, our OpST can still pro- +vide a higher compression ratio than NaST. This is because +our proposed optimization will generate larger sub-blocks, +which provide more information for prediction-based lossy +compressors such as SZ to achieve better rate-distortion. A +detailed evaluation will be shown in Section 4. +3.3 +Adaptive k-D Tree for Medium-density Data +The OpST approach proposed for low-density data, how- +ever, has a high computation overhead, especially when the +data is relatively dense. This is because, on one hand, OpST +needs to update BS based on maxSide for each extraction of +a sub-block, while the larger the maxSide, the more values in +BS that need to be updated; on the other hand, maxSide is the +dimension size of the largest non-empty cube in the dataset, +Fig. 9: 2D example of adaptive k-D tree. Sub-block will be adaptively +split to effectively remove empty regions and get bigger full sub-blocks. +which is highly related to the density of the dataset. Thus, +the time complexity of OpST can be expressed as O(N 2 · d), +where N is the unit block number and d is the density. Note +that here density describes how dense the data is. For exam- +ple, the density of 77% means that 23% of the data is empty. +Clearly, when the density of an AMR level is relatively high, +using OpST will be relatively time-consuming. +Algorithm 3: Dynamic k-D Tree +Input: data block d, counts information +Output: k-d tree +1 node.count ← counts information; +2 if d is empty or d is full then +3 +continue ; +/* stop splitting */ +4 else +5 +if d is a cube then +6 +split d equally into 8 oct-blocks: s1, · · · , s8; +7 +get the counts c1, ...c8 for s1, · · · , s8; +8 +find the maxDiff partition d1,d2; +9 +node.left += AKDTree (d1, four ci of d1); +10 +node.right = AKDTree (d2, four ci of d2); +11 +else if d is a flat cuboid then +12 +get the counts c1, · · · , c4 from counts +information; +13 +find the maxDiff partition d1, d2; +14 +node.left += AKDTree (d1, two ci of d1); +15 +node.right = AKDTree (d2, two ci of d2); +16 +else if d is a slim cuboid then +17 +get the counts c1, c2 from counts information; +18 +split d along the largest dimension to get +d1,d2; +19 +node.left += AKDTree (d1, c1); +20 +node.right = AKDTree (d2, c2); +21 end +22 return node; +To address the above high overhead issue of OpST, we +propose an adaptive k-d tree, called AKDTree, to remove +empty regions and extract sub-blocks (containing multi- +ple unit blocks). AKDTree has a lower time complexity of +O( 1 +3N · log N) (will be discussed later). Figure 9 shows a +simple 2D example. Specifically, (1) we partition the data +into small unit blocks. (2) We use a tree to hierarchically +represent the whole data. Each node in the tree is associated +with a sub-block of the data. Moreover, each node stores the +number of non-empty unit blocks in the sub-block associated +with the node. (3) For each node, we split its associated +sub-block from the middle along one dimension to form +6 + +n[21[2] +non-adaptive splittwo sub-blocks for its two children. Note that we select one +dimension which can maximize the difference of the num- +bers of non-empty unit blocks of the two children (will be +discussed in the next two paragraphs). (4) We keep splitting +a node until it has no empty unit block or itself is empty. +(5) Once finishing the construction of the tree, we collect all +the leaf nodes and send them to the compressor. Note that +a non-empty leaf node does not have any empty unit block; +otherwise, it will keep splitting. Thus, a leaf node must be +an empty or full node, as shown in Figure 9. The detailed +algorithm is described in Algorithm 3. +As mentioned in step (3), we are distributing the non- +empty unit blocks unevenly to two children for each node +because we attempt to get as many leaf nodes with large sub- +block sizes as possible. If we keep splitting sub-blocks in a +fixed way, for instance, first split along the x-axis, second +split along the y-axis, third split along the x-axis, fourth split +along the y-axis, and so on, we will get a 2-by-2 sub-block for +the node n[2][2] as shown in the dashed box, while its largest +possible sub-block could be 4 by 2 as shown in Figure 9. +Fig. 10: Example of adaptive splitting, different shapes will have a +different number of choices for splitting. +To select one of the dimensions to unevenly distribute its +non-empty unit blocks to the two children, we now present +our dynamic splitting approach. We categorize nodes into +three different types: “cube” nodes, “flat” nodes, and “slim” +nodes, whose dimension ratios are 1:1:1, 2:2:1, 2:1:1, respec- +tively. First of all, for the cube node d, we first divide it into +eight oct-blocks, i.e., s1, s2, · · · , s8 (as shown in Figure 10), +each sized n +2 +3. Here n is the dimension size of the original +data. Then, we can get the counts of non-empty unit blocks +of the eight oct-blocks, i.e., c1, c2, · · · , c8. After that, we will +decide along which dimension to split the cube node d based +on the counts. Specifically, we can calculate the following +three differences: +diffx = |c1 + c3 + c5 + c7 − c2 − c4 − c6 − c8|, +diffy = |c1 + c2 + c5 + c6 − c3 − c4 − c7 − c8|, +diffz = |c1 + c2 + c3 + c4 − c5 − c6 − c7 − c8|. +Finally, we compare these three values and choose the +dimension with the maximum difference to split. For exam- +ple, if the maximum difference is diffz, we will split d along +the z-axis (i.e., the pink 2D plane shown in Figure 10) and +get two flat nodes d1 and d2. For the flat nodes such as d1, +we can reuse c1, · · · , c4 to decide whether to split d1 along +the x-axis or y-axis by choosing the larger one among the +following two differences. +diffx = |c1 + c3 − c2 − c4|, diffy = |c1 + c2 − c3 − c4|. +For the slim nodes such as d11, we simply split it along +the x-axis to get two cube nodes s1 and s2. This process +(i.e., cube nodes→flat nodes→slim nodes) in step (3) will be +looped until the node becomes a leaf node (empty or full). +Algorithm 4: SZ compression with SHE +Input: multiple data block D1...n +Output: compressed data S +1 quantCode Q, regreCoeff R, compressed data S; +2 for each data block Di do +3 +quantCode block qi, regreCoeff block ri; +4 +qi, ri = SZ.compress(Di); +5 +Q.append(qi); +6 +R.append(ri); +7 end +8 S ← HuffmanEncode(Q); +9 S ← HuffmanEncode(R); +/* end compression */ +10 return S; +Note that based on the above description, the counting +process is required for every three nodes in each three path +(i.e., only for the “cube” nodes). Thanks to this dynamic +splitting approach, we can lower the time complexity of the +AKDTree algorithm to O +� 1 +3 · N · log N +� +, where N is the +number of unit blocks while extracting as many relatively +large sub-blocks without empty unit block as possible. +In addition, after the dynamic splitting, we will have a +series of sub-blocks with the same size but in different di- +rections (e.g., 2:2:1, 2:1:2, 1:2:2). We will align the sub-blocks +with the same size based on their splitting dimensions (in- +stead of in-memory transposing them), merge them into an +array, and feed multiple merged arrays to the compression. +3.4 +Shared Huffman encoding +As mentioned in Section 1, +3.3 and 3.2, the OpST and +AKDtree methods collect and linearize the data blocks with +the same shape into a 4D array and send it to SZ. However, +poor data locality/smoothness between non-neighbored +data blocks can negatively impact prediction accuracy, re- +sulting in low data quality. A potential solution could be +compressing each data block individually using SZ. How- +ever, this approach would result in low Huffman encoding +efficiency because the entire dataset would be divided into +many small blocks, requiring SZ to build a separate Huff- +man tree for each of these small blocks. In other words, the +original SZ method either requires predicting and encoding +small blocks together (by forcing them merged into 4D ar- +rays), leading to low prediction accuracy or predicting and +encoding each small block separately, which results in high +Huffman encoding overhead. Furthermore, even if the data +blocks with the same shape can be compressed together, the +data blocks with different shapes still need to be compressed +separately, resulting in low Huffman encoding performance +and a high time cost of launching SZ multiple times. +To this end, for OpST and AKDtree, we propose a shared +Huffman encoding technique to predict data blocks sepa- +rately while encoding them together using a single shared +Huffman tree. The detail is shown in Algorithm 4. Each data +block is first predicted and quantized separately. Then, the +quantization codes and regression coefficients of each data +block are aggregated to build a shared Huffman tree and +encoded at one time. This approach improves the prediction +performance of SZ without introducing high time overhead +to the encoding of SZ. As shown in Figure 3.4, compared to +the original AKDtree, AKDtree with SHE can significantly +reduce the overall compression error, especially for the data +located at the boundary of data blocks, leading to significant +7 + +(a) d = 23 +(b) d = 58 +(c) d = 64 +(d) d = 42 +(e) d = 77 +(f) d = 99 +Fig. 11: Compression performance comparison of GSP, OpST, and AKDTree on six datasets with different densities. +(a) AKDtree +(CR=222, PSNR=78.5dB) +(b) AKDtree + SHE +(CR=231, PSNR=79.6dB) +Fig. 12: Visual comparison (one slice) of compression errors of two +approaches using SZ based on Nyx’s “baryon density” field (i.e., z10’s +fine level, 23% density). Brighter means higher compression error. The +error bound is the relative error bound of 4.8 × 10−4. +Fig. 13: Comparison of original AKDTree and AKDTree with SHE on +Nyx’s “baryon density” field (i.e., z10’s fine level, 23% density). +PSNR improvement, as shown in Figure 13. Furthermore, +the use of SHE reduces the number of Huffman trees needed +for the data (since we do not need separate Huffman trees +for different block shapes), thereby improving encoding effi- +ciency and compression ratio. +3.5 +Hybrid Compression Strategy +In this section, we propose a solution to adaptively choose a +best-fit compression strategy from our proposed OpST with +SHE (OpST+), AKDTree with SHE (AKDTree+), and GSP based +on the data characteristics (i.e., data density). According to +Section 3.2 and 3.3, OpST+ is more suitable for sparse (i.e., +low-density) data, while AKDTree+ is designed to address +the high time overhead of OpST+ when the density of data +increases. Thus, there should be a data-density threshold to +determine when to use OpST+ or AKDTree+. +To decide the threshold T for switching between OpST+ +and AKDTree+, we perform a series of experiments, as +shown in Figure 11. The figure shows that OpST+ and +AKDTree+ have almost identical compression performance +in terms of bit-rate and PSNR on all six datasets/levels +(from different timesteps) with different densities. Moreover, +Figure 14 shows the time costs of OpST+ and AKDTree+ +(excluding compression). The figure demonstrates that the +time of AKDTree+ is relatively stable, while the time of +OpST+ increases linearly with the increase of data density. +Overall, the only criterion for selecting OpST or AKDTree is +the time cost rather than the compression performance. This +is consistent with our previous design aim, that is, AKDTree +is mainly designed to address the high time overhead issue +of OpST+. Since OpST+ and AKDTree+ have a similar speed +when the density is around 50%, we propose to choose T = +50 for choosing OpST+ or AKDTree+. +According to Section 3.1, GSP is designed to effectively +handle the AMR level with high data density to prevent the +negative impact of data partitioning without the use of SHE. +In contrast, the data partition methods such as AKDTree +require small blocks to be compressed together without +SHE, which can significantly decrease prediction accuracy, +as shown in Section 3.4. However, the negative impact on +prediction accuracy caused by the partition can be elimi- +nated by using SHE to compress each small block produced +by the partition, which incurs little overhead, while GSP +introduces significant overhead to the data size. As shown in +Figure 11, OpST+ and AKDTree+ outperform GSP across all +the densities. As a result, the improved partition strategies +using SHE can be a viable alternative to GSP for all levels. +In summary, our proposed hybrid compression approach +Fig. 14: Time overhead comparison of OpST and AKDTree on different +datasets with different densities. +8 + +-KD+ +-OpST+ +-GSP +105 +95 +PSNR +85 +75 +0 +1 +2 +3 +4 +5 +bit-rate→KD+ +-OpST+ +-GSP +150 +PSNR +130 +110 +0 +1 +2 +3 +4 +5 +bit-rate→-KD+ +-OpST+ +-GSP +165 +PSNR +145 +125 +0 +1 +2 +3 +4 +5 +CR←-KD+ +-OpST+ +OGSP +160 +145 +SNR +130 +115 +0 +1 +2 +3 +4 +5 +6 +bit-rateKD+ +-OpST+ +-GSP +110 +100 +PSNR +90 +80 +0 +1 +2 +3 +4 +bit-rateKD+ +-OpST+ +-GSP +115 +105 +PSNR +95 +85 +75 +0 +1 +2 +3 +4 +bit-rate-AKDTree +- AKDTree + SHE +105 +95 +PSNR +85 +75 +0 +50 +100 +150 +200 +250 +CR4.5 +4 +3.5 +3 +CPU time (ms) +2.5 +2 +1.5 +1 +0.5 +-tree +opst +opst (theoretical) +0 +20 +30 +40 +50 +60 +70 +Densityis described as follows. +1) When the density is smaller than T = 50%, we use +OpST+ to remove empty regions and then compress; +2) When the density is larger than T += 50%, we use +AKDTree+ to remove empty regions and then compress. +4 +EXPERIMENTAL EVALUATION +4.1 +Experimental Setup +Test data. Our evaluation mainly focuses on the AMReX +framework [3], particularly the Nyx cosmology simula- +tion [9]. Nyx is a state-of-the-art extreme-scale cosmology +code using AMReX, which generates six fields including +baryon density, dark matter density, temperature, and veloc- +ities (x, y, and z). We use nine datasets generated by three +real-world simulation runs with different numbers of AMR +levels, simulating a region of 64 megaparsecs (Mpc). For this +data, Z is equal to the redshift, i.e. the displacement distant +galaxies and celestial objects, as seen in Tab 1. +Specifically, the first run has two levels of refinement, +with the coarse level of 2563 grids and the fine level of 5123 +grids. We’ve collected four timesteps with the finest level +density from 23% to 64%. The second run has a maximum +of four levels of refinement. It was initially configured at the +resolution of 1283 and gradually refined to 10243. This run +collected three timesteps with the finest-level resolution of +2563 (two levels), 5123 (three levels), and 10243 (four levels), +respectively. The density of the finest level varies from 0.2% +to 0.003%. The third run has three levels of refinement with +the grid sizes of 1283 at the coarsest level, 2563 at the inter- +mediate level, and 5123 at the finest level. The density of the +finest level ranges from 0.87% to 0.90%. +Note that the density of the finest level describes how +much of the data in the dataset is at the highest resolution; +a higher density of the finest level means that more data is +refined to the highest resolution. Usually, the data density is +gradually increasing at the finest level, within a single run. +Evaluation platform and compressor. The test platform is +equipped with two 28-core Intel Xeon Gold 6238R processors +and 384 GB of memory. We use SZ3 [40], which is easy to +(de)couple lossless encoding from SZ compression, due to +its high modularity. +TABLE 1: Our tested datasets. +Dataset +# Levels +Grid Size of Each Level +(Fine to Coarse) +Density of Each Level +(Fine to Coarse) +Nyx_Run1_Z10 +2 +512, 256 +23%, 77% +Nyx_Run1_Z5 +2 +512, 256 +58%, 42% +Nyx_Run1_Z2 +2 +512, 256 +63%, 37% +Nyx_Run2_T2 +2 +256, 128 +0.2%, 99.8% +Nyx_Run2_T3 +3 +512, 256, 128 +0.02%, 0.56%, 99.42% +Nyx_Run2_T4 +4 +1024, 512, 256, 128 +3E-5, 0.02%, 2.2%, 97.7% +Nyx_Run3_Z1.5 +3 +512, 256, 128 +0.87%, 13.88%, 85.25% +Nyx_Run3_Z1 +3 +512, 256, 128 +0.90%, 14.70%, 84.40% +Comparison baselines. As discussed in Section 2, we have +three 1D or 3D comparison baselines. Specifically, (1) the 1D +baseline (naive): each AMR level is compressed separately as +a 1D array; (2) the 1D baseline (zMesh) [27]: we refer readers +to Section 2 for more details about how the zMesh approach +reorganize the AMR data for 1D compression; and (3) the 3D +baseline: Different AMR levels are unified to the same reso- +lution for 3D compression; (4) the TAC baseline: each AMR +level is compressed using OpST (without SHE), AKDTree +(a) Run1_Z10 (finest-level density = 23%) +(b) Run1_Z5 (finest-level density = 58%) +(c) Run1_Z2 (finest-level density = 63%) +Fig. 15: Rate-distortion comparison of our approach and baselines on +the early time-step (Z10) to the late time-step (Z2) from Nyx run1. +(without SHE), and GSP based on the data density. For more +information, we refer readers to [41] for more details. +4.2 +Evaluation Metrics +We will evaluate the compression performance based on the +following metrics: (1) compression ratio or bit-rate (generic), +(2) distortion quality (generic), (3) compression throughput +(generic), (4) rate-distortion (generic), (5) power spectrum +(cosmology specific), (6) Halo finder (cosmology specific). +Metric 1: To evaluate the size reduction as a result of the +compression, we use the compression ratio, defined as the +ratio of the original data size compared to the compressed +data size, or bit-rate (bits/value), representing the amortized +storage cost of each value. For single/double floating-point +data, the bit-rate is 32/64 bits per value before compression. +The compression ratio and bit-rate have a mathematical re- +lationship as their multiplication is 32/64 so that a lower bit +rate means a higher compression ratio. +Metric 2: Distortion is another important metric used to +evaluate lossy compression quality. We use the peak signal- +to-noise ratio (PSNR) to measure the distortion quality. +PSNR = 20 · log10 (RX) − 10 · log10 +��N +i=1 e2 +i /N +� +, +where ei is the difference between the original and decom- +pressed values for the point i, N is the number of points, and +RX is the value range of X. Higher PSNR less error. +Metric 3: (De)compression throughputs are critical to im- +proving the I/O performance. We calculate the throughput +based on the original data size and (de)compression time. +Metric 4: Similar to prior work [14], [15], [42], [32], [20], +[43], [35], we plot the rate-distortion curve to compare the +9 + +-naive1D --zMesh --3D +TAC -O-TAC+ +105 +95 +PSNR +85 +75 +0 +2 +4 +6 +bit-ratenaive1D -zMesh --3D --TAC -TAC+ +150 +PSNR +130 +110 +0 +2 +4 +6 +bit-rate-naivelD -O-zMesh --3D +TAC -O-TAC+ +170 +PSNR +150 +130 +0 +2 +4 +6 +bit-rate(a) Run2_T2 (finest-level density = 0.2%) +(b) Run2_T3 (finest-level density = 0.02%) +(c) Run2_T4 (finest-level density = 3E-5) +Fig. 16: Rate-distortion comparison of TAC+ (top-left) and baselines on +different time-steps from run2. +distortion quality with the same bit-rate, for a fair compar- +ison between different compression approaches, taking into +account diverse compression algorithms. +Metric 5: Matter distribution in the Universe has evolved +to form astrophysical structures on different physical scales, +from planets to larger structures such as superclusters and +galaxy filaments. The two-point correlation function ξ(r), +which gives the excess probability of finding a galaxy at +a certain distance r from another galaxy, statistically de- +scribes the amount of the Universe at each physical scale. +The Fourier transform of ξ(r) is called the matter power +spectrum P(k), where k is the comoving wavenumber. The +matter power spectrum describes how much structure exists +at each physical scale. We run the power spectrum on the +baryon density field by using a cosmology analysis tool +called Gimlet. We compare the power spectrum p′(k) of +decompressed data with the original p(k) and accept a max- +imum relative error within 1% for all k < 10. +Metric 6: Halo finder aims to find the halos (over- +densities) in the dark matter distribution and output the +positions, the number of cells, and the mass for each +halo it finds, respectively. Specifically, the halo-finder algo- +rithm [44] searches for the halos from all the simulated data, +with the following two criteria: (1) the mass of a data point +must be greater than a threshold (e.g., 81.66× of the average +mass of the whole dataset) to become a halo cell candi- +date [20], [43], [45], and (2) there must be enough halo cell +candidates in a certain area to form a halo. For decompressed +data, some of the information (mass and cells of halos) can +be distorted from the original. +(a) Run3_Z1.5 (density for each level: 0.87%, 13.88%, 85.25% ) +(b) Run3_Z1 (density for each level: 0.90%, 14.70%, 84.40%) +Fig. 17: Rate-distortion comparison of our approach and baselines on +different time-steps from Nyx run3. +4.3 +Evaluation on Rate-distortion +We first evaluate the rate-distortion of our proposed com- +pression approach and compare it with the baselines on +different datasets. As shown in Figure 15, 16 and 17, our +new approach with SHE, TAC+ (represented by the pink +curve) yields better performance than the original TAC (the +yellow curve) without SHE. Also, for the 1D baseline, our +approach (top-left curve) outperforms the 1D baseline across +all 8 datasets. Furthermore, the performance of our approach +is more stable (i.e., smoother curve) than the 1D baseline. +We can also find that zMesh is slightly worse than the 1D +baseline on our tested data as shown in Figure 15, which will +be explained in the next section. +For the 3D baseline, we observe that our solution has +much better performance when the finest level has a rel- +atively low density or the decompressed data has a high +PSNR, as shown in Figure 15, 16 and 17. However, our +approach cannot dominate the 3D baseline as shown in +Figure 15a, 15b, and 16a, when the following criteria are +satisfied: (1) the AMR data has only two levels of refinement, +(2) the finest level has a relatively low density, and (3) the +decompressed data has low PSNR/bitrate. +We also find that as the finest level density increases, +the bitrate threshold where the 3D baseline performs better +than TAC+ is increasing, while the advantage of the 3D +baseline over TAC+ is decreasing, as shown in Figure 15 +and 16a. Specifically, in Figure 16a (the finest level density +is 0.2%), TAC+ outperforms the 1D baseline when the bit- +rate is above 1.2; in Figure 15a (the finest level density is +23%), the intersection is the bit-rate of 1.25; as the finest level +density continues to grow up to 58% in Figure 15c, TAC+ +is slightly worse than the 3D baseline until the bit-rate is +larger than 1.5; when the finest level density reaches 63% +in Figure 15c, TAC+ and the 3D baseline yield (almost) the +same performance. +In the next section, we will discuss how the number +of AMR levels, the density of the finest level, and the bit- +rate/PSNR of decompressed data affect the performance of +the 3D baseline and TAC+ in detail. +10 + +-naiveiD +3D +TAC +-TAC+ +90 +PSNR +80 +70 +60 +0 +2 +4 +6 +bit-rate-naivelD +-3D +TAC +-TAC+ +100 +PSNR +85 +70 +0 +2 +4 +6 +bit-rate←-naivelD +-3D +TAC +-TAC+ +115 +100 +PSNR +85 +70 +0 +2 +4 +bit-rate→-naiveD +3D +TAC +-TAC+ +140 +PSNR +125 +110 +0 +2 +4 +6 +bit-rate→naivelD +3D +-TAC +-TAC+ +130 +PSNR +115 +100 +0 +2 +4 +6 +bit-rate(a) Tree-based AMR data (b) Patch-based AMR data +Fig. 18: An example of how the 1D baseline, zMesh, and original z-order +reorder a simple 2D AMR data without and with redundancy. Orange: +coarse level, blue: fine level, red: redundant data. +4.4 +Discussion on Comparison with Baselines +On compression, zMesh is meant to improve the smoothness +of the patch-based AMR datasets by taking advantage of the +data redundancy between each AMR level (as described in +the introduction). Thus, zMesh cannot improve the smooth- +ness if there is no data redundancy in the tree-structured +AMR datasets (i.e., our tested datasets). A simple example +is used to illustrate this in Figure 18b, where the finer-level +data has higher values because a grid will be refined only if +its value is larger than a certain threshold. For block-based +AMR, when a grid needs to be refined because of its high +value, the value will still remain in the level, resulting in +a redundant value saved (i.e., the red 8). If one uses the +original z-ordering to traverse the data level-by-level (shown +in Figure 18b), the reordered data will have three significant +value changes (i.e., from 2 to 8, from 8 to 1, and from 1 to +9). To solve this issue, zMesh traverses the two AMR levels +together based on the layout of the 2D array. The reordered +data are “1-2-8-9-8-7-8-1”, which only has two significant +value changes (i.e., from 2 to 8 and from 8 to 1). Thus, zMesh +can improve the smoothness of patch-based AMR data. +However, as shown in Figure 18a, for tree-structured +AMR data (without saving a redundant “8”), compared to +the 1D baseline that compresses each level separately, zMesh +introduces two significant data changes (i.e., from 2 to 9 and +from 8 to 1) as it traverses between two AMR levels. This +explains why zMesh is slightly worse than the 1D baseline +on our data. +When considering the 3D baseline, we observe that it +works slightly better than TAC+ in the following circum- +stances: (1) the AMR data has only two levels of refinement, +(2) the decompressed data has a low PSNR/bit-rate, and (3) +the finest level of the data has a relatively low density. We +also observe that (4) when the finest level density increases, +the bit-rate intersection point where the 3D baseline per- +forms better than TAC+ shifts to a higher value, (5) while the +advantage of the 3D baseline over TAC+ becomes smaller. +We now explain each observation point individually. As +mentioned in Section 2.4, the main disadvantage of the 3D +baseline is the redundant data generated by the up-sampling +process, which becomes more significant for the AMR data +with more than two levels of refinement, resulting in poor +compression performance (point 1). In contrast, the 3D base- +line has better data locality and smoothness over TAC+ +because it compresses the entire dataset together without +partitioning the data. This leads to better compressibility +for the 3D baseline, resulting in faster bitrate reduction and +Fig. 19: Bit-rates with different error bounds using SZ lossy compression +for fine and coarse levels on Run1_Z2 dataset. +slower PSNR degradation when the error bound increases, +compared to TAC+. As a result, the 3D baseline performs +better at a low bit-rate (point 2). +However, when the finest level density increases, the +smoothness advantage of the 3D baseline will decrease. This +is because TAC+ will have more data located in the finest +level, and the partition overhead of TAC+ is lower at the +finest level compared to coarser levels due to the larger (unit) +block size. This explains why the 3D baseline only outper- +forms TAC+ for AMR datasets with low density (points 3 +& 5). Also, note that for the dataset with low density in +the finest level, the overhead of redundant up-sampled data +will be high for the 3D baseline, especially for high bit- +rates (because the 3D baseline only has better smoothness +in low bit-rates). Thus, the compression performance of the +3D baseline will decrease rapidly as we aim for relatively +high PSNRs/bit-rates, leading to a smaller intersection bit- +rate between the 3D baseline and TAC+. Clearly, as the +finest level density increases, the overhead of the redundant +data will be smaller, resulting in a larger intersection bit-rate +between the 3D baseline and TAC+ (point 4). +4.5 +Post-analysis Quality with Adaptive Error Bound +When factoring level-wise compression, our approach can +apply different error bounds to different AMR levels based +on (1) the post-analysis metrics, (2) the up-sampling rates of +coarse levels, and (3) the rate-distortion trade-off between +different AMR levels. We now evaluate our approach with +the two cosmology-specific post-analysis metrics (i.e., power +spectrum and halo finder) to demonstrate the benefit of the +adaptive error bound method. We choose the dataset run1- +Z2 for evaluation because TAC+ has a similar performance +as the 3D baseline on this dataset. +Figure 19 shows the motivation of performing rate- +distortion trade-off between different AMR levels. As the +error bounds for the fine and coarse levels increase, their bit +rates will converge to a similar value. This means that when +the error bound is relatively large, the reduction in data +size will be insignificant compared to the compression error +increment (i.e., the slopes of both curves are very small). +Therefore, we can say that when the error is large, it is not +worth trading data quality for size reduction. +Power Spectrum Figure 20a shows that under the (al- +most) same compression ratio, TAC+ (with the uniform er- +ror bound) has a better power-spectrum error compared to +the 3D baseline. Also note that TAC+ yields nearly lossless +power-spectrum distortion (less than 0.01%) under the com- +pression ratio of 42×, as shown in Figure 20a’s gray curve. +Now, let us follow the three steps mentioned at the begin- +ning of this section to adjust the error bound for each AMR +11 + +2 +9 +8 +8 +2 +9 +8 +78 +8 +82 +9 +8 +8 +1 +7 +8 +128987 +28Fine lyl +-Coarse lyl +2 +Bitrate +0 +0 +1E+09 +2E+09 +3E+09 +4E+09 +5E+09 +ABS error boundTABLE 2: Overall compression/decompression throughput (MB/s) of different approaches with different absolute error bounds. +EB_abs +Run1_Z10 +Run1_Z2 +Run2_T2 +Run2_T4 +Run3_Z1 +1D +3D +TAC +TAC+ +1D +3D +TAC +TAC+ +1D +3D +TAC +TAC+ +1D +3D +TAC +TAC+ +1D +3D +TAC +TAC+ +1E+08 +51 +37 +79 +77 +51 +79 +85 +84 +71 +15 +71 +66 +49 +0.32 +24 +24 +49 +5.6 +63 +63 +1E+09 +81 +48 +94 +93 +55 +92 +103 +97 +79 +21 +85 +79 +53 +0.39 +26 +26 +53 +6.5 +78 +75 +1E+10 +89 +53 +107 +102 +58 +100 +111 +104 +86 +23 +98 +87 +84 +0.41 +28 +27 +57 +7.1 +91 +84 +(a) Power spectrum error under 1% limit +(b) Power spectrum error under 0.01% limit +Fig. 20: Power spectrum error (in relative) of the 3D baseline and TAC+ +(uniform error bound) and TAC+ (adaptive error bound) on baryon +density field on run1-Z2. The red and blue dashed line is the 1% and +0.01% limit of acceptable power spectrum error. +level. First, the post-analysis metric–power spectrum— +needs to be run on the uniform-resolution data and focuses +on the global quality of data. Thus, the ideal error-bound +configuration/ratio for the fine and coarse levels on the +uniform-resolution data would be 1:1. +As aforementioned, the coarse level of the AMR dataset +needs to be up-sampled to uniform the resolution. As a +result, the compression error of the coarse level will be up- +sampled as well, resulting in more error in the post-analysis. +Thus, we then need to give the coarse level a smaller error +bound based on the up-sample rate. Here the up-sample rate +for Z2’s coarse level is 23, leading to an ideal error-bound +ratio of the fine and coarse levels changed to 8:1. +Finally, this 8:1 ratio needs to be adjusted based on the +rate-distortion trade-off as aforementioned. As shown in Fig- +ure 19, when using the error-bound ratio of 8:1 (e.g., 4E+9 +for the fine level and 5E+8 for the coarse level), the error +bound of the fine level is too large, resulting in an ineffective +rate-distortion trade-off. Thus, we can balance two levels +by increasing the error bound for the coarse level (to gain +compression ratio) and decreasing the error bound for the +fine level (to add compression error), which can achieve an +overall rate-distortion benefit. Based on our experiments, we +adjust the error-bound ratio from 8:1 to 3:1, As shown in Fig- +ure 20b, TAC+ with adaptive error bound can significantly +improve the compression ratio compared with uniform error +bound under similar power spectrum error. +Halo finer We evaluate the mass change, and the number +of cells change for the three largest halos identified using the +3D baseline, TAC+ (with uniform error bound), and TAC+ +(with adaptive error bound), as shown in Table 3. We can see +that TAC+ with uniform error bound produces better halo- +finer analysis quality than the 3D baseline. +Similar to the error-bound configuration analysis done +for the power spectrum, let us now adjust the error-bound +ratio between the fine and coarse levels for halo finder. The +TABLE 3: Halo finder analysis with different methods. +CR +Avg Rel Mass Diff +Avg Rel Cells Diff +3D baseline +188.7 +2.7E-04 +2.2E-03 +TAC+ (1:1) +189.1 +2.2E-04 +2.1E-03 +TAC+ (2:1) +192.5 +1.1E-04 +9.0E-04 +halo-finer analysis also requires uniform-resolution data as +input. However, different from the power-spectrum anal- +ysis, the halo-finder analysis focuses more on high-value +points at the fine level, since only high-value data points +qualify as halo candidates, as described in Section 4.2. Note +that this does not mean we can directly discard the coarse- +level data with small values as they still contribute to the +average value of the dataset, which is also an important +parameter for the halo finder [44]. Therefore, we set the ideal +error-bound ratio to 1:2 (i.e., fine level vs coarse level) for the +uniform-resolution data based on our massive experiments. +After that, considering the up-sampling rate of 23, the error- +bounded ratio is changed to 4:1. Finally, we adjust the ratio +to 2:1 based on the rate-distortion trade-off. Table 3 shows +that TAC+ with adaptive error bound obtains the minimal +differences of the mass and cell numbers. +4.6 +Evaluation on Time Overhead +We evaluate the overall throughput (including preprocess- +ing and (de)compression) on the datasets with different error +bounds. Compared to the 3D baseline, the throughput of +TAC+ is up to 11.8× higher on the Run3 dataset, 75× higher +on the Run2 datasets, and 2× higher on the Run1 datasets. +This is because the Run2 and Run3 datasets have a lower +density than the Run1 datasets at the finest level, resulting +in a higher overhead of redundant data for the 3D baseline, +which is consistent with our discussion in Section 4.4. +We note that TAC+ performs better than the 1D baseline +on all the tested datasets except the T4 datasets, due to +its relatively long data-partition time compared to the total +time on the small-sized datasets. TAC+ has almost the same +throughput as TAC. The very slight decrease is because +TAC+ compresses the data in a more fine-grained manner. +We exclude zMesh in this evaluation as it is theoretically +slower than the 1D baseline due to the extra z-ordering and +provides worse rate-distortion according to our evaluation. +5 +CONCLUSION AND FUTURE WORK +In conclusion, this paper leverages 3D compression for AMR +data on a systemic level. We propose three pre-processing +strategies that can adapt based on the density of each AMR +level. Our approach improves the compression ratio com- +pared to the state-of-the-art approach by up to 4.9× under +the same data quality loss. With our level-wised compres- +sion approach, we are able to tune the error-bound ratio of +fine and coarse levels to be 3:1 and 2:1 for better power- +spectrum and halo-finder analyses, respectively, under the +same compression ratio. 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Ahrens, “Adaptive +configuration of in situ lossy compression for cosmology simu- +lations via fine-grained rate-quality modeling,” in Proceedings of +the 30th International Symposium on High-Performance Parallel and +Distributed Computing, 2021, pp. 45–56. +[44] M. Davis, G. Efstathiou, C. S. Frenk, and S. D. White, “The evolu- +tion of large-scale structure in a universe dominated by cold dark +matter,” The Astrophysical Journal, vol. 292, pp. 371–394, 1985. +[45] B. Fang, D. Wang, S. Jin, Q. Koziol, Z. Zhang, Q. Guan, S. Byna, +S. Krishnamoorthy, and D. Tao, “Characterizing impacts of storage +faults on hpc applications: A methodology and insights,” in 2021 +IEEE International Conference on Cluster Computing (CLUSTER). +IEEE, 2021, pp. 409–420. +Daoce Wang is a PhD student in Intelligent Sys- +tems Engineering at Indiana University Bloom- +ington. He received his bachelor’s degree in +Computer Science from University of Electronic +Science and Technology of China in 2018 and +his master’s degree in Computer Science from +University of Florida in 2020. His research in- +terests include scientific data reduction, AMR +simulations, and parallel algorithms. +Jesus Pulido is a research scientist in the Data +Science at Scale Team at Los Alamos National +Laboratory. Pulido received his Ph.D. in Com- +puter Science at University of California, Davis +in 2019. Pulido specializes in data analysis, data +reduction, visualization, high performance com- +puting, wavelets and multi-resolution methods. +He has experience in applications of image sen- +sors, astronomy, turbulence and cosmology. +Pascal Grosset is a scientist in the Data Sci- +ence at Scale team at Los Alamos National Lab- +oratory. His primary research interests are large- +scale data analysis and visualization, and data +reduction. He received his Ph.D. in Computing: +Graphics and Visualization from the University +of Utah in 2016 where his research focused on +large scale visualization. +Sian Jin is currently working toward the PhD +degree majoring in computer engineering at In- +diana University Bloomington. His research in- +terests include High Performance Computing, +Compression Algorithms, Artificial Neural Net- +works, and Parallel Computing. He has pub- +lished several papers in major journals and inter- +national conferences including the SC, PPoPP, +VLDB, HPDC, IPDPS, and ICDE. +Jiannan Tian is a PhD candidate in Intelli- +gent Systems Engineering at Indiana Univer- +sity Bloomington. His research interests in- +clude lossy compression for scientific data +and error analysis, and GPU-centric computing. +His ongoing project including developing GPU- +accelerated compression algorithm and system +design optimization of lossy compression. +Kai Zhao is an assistant professor of computer +science at University of Alabama at Birming- +ham. He received his Ph.D. in computer sci- +ence from University of California, Riverside in +2022. He received his bachelor’s degree from +Peking University in 2014. His research interests +include high-performance computing, scientific +data management and reduction, and resilient +machine learning. +James P. Ahrens received the BS degree +in computer science from the University of +Massachusetts +at +Amherst, +Amherst, +Mas- +sachusetts, in 1989, and the PhD degree in com- +puter science from the University of Washington, +Seattle, Washington, in 1996. He is a senior +scientist with Applied Computer Science Group, +Los Alamos National Laboratory. His primary re- +search interests include visualization, computer +graphics, data science, and parallel systems. He +is author of more than 100 peer reviewed papers +and the founder/design lead of ParaView, an open-source visualization +tool designed to handle extremely large data. ParaView is broadly used +for scientific visualization and is in use at supercomputing and scien- +tific centers worldwide. He is the chair of the IEEE Computer Society +Technical Committee on Visualization and Graphics (VGTC). +Dingwen Tao is an associate professor at In- +diana University Bloomington, where he directs +the High-Performance Data Analytics and Com- +puting Lab. He received his Ph.D. in Computer +Science from University of California, Riverside +in 2018 and B.S. in Mathematics from University +of Science and Technology of China in 2013. +He is the recipient of various awards including +NSF CAREER Award (2023), Amazon Research +Award (2022), Meta Research Award (2022), +R&D100 Awards Winner (2021), IEEE Computer +Society TCHPC Early Career Researchers Award for Excellence in +HPC (2020), NSF CRII Award (2020), and IEEE CLUSTER Best Paper +Award (2018). He is serving on the Technical Review Board of IEEE +Transactions on Parallel and Distributed Systems. He served as the +Program Co-chair of 2021 IEEE International Conference on Scalable +Computing and Communications and International Workshops on Big +Data Reduction. He is also a reviewer, program committee member, or +session chair of major HPC venues, such as SC, HPDC, ICS, IPDPS, +CLUSTER, ICPP, CCGrid, and HiPC. Email: ditao@iu.edu. +14 + diff --git a/zNAzT4oBgHgl3EQf8P5i/content/tmp_files/load_file.txt b/zNAzT4oBgHgl3EQf8P5i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..323be550d4a47edbe7d61b134f5d0ba7b96bf2d6 --- /dev/null +++ b/zNAzT4oBgHgl3EQf8P5i/content/tmp_files/load_file.txt @@ -0,0 +1,1134 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf,len=1133 +page_content='1 TAC+: Drastically Optimizing Error-Bounded Lossy Compression for 3D AMR Simulations Daoce Wang, Jesus Pulido, Pascal Grosset, Sian Jin, Jiannan Tian, Kai Zhao, James Arhens, and Dingwen Tao Abstract—Today’s scientific simulations require a significant reduction of data volume because of extremely large amounts of data they produce and the limited I/O bandwidth and storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Error-bounded lossy compression has been considered one of the most effective solutions to the above problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, little work has been done to improve error-bounded lossy compression for Adaptive Mesh Refinement (AMR) simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Unlike the previous work that only leverages 1D compression, in this work, we propose an approach (TAC) to leverage high-dimensional SZ compression for each refinement level of AMR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To remove the data redundancy across different levels, we propose several pre-process strategies and adaptively use them based on the data characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We further optimize TAC to TAC+ by improving the lossless encoding stage of SZ compression to efficiently handle many small AMR data blocks after the pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Experiments on 8 AMR datasets from a real-world large-scale AMR simulation demonstrate that TAC+ can improve the compression ratio by up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='9× under the same data distortion, compared to the state-of-the-art method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In addition, we leverage the flexibility of our approach to tune the error bound for each level, which achieves much lower data distortion on two application-specific metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 1 INTRODUCTION T He increase in supercomputer performance over the past decades has been insufficient to solve many chal- lenging modeling and simulation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For example, the complexity of solving evolutionary partial differential equations scales as Ω(n4), where n is the number of mesh points per dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, the performance improvement of about three orders of magnitudes over the past 30 years has meant just a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='6× gain in spatio-temporal resolution [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To address this issue, many high-performance computing (HPC) simulation packages [2] (such as AMReX [3] and Athena++ [4]) use Adaptive Mesh Refinement (AMR)— which applies computation to selective regions of most interest—to increase resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Compared to the method where a high resolution is applied everywhere, the AMR method greatly reduces the computational complexity and storage overhead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' thus, it is one of the most widely used frameworks for many HPC applications [5], [6], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Although AMR can save storage space to some extent, AMR applications running on supercomputers still generate large amounts of data, bringing challenges to data transmis- sion and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For example, one Nyx simulation [9] with a resolution of 40963 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5×20483 mesh points in the coarse level and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5×40963 in the fine level ) can generate up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='8 TB of data for a single snapshot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' a total of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='8 PB of disk stor- age is needed assuming running the simulation 5 times with 200 snapshots dumped per simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Therefore, reducing data size is necessary to lower the storage overhead and I/O cost and improve the overall application performance for running large-scale AMR simulations on supercomputers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Daoce Wang, Sian Jin, Jiannan Tian, and Dingwen Tao (corresponding author) are with Indiana University, Bloomington, IN 47405, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Jesus Pulido, Pascal Grosset, and James Arhes are with Los Alamos Na- tional Laboratory, Los Alamos, NM 87545, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Kai Zhao is with University of Alabama at Birmingham, AL 35294, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A straightforward way to address this issue is to use data compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, traditional lossless compres- sion techniques such as GZIP [10] and Zstandard [11] can only provide a compression ratio by up to 2× for scientific data [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' On the other hand, a new generation of lossy compressors that can provide strict error control (called “error-bounded” lossy compression) has been developed, such as SZ [13], [14], [15], ZFP [16], MGARD [17], and TTHRESH [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Using those error-bounded lossy compres- sors, scientists can achieve relatively high compression ratios while minimizing the quality loss of reconstructed data and post-analysis, as demonstrated in [19], [20], [21], [22], [23], [24], [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Only a few existing contributions have investigated error-bounded lossy compression for AMR applications and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A common approach is to generate uniform- resolution data by up-sampling the coarse-level data and merging them with the finest-level data and then performing compression on the merged data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, this approach introduces redundant information to the data, which signif- icantly degrades the compression ratio, especially when the up-sampling rate is high or there are multiple coarse levels to up-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Recently, Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' introduced zMesh [27], a technique that groups data points that are mapped to the same or adja- cent geometric coordinates such that the dataset is smoother and more compressible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, since zMesh maps data points from different AMR levels to adjacent geometric co- ordinates and generates a 1D array, it cannot adopt 3D com- pression which most HPC simulations use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Moreover, zMesh is designed for patch-based AMR data1 with redundancy across different AMR levels to improve the compression 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The patch-based AMR data redundantly saves the data block to be refined at the next finer level in the current coarse level (will be introduced in detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='01901v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='DC] 5 Jan 2023 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, these coarser levels of redundant data are of- ten not used for post-analysis or visualization and hence can be directly discarded to improve the compression ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For this case, the reorganization approach proposed by zMesh cannot improve the data smoothness appropriately (will be illustrated in Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To solve these issues, we propose TAC that removes the redundant data in coarser level(s) and employs 3D lossy compression for each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We note that each level may contain many empty/zero regions, where data points are saved in other levels, which may significantly decrease the data smoothness and hence reduce the compression ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To this end, TAC either removes these empty regions us- ing adaptive partition strategy or partially pads them with appropriate values, based on the density of empty regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' TAC also has an optimization to reduce the time complexity of removing empty regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Although our partition strategies can remove empty re- gions, there are still several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, the par- tition strategies can generate many (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 3,000+) small data blocks with totally different shapes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 10 different shapes), whereas the SZ compressor performs poorly on relatively small data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This is because the SZ compressor uses thou- sands of Huffman trees to encode these small blocks sepa- rately, leading to a low encoding efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A naive solution to the heavy Huffman encoding cost is to linearize/merge the thousands of small blocks into fewer larger blocks and then pass these larger blocks to SZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This approach can reduce the overhead of the Huffman trees for encoding and hence increase the amount of data encoded together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, TAC still has two limitations: (1) Most of the merged small blocks are not adjacent in the original dataset, leading to rapid changes in the data values at the boundaries of these non-neighboring blocks, which impacts the accuracy of SZ’s predictor negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (2) Although we can compress data blocks with the same shape together, the SZ compressor must be called multiple times for each shape, resulting in the inevitable low performance of Huffman encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To address these limitations, we further optimize TAC to TAC+ by designing a Shared Huffman Encoding (SHE) approach for the SZ compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This approach allows individual pre- dictions for each small block while being encoded using a single shared Huffman tree, which can improve the predic- tion accuracy and compression ratio accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The main contributions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We propose to leverage 3D SZ compression to compress each level of an AMR dataset separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We propose a hybrid compression approach based on the following three pre-process strategies and data characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We propose an optimized sparse tensor representation to efficiently partition data and remove empty regions for sparse AMR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We propose an enhanced k-d tree approach to reduce the time overhead of removing empty regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We propose a padding approach to improve the smooth- ness and compressibility of dense AMR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We employ the SHE approach in the SZ compressor to reduce the high time and storage costs of compressing multiple small blocks after the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We tune the error bound for each AMR level to fur- ther improve the compression quality in terms of two application-specific post-analysis metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Experiments show that, compared to the state-of-the-art approach zMesh, our proposed AMR compression can improve the compression ratio by up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='9× under the same data distortion on the tested datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We evaluate our proposed compression method on eight datasets from three real-world AMR simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The AMR simulations are well-known, open-source cosmology simulations—Nyx [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We compare our method with four baselines including zMesh using generic metrics such as compression ratio and peak signal-to-noise ratio (PSNR) and application-specific metrics such as power spectrum and halo finder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Our code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='com/ FabioGrosso/3dAMRcomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In Section 2, we present background information about error-bounded lossy compression, AMR method, k-d tree, and related work on AMR data compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In Section 3, we describe our proposed pre-process strategies, SHE ap- proach, and hybrid compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In Section 4, we show the experimental results on different AMR datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In Section 5, we conclude our work and discuss the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2 BACKGROUND AND RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1 Lossy Compression for Scientific Data There are two main categories for data compression: lossless and lossy compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Compared to lossless compression, lossy compression can offer a much higher compression ra- tio by trading a little bit of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' There are some well- developed lossy compressors for images and videos such as JPEG [28] and MPEG [29], but they do not have a good performance on the scientific data because they are mainly designed for integers rather than floating points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In recent years there is a new generation of lossy com- pressors that are designed for scientific data, such as SZ [13], [14], [15], ZFP [16], MGARD [17], and TTHRESH [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' These lossy compressors provide parameters that allow users to finely control the information loss introduced by lossy compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Unlike traditional lossy compressors such as JPEG [28] for images (in integers), SZ, ZFP, MGARD, and TTHRESH are designed to compress floating-point data and can provide a strict error-controlling scheme based on the user’s requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Generally, lossy compressors provide multiple compression modes, such as error-bounding mode and fixed-rate mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Error-bounding mode requires users to set an error type, such as the point-wise absolute error bound and point-wise relative error bound, and an error bound level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The compressor ensures that the differences between the original data and the reconstructed data do not exceed the user-set error bound level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In this work, we focus on the SZ lossy compression (2021 R&D 100 Award Winner [30]) because SZ typically provides a higher compression ratio than ZFP [22], [31] and higher (de)compression speeds than MGARD [31], [32] and TTHRESH [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' SZ is a prediction-based error-bounded lossy compressor for scientific data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' It has three main steps: (1) predict each data point’s value based on its neighboring points by using an adaptive, best-fit prediction method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (2) quantize the difference between the real value and predicted value based on the user-set error bound;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' and (3) apply a customized Huffman coding and lossless compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 1: Visualization (one zoom-in 2D slice) of three key timesteps gen- erated from an AMR-based cosmology simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The grid structure changes with the universe’s evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The red boxes indicate different resolutions within one AMR level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2 AMR Method and AMR data AMR is a method of adapting the accuracy of a solution by using a non-uniform grid to increase computational and storage savings while still achieving the desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' AMR applications change the mesh or spatial resolution based on the level of refinement needed by the simulation and use finer mesh in the regions with more importance/interest and coarser mesh in the regions with less importance/interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Figure 1 shows that during an AMR run, the mesh will be refined when the value meets the refinement criteria, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', refining a block when its norm of the gradients or maximum value is larger than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2: A typical example of AMR data storage and usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Clearly, the data generated by an AMR application are hierarchical data with different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The data of each AMR level are usually stored separately (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', in a 1D array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For example, Figure 2 (left) shows a simple example of two- level AMR data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' “0” means high resolution (the fine level) and “1” for low resolution (the coarse level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' When the AMR data are needed for post analysis or visualization, users will typically convert the data from different levels to a uniform resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In the previous example, we will up-sample the data in the coarse level and combine it with the data in the fine level, as shown in Figure 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='3 Tree-based and Patch-based AMR Data There are two types of techniques to represent AMR data: patch-based AMR and tree-based AMR [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The main dif- ference between them is that the patch-based AMR tech- nique generates AMR data with redundancy across different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In other words, the patch-based AMR data structure redundantly saves data blocks to be refined at the next level in the current level, simplifying the computation in the re- finement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' By comparison, the tree-based AMR tech- nique organizes the grids on the tree leaves, so there is no redundant data across different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' But tree-based AMR data is more complex for post analysis and visualization compared to patch-based AMR data [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In this work, we focus on a state-of-the-art patch-based AMR framework AMReX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that since the redundant coarser-level data in the patch-based AMR will not often be used in post-analysis, we discard them during compression to improve the compression ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='4 Existing AMR Data Compression 1D AMR Compression: The main challenge for AMR data compression is that the AMR data is comprehensive and hierarchical with different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A naive approach is to compress the 1D data of each AMR level separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' How- ever, this approach loses most of the topological/spatial in- formation, which is critical for data compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' zMesh [27] is a state-of-the-art AMR data compression based on the 1D approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Different from the naive 1D approach, zMesh re- organizes the 1D data based on each point’s coordinate in the 2D layout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' in other words, zMesh puts the points neigh- bored in the 2D layout closer in the 1D array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' It can increase the data smoothness/compressibility to benefit the follow- ing 1D compression such as SZ on patch-based AMR data with redundancy across different AMR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, zMesh does not leverage high-dimensional compression, while many previous studies [14], [35] proved that lever- aging more dimensional information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', spatial/temporal information) can significantly improve the compression per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Moreover, it only focuses on 2D AMR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Our work aims to leverage high-dimensional data compression and supports 3D AMR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' High-dimensional AMR Compression: Similar to the idea described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2, a straightforward way to leverage 3D compression on 3D AMR data is to compress different levels together by up-sampling coarse levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, this approach must handle extra redundant data generated by the up-sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As shown in Figure 2, 1A, 1B, and 1C are redundant points in the compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that the storage overhead of these redundant points will be higher when more data are in the coarse levels or the up-sampling rate is higher, especially for 3D AMR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This is because we only need to duplicate one point from the coarse level 4 times for 2D AMR data but 8 times for 3D AMR data, with an up- sampling rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Another limitation of this approach is that it cannot apply different compression configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', error bound) to different AMR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This is because after up-sampling all data points will have the same importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, the purpose of using the AMR method is to set different interests to different AMR levels, so the error bound for each AMR level can be chosen adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 k-D Tree for Particle Data Compression k-d tree [36] is a binary tree in which every node represents a certain space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Without loss of generality, for the 3D case, ev- ery non-leaf node in a k-d tree splits the space into two parts by a 2D plane associated with one of the three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The left subspace is associated with the left child of the node, while the right subspace is associated with the right child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' k- d tree is commonly used in particle data compression [37], [38], [39] to locate each particle and remove empty regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, a k-d tree keeps dividing the space in between 3 口1B 1B Ivl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='bin (0A,0B,0C,0D) A 1A 1B 1B 1B Ivl 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='bin(1A,1B,1C) 1C 0A 0B 1C 1C OA 0B OC OD 1C 1C oC OD(a) z10 fine level (b) z10 coarse level Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 3: Visualization of data distributions of an example AMR data “z10”, where z = redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Non-empty regions are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' along one dimension until the space is empty or contains only one particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We will optimize the classic k-d tree and use it to remove empty regions and increase the compress- ibility for each AMR level (to be detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 3 OUR PROPOSED DESIGN In this section, we propose a pre-processing approach for AMR data to leverage high-dimensional SZ lossy compres- sion at each AMR level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, we propose three pre- process strategies to mitigate the issue of irregular data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We further propose Shared Huffman Encoding (SHE) and integrate it into the SZ compressor to further improve the compression performance for AMR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We also propose an adaptive approach to select the best-fit pre- processing strategy based on the data density of each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1 Ghost-Shell Padding for High-density Data To compress the AMR data in 3D, besides the aforemen- tioned 3D baseline, we can also compress each level sepa- rately in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In that way, however, the data will be split into multiple levels, and each level will have many empty regions and an irregular data distribution, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A naive solution to handle the irregular 3D data is to fill the empty regions with zeros and pass a large 3D block to the compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Although the padded zeros will increase the size of data for compression, for high-density data such as z10’s coarse level shown in Figure 3b (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', about 77% density), the size overhead will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, these padded zeros can also greatly reduce the performance of compression, especially for prediction- based lossy compression such as SZ, because these zeros can significantly affect the prediction accuracy of SZ, resulting in high compression errors on the boundaries, as shown in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' More specifically, as mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2, SZ uses each point’s neighboring points’ values to predict its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, for those boundary points which are adjacent to padded zeros, SZ will involve zero(s) in the prediction, while the actual values of these empty regions are typically non-zeros (saved in other AMR levels), which will seriously mislead the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To eliminate the above issue of padding zeroes, we pro- pose to use a ghost-shell padding strategy (GSP) to diffuse neighboring values to a padding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Figure 4 illustrates the high-level idea, and the detailed algorithm is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, we first partition the data into Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 4: A 2D example of GSP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Non-empty blocks are in navy blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' padded blocks are in light blue/red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' padded blocks based on more than one non-empty neighbor are in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Algorithm 1: Proposed Ghost Shell Padding Method Input: Data, x, y Output: Data after padding 1 for each unit block bi do 2 if bi is empty and bi has non-empty neighbor then 3 for each non-empty neighbor nj do 4 pad slice = avg (first y slices of nj next to bi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 5 if overlap edge then 6 pad = pad/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 7 else if overlap corner then 8 pad = pad/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 9 else 10 continue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 11 end 12 add an x-layers pad slice to bi next to nj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 13 end 14 end 15 end 16 return padded Data unit blocks and then pad each empty unit block by using the average of its non-empty neighbors’ boundary data values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that some empty unit blocks have more than one non-empty neighbor such as the red box shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For these blocks, we will use the average value of all its neighbors for padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Correspondingly, we will remove these padded values in the decompression based on the saved padding information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that since the padding pro- cess is only for non-empty blocks, this metadata overhead is almost negligible for high-density data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (a) ZF (CR=156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='7, PSNR=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='8dB) (b) GSP (CR=161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='3, PSNR=33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5dB) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 5: Visual comparison (one slice) of compression errors of two ap- proaches using SZ based on Nyx’s “baryon density” field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', z10’s coarse level, 77% density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Brighter means higher compression error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The error bound is the relative error bound of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='7 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 4 口After padding, each boundary point will be predicted us- ing the average of all the boundary data in the unit block(s) to which it belongs or is neighbored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As shown in Figure 5, compared to the zero-filling (ZF) approach, GSP can signif- icantly reduce the overall compression error, especially for the boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Moreover, the GSP approach can provide a similar compression ratio to the ZF approach on this high- density data and hence a better rate-distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A detailed evaluation will be presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2 Optimized Sparse Tensor Representation for Low- density Data When most of the regions in the data are empty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', about 77% of the data is empty in Figure 3a), the large amount of padded data would greatly increase the size of data for compression, resulting in a low compression ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To solve this issue, we propose to use a naive sparse- tensor-based approach (called NaST) to remove the empty regions, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' NaST includes four main steps in the compression process: (1) partition the 3D data into multiple unit blocks, (2) remove the empty blocks, (3) linearize the remaining 3D blocks into a 4D array, and (4) pass the 4D array to the compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that in the de- compression process, we will put the unit blocks from the decompressed 4D array back into the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 6: Workflow of the naive sparse tensor (NaST) method (empty regions marked in pink and non-empty regions marked in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, in order to completely remove the empty re- gions to form a sparse representation, the unit block size needs to be relatively small compared to the input data size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 163 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 5123), resulting in a high proportion of data on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' While the linearized unit blocks are usually not adjacent in the original data, so the boundary data be- tween them are not smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, it is harder for prediction- based compressors such as SZ to predict the boundary data values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As a result, the NaST method without optimizing the boundary data would have low compression performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To address the above problem, we propose an optimized sparse tensor representation (called OpST) to effectively re- move the empty regions as well as maintain a relatively large unit block size so as to reduce the portion of boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A detailed description of our algorithm can be found in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We use a 2D example to demonstrate our approach, as illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, (1) we par- tition the data into many small unit blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (2) For each unit block, we use the dynamic programming method to initiate an array BS to save the dimension/size of the maximum square whose bottom-right corner is that unit block (line 6, which will be discussed in the next paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (3) We extract the sub-blocks (composed of multiple unit blocks) from the original data according to the sizes saved in BS (lines 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (4) Since the original data will be changed after the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 7: A 2D example of our proposed OpST approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The sub-blocks are extracted according to our optimized sizes saved in BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', a 2-by- 2 sub-block B0 is extracted according to BS1[2][1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' extraction, we need to partially update BS based on maxSide (lines 14, will be discussed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We loop (3) and (4) from the bottom-right corner to the top-left corner until the original data is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (5) After extracting all the sub-blocks, we put them into multiple 3D arrays (to be compressed) based on their sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that the sub-blocks with the same size will be merged into the same array for easy compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Algorithm 2: Proposed Optimized Sparse Tensor Method Input: Sparse 3D data S Output: multiple 4D array Dn 1 for each unit block b(x, y, z) do 2 if b(x, y, z) is non-empty then 3 if x is 0 or y is 0 or z is 0 then 4 BS(x, y, z) = 1 5 else 6 BS(x, y, z) = min(BS(x − 1, y, z), BS(x, y − 1, z), BS(x, y, z − 1), BS(x − 1, y − 1, z), BS(x, y − 1, z − 1), BS(x − 1, y, z − 1), BS(x − 1, y − 1, z − 1)) + 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' /* BS(x,y,z) is the dimension size of the maximum cube whose bottom right rear corner is the unit block with index (x,y,z) in the original data */ 7 maxSide = max(maxSide, BS(x, y, z)) 8 end 9 end 10 end 11 for each unit block b(x, y, z) do 12 if BS(x, y, z) ≥ 1 then 13 size = BS(x, y, z) Dsize ← S((x − size : x) ∗ blkSize, (y − size : y) ∗ blkSize, (z − size : z) ∗ blkSize) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' /* put the sub-block to the according to 4D array */ 14 b(x − size : x, y − size : y, z − size : z) ← empty BS(x − size : x, y − size : y, z − size : z) = 0 BS = updateBs(BS, x, y, z, maxSide) 15 end 16 end 17 return Dn When initializing the BS in step (2), we start with the b′[i][j] with i = 0 or j = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', on the top-left edge), where b′[·][·] are the unit blocks: if b′[i][j] is empty, we will set BS[i][j] to 0 otherwise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For the remaining unit blocks, if it is empty, BS[i][j] will be 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' otherwise, BS[i][j] will be set to 1 plus the minimum value among its three neighboring blocks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', upper block, left block, and upper-left block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In other words, we have BS[i][j] = 1+min(BS[i][j−1], BS[i− 1][j], BS[i−1][j−1]) for the 2D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For example, BS1[2][1] is 2 because all its upper-left neighbors are 1 (as shown in Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, both BS1[1][1] and BS2[1][2] can only reach 1 because one of their neighbors is set to 0, having no chance to form a sub-block with the size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Moreover, as mentioned in step (3), we need to update 5 Bo 0 1 1 0 1 0 0 1 1 1 2 0 0 1 0 0 0 1 2 0 0 0 0 0 0 0 BS2 BS3 BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (a) NaST(CR=245, PSNR=77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5dB) (b) OpST(CR=248, PSNR=78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='0dB) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 8: Visual comparison (one slice) of compression errors of two ap- proaches using SZ based on Nyx’s “baryon density” field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', z10’s fine level, 23% density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Brighter means higher compression error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The error bound is the relative error bound of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' BS after each extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, for each sub-block we extract, we have to set its corresponding values in BS to zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For instance, as shown in Figure 7, after we extract a 2-by-2 sub-block B0 at BS1[2][1], we need to set BS2[1][0], BS2[1][1], BS2[2][0], and BS2[2][1] to zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In addition, we also need to recalculate a part of BS (line 17 in Algorithm 2) because the extraction could influence other BS values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For example, we need to recalculate BS2[1][2] (marked in bold orange) after extracting B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that this update is a partial update as the BS values to be updated will be bounded by maxSide which is the dimension size of the largest cube in the dataset (line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Similar to NaST, in decompression, we will put the sub- blocks back to reconstruct the data based on the saved co- ordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that after our optimization, each sub-block size will be relatively large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 963 vs the original data size of 5123), the overhead of saving the coordinates of all the sub-blocks will be negligible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Finally, we show a visual comparison of the compression quality between NaST and OpST in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that both use SZ with the same error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Brighter means more errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We can observe that compared to the NaST method, OpST can significantly reduce the overall compression error, especially for the data points on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' It is worth noting that even with a lower error, our OpST can still pro- vide a higher compression ratio than NaST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This is because our proposed optimization will generate larger sub-blocks, which provide more information for prediction-based lossy compressors such as SZ to achieve better rate-distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A detailed evaluation will be shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='3 Adaptive k-D Tree for Medium-density Data The OpST approach proposed for low-density data, how- ever, has a high computation overhead, especially when the data is relatively dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This is because, on one hand, OpST needs to update BS based on maxSide for each extraction of a sub-block, while the larger the maxSide, the more values in BS that need to be updated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' on the other hand, maxSide is the dimension size of the largest non-empty cube in the dataset, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 9: 2D example of adaptive k-D tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Sub-block will be adaptively split to effectively remove empty regions and get bigger full sub-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' which is highly related to the density of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, the time complexity of OpST can be expressed as O(N 2 · d), where N is the unit block number and d is the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that here density describes how dense the data is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For exam- ple, the density of 77% means that 23% of the data is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Clearly, when the density of an AMR level is relatively high, using OpST will be relatively time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Algorithm 3: Dynamic k-D Tree Input: data block d, counts information Output: k-d tree 1 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='count ← counts information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2 if d is empty or d is full then 3 continue ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' /* stop splitting */ 4 else 5 if d is a cube then 6 split d equally into 8 oct-blocks: s1, · · · , s8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 7 get the counts c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='c8 for s1, · · · , s8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 8 find the maxDiff partition d1,d2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 9 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='left = AKDTree (d1, four ci of d1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 10 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='right = AKDTree (d2, four ci of d2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 11 else if d is a flat cuboid then 12 get the counts c1, · · · , c4 from counts information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 13 find the maxDiff partition d1, d2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 14 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='left = AKDTree (d1, two ci of d1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 15 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='right = AKDTree (d2, two ci of d2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 16 else if d is a slim cuboid then 17 get the counts c1, c2 from counts information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 18 split d along the largest dimension to get d1,d2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 19 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='left = AKDTree (d1, c1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 20 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='right = AKDTree (d2, c2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 21 end 22 return node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To address the above high overhead issue of OpST, we propose an adaptive k-d tree, called AKDTree, to remove empty regions and extract sub-blocks (containing multi- ple unit blocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' AKDTree has a lower time complexity of O( 1 3N · log N) (will be discussed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Figure 9 shows a simple 2D example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, (1) we partition the data into small unit blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (2) We use a tree to hierarchically represent the whole data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Each node in the tree is associated with a sub-block of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Moreover, each node stores the number of non-empty unit blocks in the sub-block associated with the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (3) For each node, we split its associated sub-block from the middle along one dimension to form 6 n[21[2] non-adaptive splittwo sub-blocks for its two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that we select one dimension which can maximize the difference of the num- bers of non-empty unit blocks of the two children (will be discussed in the next two paragraphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (4) We keep splitting a node until it has no empty unit block or itself is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (5) Once finishing the construction of the tree, we collect all the leaf nodes and send them to the compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that a non-empty leaf node does not have any empty unit block;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' otherwise, it will keep splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, a leaf node must be an empty or full node, as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The detailed algorithm is described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As mentioned in step (3), we are distributing the non- empty unit blocks unevenly to two children for each node because we attempt to get as many leaf nodes with large sub- block sizes as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' If we keep splitting sub-blocks in a fixed way, for instance, first split along the x-axis, second split along the y-axis, third split along the x-axis, fourth split along the y-axis, and so on, we will get a 2-by-2 sub-block for the node n[2][2] as shown in the dashed box, while its largest possible sub-block could be 4 by 2 as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 10: Example of adaptive splitting, different shapes will have a different number of choices for splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To select one of the dimensions to unevenly distribute its non-empty unit blocks to the two children, we now present our dynamic splitting approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We categorize nodes into three different types: “cube” nodes, “flat” nodes, and “slim” nodes, whose dimension ratios are 1:1:1, 2:2:1, 2:1:1, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' First of all, for the cube node d, we first divide it into eight oct-blocks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', s1, s2, · · · , s8 (as shown in Figure 10), each sized n 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Here n is the dimension size of the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Then, we can get the counts of non-empty unit blocks of the eight oct-blocks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', c1, c2, · · · , c8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' After that, we will decide along which dimension to split the cube node d based on the counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, we can calculate the following three differences: diffx = |c1 + c3 + c5 + c7 − c2 − c4 − c6 − c8|, diffy = |c1 + c2 + c5 + c6 − c3 − c4 − c7 − c8|, diffz = |c1 + c2 + c3 + c4 − c5 − c6 − c7 − c8|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Finally, we compare these three values and choose the dimension with the maximum difference to split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For exam- ple, if the maximum difference is diffz, we will split d along the z-axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', the pink 2D plane shown in Figure 10) and get two flat nodes d1 and d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For the flat nodes such as d1, we can reuse c1, · · · , c4 to decide whether to split d1 along the x-axis or y-axis by choosing the larger one among the following two differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' diffx = |c1 + c3 − c2 − c4|, diffy = |c1 + c2 − c3 − c4|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For the slim nodes such as d11, we simply split it along the x-axis to get two cube nodes s1 and s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', cube nodes→flat nodes→slim nodes) in step (3) will be looped until the node becomes a leaf node (empty or full).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Algorithm 4: SZ compression with SHE Input: multiple data block D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='n Output: compressed data S 1 quantCode Q, regreCoeff R, compressed data S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2 for each data block Di do 3 quantCode block qi, regreCoeff block ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 4 qi, ri = SZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='compress(Di);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 5 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='append(qi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='append(ri);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 7 end 8 S ← HuffmanEncode(Q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 9 S ← HuffmanEncode(R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' /* end compression */ 10 return S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that based on the above description, the counting process is required for every three nodes in each three path (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', only for the “cube” nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thanks to this dynamic splitting approach, we can lower the time complexity of the AKDTree algorithm to O � 1 3 · N · log N � , where N is the number of unit blocks while extracting as many relatively large sub-blocks without empty unit block as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In addition, after the dynamic splitting, we will have a series of sub-blocks with the same size but in different di- rections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 2:2:1, 2:1:2, 1:2:2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We will align the sub-blocks with the same size based on their splitting dimensions (in- stead of in-memory transposing them), merge them into an array, and feed multiple merged arrays to the compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='4 Shared Huffman encoding As mentioned in Section 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2, the OpST and AKDtree methods collect and linearize the data blocks with the same shape into a 4D array and send it to SZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, poor data locality/smoothness between non-neighbored data blocks can negatively impact prediction accuracy, re- sulting in low data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A potential solution could be compressing each data block individually using SZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' How- ever, this approach would result in low Huffman encoding efficiency because the entire dataset would be divided into many small blocks, requiring SZ to build a separate Huff- man tree for each of these small blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In other words, the original SZ method either requires predicting and encoding small blocks together (by forcing them merged into 4D ar- rays), leading to low prediction accuracy or predicting and encoding each small block separately, which results in high Huffman encoding overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Furthermore, even if the data blocks with the same shape can be compressed together, the data blocks with different shapes still need to be compressed separately, resulting in low Huffman encoding performance and a high time cost of launching SZ multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To this end, for OpST and AKDtree, we propose a shared Huffman encoding technique to predict data blocks sepa- rately while encoding them together using a single shared Huffman tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The detail is shown in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Each data block is first predicted and quantized separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Then, the quantization codes and regression coefficients of each data block are aggregated to build a shared Huffman tree and encoded at one time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This approach improves the prediction performance of SZ without introducing high time overhead to the encoding of SZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='4, compared to the original AKDtree, AKDtree with SHE can significantly reduce the overall compression error, especially for the data located at the boundary of data blocks, leading to significant 7 (a) d = 23 (b) d = 58 (c) d = 64 (d) d = 42 (e) d = 77 (f) d = 99 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 11: Compression performance comparison of GSP, OpST, and AKDTree on six datasets with different densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (a) AKDtree (CR=222, PSNR=78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5dB) (b) AKDtree + SHE (CR=231, PSNR=79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='6dB) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 12: Visual comparison (one slice) of compression errors of two approaches using SZ based on Nyx’s “baryon density” field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', z10’s fine level, 23% density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Brighter means higher compression error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The error bound is the relative error bound of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='8 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 13: Comparison of original AKDTree and AKDTree with SHE on Nyx’s “baryon density” field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', z10’s fine level, 23% density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' PSNR improvement, as shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Furthermore, the use of SHE reduces the number of Huffman trees needed for the data (since we do not need separate Huffman trees for different block shapes), thereby improving encoding effi- ciency and compression ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 Hybrid Compression Strategy In this section, we propose a solution to adaptively choose a best-fit compression strategy from our proposed OpST with SHE (OpST+), AKDTree with SHE (AKDTree+), and GSP based on the data characteristics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', data density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' According to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='3, OpST+ is more suitable for sparse (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', low-density) data, while AKDTree+ is designed to address the high time overhead of OpST+ when the density of data increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, there should be a data-density threshold to determine when to use OpST+ or AKDTree+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To decide the threshold T for switching between OpST+ and AKDTree+, we perform a series of experiments, as shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The figure shows that OpST+ and AKDTree+ have almost identical compression performance in terms of bit-rate and PSNR on all six datasets/levels (from different timesteps) with different densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Moreover, Figure 14 shows the time costs of OpST+ and AKDTree+ (excluding compression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The figure demonstrates that the time of AKDTree+ is relatively stable, while the time of OpST+ increases linearly with the increase of data density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Overall, the only criterion for selecting OpST or AKDTree is the time cost rather than the compression performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This is consistent with our previous design aim, that is, AKDTree is mainly designed to address the high time overhead issue of OpST+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Since OpST+ and AKDTree+ have a similar speed when the density is around 50%, we propose to choose T = 50 for choosing OpST+ or AKDTree+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' According to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1, GSP is designed to effectively handle the AMR level with high data density to prevent the negative impact of data partitioning without the use of SHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In contrast, the data partition methods such as AKDTree require small blocks to be compressed together without SHE, which can significantly decrease prediction accuracy, as shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, the negative impact on prediction accuracy caused by the partition can be elimi- nated by using SHE to compress each small block produced by the partition, which incurs little overhead, while GSP introduces significant overhead to the data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As shown in Figure 11, OpST+ and AKDTree+ outperform GSP across all the densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As a result, the improved partition strategies using SHE can be a viable alternative to GSP for all levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In summary, our proposed hybrid compression approach Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 14: Time overhead comparison of OpST and AKDTree on different datasets with different densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 8 KD+ OpST+ GSP 105 95 PSNR 85 75 0 1 2 3 4 5 bit-rate→KD+ OpST+ GSP 150 PSNR 130 110 0 1 2 3 4 5 bit-rate→-KD+ OpST+ GSP 165 PSNR 145 125 0 1 2 3 4 5 CR←-KD+ OpST+ OGSP 160 145 SNR 130 115 0 1 2 3 4 5 6 bit-rateKD+ OpST+ GSP 110 100 PSNR 90 80 0 1 2 3 4 bit-rateKD+ OpST+ GSP 115 105 PSNR 95 85 75 0 1 2 3 4 bit-rate-AKDTree AKDTree + SHE 105 95 PSNR 85 75 0 50 100 150 200 250 CR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 3 CPU time (ms) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 tree opst opst (theoretical) 0 20 30 40 50 60 70 Densityis described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 1) When the density is smaller than T = 50%, we use OpST+ to remove empty regions and then compress;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 2) When the density is larger than T = 50%, we use AKDTree+ to remove empty regions and then compress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 4 EXPERIMENTAL EVALUATION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1 Experimental Setup Test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Our evaluation mainly focuses on the AMReX framework [3], particularly the Nyx cosmology simula- tion [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Nyx is a state-of-the-art extreme-scale cosmology code using AMReX, which generates six fields including baryon density, dark matter density, temperature, and veloc- ities (x, y, and z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We use nine datasets generated by three real-world simulation runs with different numbers of AMR levels, simulating a region of 64 megaparsecs (Mpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For this data, Z is equal to the redshift, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' the displacement distant galaxies and celestial objects, as seen in Tab 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, the first run has two levels of refinement, with the coarse level of 2563 grids and the fine level of 5123 grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We’ve collected four timesteps with the finest level density from 23% to 64%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The second run has a maximum of four levels of refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' It was initially configured at the resolution of 1283 and gradually refined to 10243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This run collected three timesteps with the finest-level resolution of 2563 (two levels), 5123 (three levels), and 10243 (four levels), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The density of the finest level varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2% to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='003%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The third run has three levels of refinement with the grid sizes of 1283 at the coarsest level, 2563 at the inter- mediate level, and 5123 at the finest level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The density of the finest level ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='87% to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that the density of the finest level describes how much of the data in the dataset is at the highest resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' a higher density of the finest level means that more data is refined to the highest resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Usually, the data density is gradually increasing at the finest level, within a single run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Evaluation platform and compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The test platform is equipped with two 28-core Intel Xeon Gold 6238R processors and 384 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We use SZ3 [40], which is easy to (de)couple lossless encoding from SZ compression, due to its high modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' TABLE 1: Our tested datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Dataset # Levels Grid Size of Each Level (Fine to Coarse) Density of Each Level (Fine to Coarse) Nyx_Run1_Z10 2 512, 256 23%, 77% Nyx_Run1_Z5 2 512, 256 58%, 42% Nyx_Run1_Z2 2 512, 256 63%, 37% Nyx_Run2_T2 2 256, 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2%, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='8% Nyx_Run2_T3 3 512, 256, 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='02%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='56%, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='42% Nyx_Run2_T4 4 1024, 512, 256, 128 3E-5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='02%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2%, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='7% Nyx_Run3_Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 3 512, 256, 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='87%, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='88%, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='25% Nyx_Run3_Z1 3 512, 256, 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='90%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='70%, 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='40% Comparison baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As discussed in Section 2, we have three 1D or 3D comparison baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, (1) the 1D baseline (naive): each AMR level is compressed separately as a 1D array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (2) the 1D baseline (zMesh) [27]: we refer readers to Section 2 for more details about how the zMesh approach reorganize the AMR data for 1D compression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' and (3) the 3D baseline: Different AMR levels are unified to the same reso- lution for 3D compression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (4) the TAC baseline: each AMR level is compressed using OpST (without SHE), AKDTree (a) Run1_Z10 (finest-level density = 23%) (b) Run1_Z5 (finest-level density = 58%) (c) Run1_Z2 (finest-level density = 63%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 15: Rate-distortion comparison of our approach and baselines on the early time-step (Z10) to the late time-step (Z2) from Nyx run1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (without SHE), and GSP based on the data density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For more information, we refer readers to [41] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2 Evaluation Metrics We will evaluate the compression performance based on the following metrics: (1) compression ratio or bit-rate (generic), (2) distortion quality (generic), (3) compression throughput (generic), (4) rate-distortion (generic), (5) power spectrum (cosmology specific), (6) Halo finder (cosmology specific).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Metric 1: To evaluate the size reduction as a result of the compression, we use the compression ratio, defined as the ratio of the original data size compared to the compressed data size, or bit-rate (bits/value), representing the amortized storage cost of each value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For single/double floating-point data, the bit-rate is 32/64 bits per value before compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The compression ratio and bit-rate have a mathematical re- lationship as their multiplication is 32/64 so that a lower bit rate means a higher compression ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Metric 2: Distortion is another important metric used to evaluate lossy compression quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We use the peak signal- to-noise ratio (PSNR) to measure the distortion quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' PSNR = 20 · log10 (RX) − 10 · log10 ��N i=1 e2 i /N � , where ei is the difference between the original and decom- pressed values for the point i, N is the number of points, and RX is the value range of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Higher PSNR less error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Metric 3: (De)compression throughputs are critical to im- proving the I/O performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We calculate the throughput based on the original data size and (de)compression time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Metric 4: Similar to prior work [14], [15], [42], [32], [20], [43], [35], we plot the rate-distortion curve to compare the 9 naive1D --zMesh --3D TAC -O-TAC+ 105 95 PSNR 85 75 0 2 4 6 bit-ratenaive1D -zMesh --3D --TAC -TAC+ 150 PSNR 130 110 0 2 4 6 bit-rate-naivelD -O-zMesh --3D TAC -O-TAC+ 170 PSNR 150 130 0 2 4 6 bit-rate(a) Run2_T2 (finest-level density = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2%) (b) Run2_T3 (finest-level density = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='02%) (c) Run2_T4 (finest-level density = 3E-5) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 16: Rate-distortion comparison of TAC+ (top-left) and baselines on different time-steps from run2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' distortion quality with the same bit-rate, for a fair compar- ison between different compression approaches, taking into account diverse compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Metric 5: Matter distribution in the Universe has evolved to form astrophysical structures on different physical scales, from planets to larger structures such as superclusters and galaxy filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The two-point correlation function ξ(r), which gives the excess probability of finding a galaxy at a certain distance r from another galaxy, statistically de- scribes the amount of the Universe at each physical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The Fourier transform of ξ(r) is called the matter power spectrum P(k), where k is the comoving wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The matter power spectrum describes how much structure exists at each physical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We run the power spectrum on the baryon density field by using a cosmology analysis tool called Gimlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We compare the power spectrum p′(k) of decompressed data with the original p(k) and accept a max- imum relative error within 1% for all k < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Metric 6: Halo finder aims to find the halos (over- densities) in the dark matter distribution and output the positions, the number of cells, and the mass for each halo it finds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, the halo-finder algo- rithm [44] searches for the halos from all the simulated data, with the following two criteria: (1) the mass of a data point must be greater than a threshold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='66× of the average mass of the whole dataset) to become a halo cell candi- date [20], [43], [45], and (2) there must be enough halo cell candidates in a certain area to form a halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For decompressed data, some of the information (mass and cells of halos) can be distorted from the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' (a) Run3_Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 (density for each level: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='87%, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='88%, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='25% ) (b) Run3_Z1 (density for each level: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='90%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='70%, 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='40%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 17: Rate-distortion comparison of our approach and baselines on different time-steps from Nyx run3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='3 Evaluation on Rate-distortion We first evaluate the rate-distortion of our proposed com- pression approach and compare it with the baselines on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As shown in Figure 15, 16 and 17, our new approach with SHE, TAC+ (represented by the pink curve) yields better performance than the original TAC (the yellow curve) without SHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Also, for the 1D baseline, our approach (top-left curve) outperforms the 1D baseline across all 8 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Furthermore, the performance of our approach is more stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', smoother curve) than the 1D baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We can also find that zMesh is slightly worse than the 1D baseline on our tested data as shown in Figure 15, which will be explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For the 3D baseline, we observe that our solution has much better performance when the finest level has a rel- atively low density or the decompressed data has a high PSNR, as shown in Figure 15, 16 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, our approach cannot dominate the 3D baseline as shown in Figure 15a, 15b, and 16a, when the following criteria are satisfied: (1) the AMR data has only two levels of refinement, (2) the finest level has a relatively low density, and (3) the decompressed data has low PSNR/bitrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We also find that as the finest level density increases, the bitrate threshold where the 3D baseline performs better than TAC+ is increasing, while the advantage of the 3D baseline over TAC+ is decreasing, as shown in Figure 15 and 16a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Specifically, in Figure 16a (the finest level density is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2%), TAC+ outperforms the 1D baseline when the bit- rate is above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' in Figure 15a (the finest level density is 23%), the intersection is the bit-rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' as the finest level density continues to grow up to 58% in Figure 15c, TAC+ is slightly worse than the 3D baseline until the bit-rate is larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' when the finest level density reaches 63% in Figure 15c, TAC+ and the 3D baseline yield (almost) the same performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In the next section, we will discuss how the number of AMR levels, the density of the finest level, and the bit- rate/PSNR of decompressed data affect the performance of the 3D baseline and TAC+ in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 10 naiveiD 3D TAC TAC+ 90 PSNR 80 70 60 0 2 4 6 bit-rate-naivelD 3D TAC TAC+ 100 PSNR 85 70 0 2 4 6 bit-rate←-naivelD 3D TAC TAC+ 115 100 PSNR 85 70 0 2 4 bit-rate→-naiveD 3D TAC TAC+ 140 PSNR 125 110 0 2 4 6 bit-rate→naivelD 3D TAC TAC+ 130 PSNR 115 100 0 2 4 6 bit-rate(a) Tree-based AMR data (b) Patch-based AMR data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 18: An example of how the 1D baseline, zMesh, and original z-order reorder a simple 2D AMR data without and with redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Orange: coarse level, blue: fine level, red: redundant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='4 Discussion on Comparison with Baselines On compression, zMesh is meant to improve the smoothness of the patch-based AMR datasets by taking advantage of the data redundancy between each AMR level (as described in the introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, zMesh cannot improve the smooth- ness if there is no data redundancy in the tree-structured AMR datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', our tested datasets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' A simple example is used to illustrate this in Figure 18b, where the finer-level data has higher values because a grid will be refined only if its value is larger than a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' For block-based AMR, when a grid needs to be refined because of its high value, the value will still remain in the level, resulting in a redundant value saved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', the red 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' If one uses the original z-ordering to traverse the data level-by-level (shown in Figure 18b), the reordered data will have three significant value changes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', from 2 to 8, from 8 to 1, and from 1 to 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' To solve this issue, zMesh traverses the two AMR levels together based on the layout of the 2D array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The reordered data are “1-2-8-9-8-7-8-1”, which only has two significant value changes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', from 2 to 8 and from 8 to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, zMesh can improve the smoothness of patch-based AMR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, as shown in Figure 18a, for tree-structured AMR data (without saving a redundant “8”), compared to the 1D baseline that compresses each level separately, zMesh introduces two significant data changes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', from 2 to 9 and from 8 to 1) as it traverses between two AMR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This explains why zMesh is slightly worse than the 1D baseline on our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' When considering the 3D baseline, we observe that it works slightly better than TAC+ in the following circum- stances: (1) the AMR data has only two levels of refinement, (2) the decompressed data has a low PSNR/bit-rate, and (3) the finest level of the data has a relatively low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We also observe that (4) when the finest level density increases, the bit-rate intersection point where the 3D baseline per- forms better than TAC+ shifts to a higher value, (5) while the advantage of the 3D baseline over TAC+ becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We now explain each observation point individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='4, the main disadvantage of the 3D baseline is the redundant data generated by the up-sampling process, which becomes more significant for the AMR data with more than two levels of refinement, resulting in poor compression performance (point 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In contrast, the 3D base- line has better data locality and smoothness over TAC+ because it compresses the entire dataset together without partitioning the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This leads to better compressibility for the 3D baseline, resulting in faster bitrate reduction and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 19: Bit-rates with different error bounds using SZ lossy compression for fine and coarse levels on Run1_Z2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' slower PSNR degradation when the error bound increases, compared to TAC+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As a result, the 3D baseline performs better at a low bit-rate (point 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, when the finest level density increases, the smoothness advantage of the 3D baseline will decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This is because TAC+ will have more data located in the finest level, and the partition overhead of TAC+ is lower at the finest level compared to coarser levels due to the larger (unit) block size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This explains why the 3D baseline only outper- forms TAC+ for AMR datasets with low density (points 3 & 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Also, note that for the dataset with low density in the finest level, the overhead of redundant up-sampled data will be high for the 3D baseline, especially for high bit- rates (because the 3D baseline only has better smoothness in low bit-rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, the compression performance of the 3D baseline will decrease rapidly as we aim for relatively high PSNRs/bit-rates, leading to a smaller intersection bit- rate between the 3D baseline and TAC+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Clearly, as the finest level density increases, the overhead of the redundant data will be smaller, resulting in a larger intersection bit-rate between the 3D baseline and TAC+ (point 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 Post-analysis Quality with Adaptive Error Bound When factoring level-wise compression, our approach can apply different error bounds to different AMR levels based on (1) the post-analysis metrics, (2) the up-sampling rates of coarse levels, and (3) the rate-distortion trade-off between different AMR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We now evaluate our approach with the two cosmology-specific post-analysis metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', power spectrum and halo finder) to demonstrate the benefit of the adaptive error bound method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We choose the dataset run1- Z2 for evaluation because TAC+ has a similar performance as the 3D baseline on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Figure 19 shows the motivation of performing rate- distortion trade-off between different AMR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As the error bounds for the fine and coarse levels increase, their bit rates will converge to a similar value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This means that when the error bound is relatively large, the reduction in data size will be insignificant compared to the compression error increment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', the slopes of both curves are very small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Therefore, we can say that when the error is large, it is not worth trading data quality for size reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Power Spectrum Figure 20a shows that under the (al- most) same compression ratio, TAC+ (with the uniform er- ror bound) has a better power-spectrum error compared to the 3D baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Also note that TAC+ yields nearly lossless power-spectrum distortion (less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='01%) under the com- pression ratio of 42×, as shown in Figure 20a’s gray curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Now, let us follow the three steps mentioned at the begin- ning of this section to adjust the error bound for each AMR 11 2 9 8 8 2 9 8 78 8 82 9 8 8 1 7 8 128987 28Fine lyl Coarse lyl 2 Bitrate 0 0 1E+09 2E+09 3E+09 4E+09 5E+09 ABS error boundTABLE 2: Overall compression/decompression throughput (MB/s) of different approaches with different absolute error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' EB_abs Run1_Z10 Run1_Z2 Run2_T2 Run2_T4 Run3_Z1 1D 3D TAC TAC+ 1D 3D TAC TAC+ 1D 3D TAC TAC+ 1D 3D TAC TAC+ 1D 3D TAC TAC+ 1E+08 51 37 79 77 51 79 85 84 71 15 71 66 49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='32 24 24 49 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='6 63 63 1E+09 81 48 94 93 55 92 103 97 79 21 85 79 53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='39 26 26 53 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 78 75 1E+10 89 53 107 102 58 100 111 104 86 23 98 87 84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='41 28 27 57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1 91 84 (a) Power spectrum error under 1% limit (b) Power spectrum error under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='01% limit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 20: Power spectrum error (in relative) of the 3D baseline and TAC+ (uniform error bound) and TAC+ (adaptive error bound) on baryon density field on run1-Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The red and blue dashed line is the 1% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='01% limit of acceptable power spectrum error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' First, the post-analysis metric–power spectrum— needs to be run on the uniform-resolution data and focuses on the global quality of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, the ideal error-bound configuration/ratio for the fine and coarse levels on the uniform-resolution data would be 1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As aforementioned, the coarse level of the AMR dataset needs to be up-sampled to uniform the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As a result, the compression error of the coarse level will be up- sampled as well, resulting in more error in the post-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, we then need to give the coarse level a smaller error bound based on the up-sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Here the up-sample rate for Z2’s coarse level is 23, leading to an ideal error-bound ratio of the fine and coarse levels changed to 8:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Finally, this 8:1 ratio needs to be adjusted based on the rate-distortion trade-off as aforementioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' As shown in Fig- ure 19, when using the error-bound ratio of 8:1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', 4E+9 for the fine level and 5E+8 for the coarse level), the error bound of the fine level is too large, resulting in an ineffective rate-distortion trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Thus, we can balance two levels by increasing the error bound for the coarse level (to gain compression ratio) and decreasing the error bound for the fine level (to add compression error), which can achieve an overall rate-distortion benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Based on our experiments, we adjust the error-bound ratio from 8:1 to 3:1, As shown in Fig- ure 20b, TAC+ with adaptive error bound can significantly improve the compression ratio compared with uniform error bound under similar power spectrum error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Halo finer We evaluate the mass change, and the number of cells change for the three largest halos identified using the 3D baseline, TAC+ (with uniform error bound), and TAC+ (with adaptive error bound), as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We can see that TAC+ with uniform error bound produces better halo- finer analysis quality than the 3D baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Similar to the error-bound configuration analysis done for the power spectrum, let us now adjust the error-bound ratio between the fine and coarse levels for halo finder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The TABLE 3: Halo finder analysis with different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' CR Avg Rel Mass Diff Avg Rel Cells Diff 3D baseline 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='7E-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2E-03 TAC+ (1:1) 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2E-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1E-03 TAC+ (2:1) 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='1E-04 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='0E-04 halo-finer analysis also requires uniform-resolution data as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' However, different from the power-spectrum anal- ysis, the halo-finder analysis focuses more on high-value points at the fine level, since only high-value data points qualify as halo candidates, as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Note that this does not mean we can directly discard the coarse- level data with small values as they still contribute to the average value of the dataset, which is also an important parameter for the halo finder [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Therefore, we set the ideal error-bound ratio to 1:2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=', fine level vs coarse level) for the uniform-resolution data based on our massive experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' After that, considering the up-sampling rate of 23, the error- bounded ratio is changed to 4:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Finally, we adjust the ratio to 2:1 based on the rate-distortion trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Table 3 shows that TAC+ with adaptive error bound obtains the minimal differences of the mass and cell numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='6 Evaluation on Time Overhead We evaluate the overall throughput (including preprocess- ing and (de)compression) on the datasets with different error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Compared to the 3D baseline, the throughput of TAC+ is up to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='8× higher on the Run3 dataset, 75× higher on the Run2 datasets, and 2× higher on the Run1 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' This is because the Run2 and Run3 datasets have a lower density than the Run1 datasets at the finest level, resulting in a higher overhead of redundant data for the 3D baseline, which is consistent with our discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We note that TAC+ performs better than the 1D baseline on all the tested datasets except the T4 datasets, due to its relatively long data-partition time compared to the total time on the small-sized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' TAC+ has almost the same throughput as TAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' The very slight decrease is because TAC+ compresses the data in a more fine-grained manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We exclude zMesh in this evaluation as it is theoretically slower than the 1D baseline due to the extra z-ordering and provides worse rate-distortion according to our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 5 CONCLUSION AND FUTURE WORK In conclusion, this paper leverages 3D compression for AMR data on a systemic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' We propose three pre-processing strategies that can adapt based on the density of each AMR level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Our approach improves the compression ratio com- pared to the state-of-the-art approach by up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='9× under the same data quality loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' With our level-wised compres- sion approach, we are able to tune the error-bound ratio of fine and coarse levels to be 3:1 and 2:1 for better power- spectrum and halo-finder analyses, respectively, under the same compression ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' In future work, we will apply TAC+ to more AMR simulations and 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Byna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Krishnamoorthy, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Tao, “Characterizing impacts of storage faults on hpc applications: A methodology and insights,” in 2021 IEEE International Conference on Cluster Computing (CLUSTER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 409–420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Daoce Wang is a PhD student in Intelligent Sys- tems Engineering at Indiana University Bloom- ington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He received his bachelor’s degree in Computer Science from University of Electronic Science and Technology of China in 2018 and his master’s degree in Computer Science from University of Florida in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' His research in- terests include scientific data reduction, AMR simulations, and parallel algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Jesus Pulido is a research scientist in the Data Science at Scale Team at Los Alamos National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Pulido received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' in Com- puter Science at University of California, Davis in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Pulido specializes in data analysis, data reduction, visualization, high performance com- puting, wavelets and multi-resolution methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He has experience in applications of image sen- sors, astronomy, turbulence and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Pascal Grosset is a scientist in the Data Sci- ence at Scale team at Los Alamos National Lab- oratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' His primary research interests are large- scale data analysis and visualization, and data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' in Computing: Graphics and Visualization from the University of Utah in 2016 where his research focused on large scale visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Sian Jin is currently working toward the PhD degree majoring in computer engineering at In- diana University Bloomington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' His research in- terests include High Performance Computing, Compression Algorithms, Artificial Neural Net- works, and Parallel Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He has pub- lished several papers in major journals and inter- national conferences including the SC, PPoPP, VLDB, HPDC, IPDPS, and ICDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Jiannan Tian is a PhD candidate in Intelli- gent Systems Engineering at Indiana Univer- sity Bloomington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' His research interests in- clude lossy compression for scientific data and error analysis, and GPU-centric computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' His ongoing project including developing GPU- accelerated compression algorithm and system design optimization of lossy compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Kai Zhao is an assistant professor of computer science at University of Alabama at Birming- ham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' in computer sci- ence from University of California, Riverside in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He received his bachelor’s degree from Peking University in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' His research interests include high-performance computing, scientific data management and reduction, and resilient machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' James P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Ahrens received the BS degree in computer science from the University of Massachusetts at Amherst, Amherst, Mas- sachusetts, in 1989, and the PhD degree in com- puter science from the University of Washington, Seattle, Washington, in 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He is a senior scientist with Applied Computer Science Group, Los Alamos National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' His primary re- search interests include visualization, computer graphics, data science, and parallel systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He is author of more than 100 peer reviewed papers and the founder/design lead of ParaView, an open-source visualization tool designed to handle extremely large data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' ParaView is broadly used for scientific visualization and is in use at supercomputing and scien- tific centers worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He is the chair of the IEEE Computer Society Technical Committee on Visualization and Graphics (VGTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Dingwen Tao is an associate professor at In- diana University Bloomington, where he directs the High-Performance Data Analytics and Com- puting Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' in Computer Science from University of California, Riverside in 2018 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' in Mathematics from University of Science and Technology of China in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He is the recipient of various awards including NSF CAREER Award (2023), Amazon Research Award (2022), Meta Research Award (2022), R&D100 Awards Winner (2021), IEEE Computer Society TCHPC Early Career Researchers Award for Excellence in HPC (2020), NSF CRII Award (2020), and IEEE CLUSTER Best Paper Award (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He is serving on the Technical Review Board of IEEE Transactions on Parallel and Distributed Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He served as the Program Co-chair of 2021 IEEE International Conference on Scalable Computing and Communications and International Workshops on Big Data Reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' He is also a reviewer, program committee member, or session chair of major HPC venues, such as SC, HPDC, ICS, IPDPS, CLUSTER, ICPP, CCGrid, and HiPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' Email: ditao@iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAzT4oBgHgl3EQf8P5i/content/2301.01901v1.pdf'} diff --git a/zNE3T4oBgHgl3EQfPgnE/content/2301.04404v1.pdf b/zNE3T4oBgHgl3EQfPgnE/content/2301.04404v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ab6f0f74b3817598262ee656f51781019f4e637c --- /dev/null +++ b/zNE3T4oBgHgl3EQfPgnE/content/2301.04404v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:960a8a439b4bba206e8ce082b5d1a516f0da1d578c250b0e08819bc400212067 +size 1630959 diff --git a/zNFKT4oBgHgl3EQfMi2t/content/tmp_files/2301.11751v1.pdf.txt b/zNFKT4oBgHgl3EQfMi2t/content/tmp_files/2301.11751v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a5bcf2ee582eceed9d80c802186729a1aa63b40 --- /dev/null +++ b/zNFKT4oBgHgl3EQfMi2t/content/tmp_files/2301.11751v1.pdf.txt @@ -0,0 +1,1409 @@ +arXiv:2301.11751v1 [math.AG] 27 Jan 2023 +MANIFOLDS WITH TRIVIAL CHERN CLASSES II: +MANIFOLDS ISOGENOUS TO A TORUS PRODUCT, +COFRAMED MANIFOLDS AND A QUESTION BY +BALDASSARRI +FABRIZIO CATANESE +Abstract. Motivated by a general question addressed by Mario Bal- +dassarri in 1956, we discuss the Pseudo-Abelian Varieties introduced by +Roth, and we introduce a first new notion, of Manifolds Isogenous to +a k-Torus Product: the latter have the last k Chern classes trivial in +rational cohomology and vanishing Chern numbers. +We show that in dimension 2 the latter class is the correct substitute +for some incorrect assertions by Enriques, Dantoni, Roth and Baldas- +sarri: these are the surfaces with KX nef and c2(X) = 0 ∈ H4(X, Z). +We observe in the last section, using a construction by Chad Schoen, +that a similar picture does not hold in such a simple way in higher +dimension. +We discuss then, as a class of solutions to Baldassarri’s question, +(manifolds isogenous to) projective (respectively: +K¨ahler) manifolds +whose tangent bundle (resp. cotangent bundle) has a trivial subbun- +dle. +The former class of ‘partially framed’ projective manifolds, that is, +whose tangent bundle has a trivial subbundle, consists, in the case where +KX is nef, of the Pseudo-Abelian varieties of Roth; while the latter class +of ‘partially co-framed’ projective manifolds is not yet understood; we +can however state some new results and formulate open questions and +conjectures. +In the course of the paper we address also the general case of com- +pact complex Manifolds, introducing the new notions of suspensions over +parallelizable Manifolds, and of twisted hyperelliptic Manifolds, and de- +scribe the known results under the K¨ahler assumption. +In memory of Mario Baldassarri (1920-1964). +Contents +1. +Introduction and history of the problem. +2 +2. +Surfaces with second Chern class equal to zero, and a Chern class +characterization of Hyperelliptic and Abelian surfaces +5 +3. +Pseudo Abelian Varieties and Varieties Isogenous to a k-Torus- +Product +9 +Date: January 30, 2023. +AMS Classification: 14F, 14K, 14C25 +. +1 + +2 +FABRIZIO CATANESE +4. +On the non K¨ahler case +13 +5. +Partially framed and co-framed manifolds +16 +6. +Mathematical and Historical comments on Baldassarri’s paper +[Bald56] and the questions it suggests +20 +7. +Manifolds with vanishing Chern numbers +24 +References +26 +1. Introduction and history of the problem. +The initial purpose of this article was to discuss and reformulate a question +coming from Mario Baldassarri ’s work [Bald56]. +In order to do this we first have to explain how the question arose, relying +on generalizations of some incorrect assertions concerning the classification +of algebraic surfaces (for which we present here the correct results); then, +we explain the state of the art, prove some new results, define new classes of +compact complex Manifolds, and pose some new questions, related to these +older questions. +Baldassarri’s paper [Bald56] aims at determining the manifolds whose last +Chern classes are zero in rational cohomology, namely ci(X) = 0 ∈ H2i(X, Q) +for i ≥ k + 1: but he made the false claim that these manifolds are exactly +the Pseudo-Abelian varieties introduced by Roth [Roth54]. +A first basic principle that Baldassarri and his precedessors missed to con- +sider (as the use of topology was not sufficiently established at the time) +was the following: +Remark 1.1. (Isogeny principle): If we have a finite unramified map f : +Z → X, then ci(Z) = 0 ∈ H2i(Z, Q) if and only if ci(X) = 0 ∈ H2i(X, Q). +Defining isogeny between manifolds as the equivalence relation generated +by the existence of such finite unramified maps, we see that the set of man- +ifolds which are solutions to Baldassarri’s question consists of a union of +isogeny classes. +Baldassarri’s problem has been solved in the extremal case where k = 0: +the characterization of the compact K¨ahler Manifolds with all Chern classes +zero in rational (or real) cohomology was solved in 1978, thanks to Yau’s +celebrated theorem [Yau78] on the existence of K¨ahler-Einstein metrics on +manifolds with c1(X) = 0 ∈ H2(X, Q). +It follows that these Manifolds +are the ones isogenous to complex tori (they are the so-called Hyperelliptic +Manifolds, quotients X = T/G of a complex torus by a finite group G acting +freely on T). +More precisely, we have: +Theorem 1.2. (Yau) A compact K¨ahler manifold X such that c1(X) = +0, c2(X) = 0, in H∗(X, R), is a Hyperelliptic manifold. + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.. 3 +What happens for k ≥ 1? +The first significant case is the case where n := dim(X) = 2, k = 1. +Already to describe this case we need some definition: a compact K¨ahler +Manifold X is said to be a torus product iff X ∼= T × Y , where T is a +complex torus of dimension ≥ 1. +Whereas one says that X is strongly isogenous to a torus product if +X = (T × Y )/G, where G acts on T, Y and freely by its diagonal action +on T × Y (we shall see later that, if X is not uniruled, and it is a compact +K¨ahler Manifold, the above two notions are equivalent). +The case of Roth’s Pseudo Abelian Varieties +is the case where X = +(T × Y )/G, X is projective, and moreover G acts on T freely (hence, as a +group of translations). +In the case n = 2, k = 1 we have a complete answer (see Theorem 2.6 for +more details): +Theorem 1.3. Let X be a compact K¨ahler surface with c2(X) = 0 ∈ +H4(X, Z). Then: +(1) if KX is nef, then X is strongly isogenous to a torus product, namely +X = (T ×Y )/G, where G acts freely via a diagonal action on T ×Y ; +(2) if X is a minimal surface, but KX is not nef, then X is a P1-bundle +over an elliptic curve E; +(3) if X is non minimal, then X is a blow up of a P1-bundle over a +curve C of genus g ≥ 2. +In the first two cases c1(X)2 = c2(X) = 0, while in the last case χ(X) = +1 − g, hence c1(X)2 = K2 +X < 0. +If c1(X) = 0 ∈ H2(X, R), then S is either a complex torus, or a hyperelliptic +surface. +If c1(X) = 0 ∈ Pic(X), then X is a complex torus. +There are three features in our above theorem for n = 2: +• (I): in case (1) where KX is nef, X need not be a Pseudo-Abelian +variety; +• (II): since χ(OX) ≤ 0, then X always possesses a non zero holomor- +phic 1-form ω ∈ H0(Ω1 +X); +• (III): if moreover X is minimal, then ω is everywhere non vanishing; +• (IV): if X is minimal, then X is birationally covered by a a trivial +family of complex tori, that is, there exists a dominant rational map +ψ : T × Y ��� X. +The first feature (I) contradicts results of Dantoni [Dant43] and Enriques +[Enr05b], (indeed the error of Enriques is also reproduced in the classifica- +tion theorem of Castelnuovo and Enriques [CastEn15]). These papers put +Baldassarri and Roth on the wrong track. +The second feature (II) is the one which, as we shall see, fails to hold true +in dimension n ≥ 3 as a consequence of vanishing of the top Chern class; + +4 +FABRIZIO CATANESE +while, if (II) holds true, since (−1)ncn(X) is the expected number of zeros +of a holomorphic 1-form, then (III) looks plausible. +If one takes (III) as an assumption, that there exists a holomorphic 1-form +without zeros, then Baldassarri’s claim holds at least in dimension 3, as +shown by [HS21b] (see also a similar result in Theorem 5.3). +The failure of (I) for n ≥ 3 is due to work of Chad Schoen [Schoen88], which +shows that in dimension 3 there are simply connected manifolds, actually +Calabi Yau manifolds (KX trivial in Pic0(X)) with c3(X) = 0. +Concerning the failure of feature (IV) for n ≥ 3, our result here is that +the Schoen threefolds are not birationally covered by a family of isomorphic +Abelian surfaces, see Proposition 7.1, and that they do not admit a fibration +onto a surface with general fibres isomorphic to a fixed elliptic curve E, see +Proposition 7.2. Hence that they are also birationally far away from being +isogenous to a torus product. +What’s left is then to proceed essentially assuming feature (III). +More precisely, an easy class of solutions to Baldassarri’s question is provided +by the manifolds X which are isogenous to a partially framed or co-framed +manifold Z. +Our definition here is that Z is partially (tangentially) framed if the tangent +bundle ΘZ admits a maximal trivial subbundle Ok +X, with k > 0. +Whereas Z is said to be partially co-framed (cotangentially framed) if the +cotangent bundle Ω1 +Z admits such a maximal trivial subbundle Ok +X, with +k > 0. +The first definition brings back into play Roth’s Pseudo-Abelian Varieties, +see Theorem 5.1, which contains also more general results: +Theorem 1.4. A tangentially k-framed projective Manifold with KX nef is +a pseudo-Abelian variety, in particular ΘX ∼= Ok +X ⊕ F. +The previous classical Theorem was essentially proven by Roth and, thanks +to the work of Fujiki [Fuj78] and Lieberman [Li78], the class of partially +(tangentially) framed projective manifolds is understood. We also observe, +using results of [Li78], [AMN12] (which also contains an excellent survey of +the theory of framed K¨ahler manifolds), that the picture becomes more com- +plicated if we enlarge our consideration to the wider realm of cKM (compact +K¨ahler Manifolds), where more complicated constructions such as Seifert fi- +brations, principal torus bundles and torus suspensions enter the picture +(see section 4). +We also consider in Section 3 the non K¨ahler case, where we consider a +more general notion of suspension over a parallelizable Manifold for which +a splitting of the type ΘX ∼= Ok +X ⊕ F holds true. +And we consider all the Manifolds isogenous to a parallelizable Manifold, +we call them twisted Hyperelliptic Manifolds since they also enjoy the +property of having all the real Chern classes equal to zero (but a converse +result is missing). + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.. 5 +Passing to the consideration of the second class of partially co-framed man- +ifolds, we see how this interesting class is more mysterious and presents +some intriguing questions (see the discussion following Proposition 5.2 and +Theorem 5.3). +A new result which we prove here is the following +Theorem 1.5. (i) Assume that X is a k-coframed compact K¨ahler manifold, +and that q := h0(Ω1 +X) = k: then the Albanese map aX is a differentiable fibre +bundle. +(ii) If X is projective, k = q = n − 1, and KX is nef, then X is a pseudo- +Abelian variety. +In the projective case with KX nef we do not have yet more examples of +k-coframed Manifolds than the class of the Pseudo-Abelian varieties. +In fact, we ask two general questions such that a positive answer to both of +them would imply that we have nothing more than just the Pseudo-Abelian +varieties. +Observe that for k- framed Manifolds and for k-coframed Manifolds the last +k Chern classes are zero (in the Chow ring if the Manifolds are projective). +More generally, for the Manifolds isogenous to a k-Torus product it follows +from Remark 1.1 not only that the last k Chern classes are zero in rational +cohomology, but also that all the Chern numbers of X are equal to zero. +In the last section we briefly discuss Manifolds with KX nef and with van- +ishing Chern numbers. +For n = 2, as we already illustrated, there are only manifolds isogenous to a +torus product; but, in dimension n = 3, in spite of a quite recent result by +Hao-Schreieder [HS21a], describing threefolds with c1c2 = 0, and Kodaira +dimension 2, as being birationally isogenous to a torus product, we have also +the Schoen threefolds which do not exhibit this feature. +2. Surfaces with second Chern class equal to zero, and a +Chern class characterization of Hyperelliptic and Abelian +surfaces +The main goal of this section is to establish Theorem 1.3 mentioned in the +Introduction: this is done rerunning the proof of the classification of surfaces. +The classification theorem by Castelnuovo and Enriques [CastEn15], was ex- +tended by Kodaira [Kod68], and a crucial result is the following (see [Bea78], +Chapter 6, Theorem VI.13, Theorem 1.4 of [CatLi19], and especially the cru- +cial Theorem of [Cat22]): +Theorem 2.1. Let S be a compact smooth complex surface, minimal in the +strong sense that KS is nef, and such that χ(S) = 0 (which implies that +K2 +S = 0 and the topological Euler number e(S) = c2(S) = 0). Equivalently, +assume that KS is nef, and that e(S) = c2(S) = 0. + +6 +FABRIZIO CATANESE +Then X is strongly isogenous to a torus product, namely X = (T × Y )/G, +where T is a torus of positive dimension (1 or 2) and G acts freely via a +diagonal action on T × Y . +More precisely, either +1) pg(S) = 1, q(S) = 2, and S is a complex torus A (a hyperelliptic +surface of grade 1), or +2) pg(S) = p, q(S) = p + 1 and S is isogenous to an elliptic product, i.e. S +is the quotient (C1 × C2)/G of a product of curves of genera +g1 := g(C1) = 1, g2 := g(C2) ≥ 1, +by a free action of a finite group of product type (that is, G acts faithfully +on C1, C2 and we take the diagonal action g(x, y) := (gx, gy)), such that, if +we denote by g′ +j = g(Cj/G), then +g′ +1 + g′ +2 = p + 1. +Case 2) bifurcates into two subcases: +(2.1,p) g′ +1 = 1 (hence G acts on C1 by translations), C2/G has genus p, and +we assume 1, for p = 1, that g2 ≥ 2; or +(2.0,p) g′ +1 = 0 (hence C1/G ∼= P1), C2/G has genus p + 1 = q(S), and +we assume 2, for p = 0, that g2 ≥ 2; here the image of Albanese map +α : S → Alb(S) equals C2/G ⊂ Alb(S). +Then case +(2.1,0) with g2 = 1 is the case where S is a properly hyperelliptic (bielliptic) +surface (a hyperelliptic surface of grade ≥ 2): +S = (E1 ×C2)/G, where E1, C2 are elliptic curves, and G acts via an action +of product type, such that G acts on E1 via translations, and faithfully on +C2 with C2/G ∼= P1. +In this case all the fibres of the Albanese map are isomorphic to C2, P12(S) = +1, and S admits also an elliptic fibration ψ : S → C2/G ∼= P1. +In the other cases (2.0,p), (2.1,p), for p ≥ 1, (2.1,0) with g2 ≥ 2, S is +isogenous to a higher genus elliptic product, this means that C2 has +genus g2 ≥ 2. Here S is properly elliptic and P12(S) ≥ 2. +The cases are distinguished mainly by the geometric genus pg(S) = p. +In the case of the torus and of the hyperelliptic surfaces KS is numerically +equivalent to zero, whereas in the other cases KS is not numerically equiva- +lent to zero. +Moreover, the three cases are also distinguished (notice that c1(S) is the +class of the divisor −KS) by +• KS = 0 ∈ Pic(S) for the case 1) of a complex torus, +• c1(S) = 0 ∈ H2(S, Z) but KS ̸= 0 ∈ Pic(S) in the case of properly +hyperelliptic surfaces, +1to exclude that we are in case 1) +2to exclude that we are in case (2.1,0) + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.. 7 +• c1(S) ̸= 0 ∈ H2(S, Q) in the other cases where S is isogenous to a +higher genus elliptic product. +Proof. The part which is not contained in the cited sources concerns how to +distinguish the several cases. +Everything is straightforward, except the assertion that for hyperelliptic +surfaces (case (2.1,0) with g1 = 1)) the first integral Chern class is zero: +but this is a special case of our result on Bagnera de Franchis manifolds, +Theorem 3.1 of [Cat23]. +□ +The following can be readily checked: +Proposition 2.2. In case 1) A acts transitively and freely on A, so Aut0(S) +has dimension 2. +In case (2.1,p), Aut0(S) has dimension 1: C1 = E1 acts on S, transitively +on the orbit closures, but with stabilizer H ⊂ G ⊂ E1 for the classes of +points (x, y) ∈ E1 × C2 such that Hy = y. +Finally, in case (2.0,p ), Aut0(S) has dimension 0 : there is no action of +C1 on S, but the general fibres of the Albanese map α : S → C2/G are +isomorphic to C1 (a finite number shall only be isogenous to C1). +With a weaker notion of minimality we have a counterexample to the pre- +vious Theorem 2.1, as we shall now see. +Proposition 2.3. If the smooth surface satisfies c2 +1 = c2 = 0, that is K2 +S = +e(S) = 0, then S is minimal. +Proof. By the Noether’s formula, from K2 +S = 0 and e(S) = c2(S) = 0 follows +χ(OS) = 0. +And χ(OS) is a birational invariant. +If S is not minimal, then S is the blow-up of a minimal surface S′ with +e(S′) < 0. By Castelnuovo’s theorem S′ is a ruled surface with q(S′) ≥ 2. +But then 0 > χ(OS′) = χ(OS) = 0, a contradiction. +□ +Corollary 2.1. Consider the minimal surfaces S with Chern numbers c2 +1 = +c2 = 0, that is with K2 +S = e(S) = 0. +If KS is nef, S is isogenous to a product Y ×A, with a torus A of dimension +≥ 1. +If KS is not nef, then S is a P1-bundle over an elliptic curve. +Proof. By Theorem 2.1 there remains only to consider the case where KS +is not nef, hence S is ruled. +Therefore, since the case of S = P2 yields +e(S) = 3, we must have a P1-bundle over a curve C. In this case, since +0 = e(S) = 4(1 − g(C)), we get that C has genus g(C) = 1. +□ +Remark 2.4. If we take an elliptic curve B, and a vector bundle V of rank +2 on B, say V = L ⊕ M, where deg(L) = deg(M), then P(V ) is minimal, + +8 +FABRIZIO CATANESE +but the group of automorphisms of P(V ) consist of C∗ for general choice of +L, M (see [Mar71], also Theorem 7.3 of [CatLiu21] ). +Hence the action is not transitive on the orbit closures, because the two +sections of P(V ) are left invariant by the automorphism group. +For completeness we show that: +Proposition 2.5. Surfaces in the class (2.0,0) do exist. +Proof. Let G := (Z/2)3, and make it first act on an elliptic curve C1 as the +group of transformations +z �→ ±z + η, 2η = 0, +so that G has generators η1, η2, ǫ, where ǫ(z) = −z. +To get a second action on C2 such that C2/G =: E2 is an elliptic curve, we +take E2 to be an elliptic curve, B = {x1, x2} a branch set, so that +π1(E2 \ B) = ⟨α, β, γ1, γ2|γ1γ2 = [α, β]⟩, +hence H1(E2 \ B) = Zα ⊕ Zβ ⊕ Zγ1. +Define µ : H1(E2 \ B) → G by: +µ(α) = ǫ, µ(β) = η1, µ(γ1) = η2 ⇒ µ(γ2) = η2. +We want to prove that the product action of G on C1 × C2 is free. +To this purpose we observe that η2 has eight fixed points on C2, and, since +C2 has genus 5, E′ +2 := C2/⟨η2⟩ is an elliptic curve, so that E′ +2 → E2 is ´etale, +that is, G/⟨η2⟩ acts freely on E′ +2. The conclusion is that the only element +acting on C2 with fixed points is η2; since η2 atcs freely on C1, the product +action is free. +□ +We end this section observing that if S is a minimal surface with e(S) = 0, +then either S is a P1 bundle over an elliptic curve, or KS is nef. In the latter +case it must be K2 +S = 0, since for K2 +S < 0 S is ruled, hence KS is not nef, +and for K2 +S > 0 S is of general type, and then e(S) > 0. +The following is the correction of the theorem of Dantoni [Dant43]; it show +that if S is not a torus then the surface is ‘elliptic’ only in the weak sense that +it has a rational map with fibres elliptic curves, and not in the strong sense +that a torus of dimension at least 1 should act on S, as we saw in Proposition +2.2 and in the previous remark 2.4 (it was also stated as Theorem 1.3 in the +Introduction with only slightly different wording). +Theorem 2.6. A surface S with e(S) = 0 is either the blow up of a P1- +bundle over a curve of genus at least 2, or it is minimal, and then it is +either a complex torus, or a hyperelliptic surface, or it is isogenous to a +higher genus elliptic product, or it is a P1-bundle over an elliptic curve. +In the last three cases S contains a 1-dimensional family of isomorphic el- +liptic curves whose union is dense in S. +If the canonical divisor is numerically trivial, then S is either a complex +torus, or a hyperelliptic surface. + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.. 9 +3. Pseudo Abelian Varieties and Varieties Isogenous to a +k-Torus-Product +The first aim of this section is to show how to derive from Roth’s original +definition of the Pseudo Abelian Varieties the modern definition which we +adopt in this article; and how Roth’s definition leads, in the case of compact +K¨ahler Manifolds, to the notion of Seifert fibrations. +Later on we dwell on other notions, for instance we give a new definition, of +suspension over a parallelizable Manifold. +Leonard Roth [Roth54], [Roth55] defined the Psudo-Abelian varieties as +follows: +Definition 3.1. (Roth) +A projective variety X of dimension n is said to be Pseudo Abelian of order +k if +i) Aut0(X) contains a complex torus T, of maximal dimension = k, with +the property that its orbits are all of dimension k. +Recall in fact the following theorem by Fujiki and Lieberman [Fuj78], [Li78] +Theorem 3.1. If X is a compact K¨ahler manifold (or more generally in the +Fujiki class) then there is an exact sequence of groups +1 → L → Aut0(X) → TX → 1 +where L is a linear algebraic group, and TX is a complex torus. +The Lie Algebra of Aut0(X) is A := H0(ΘX), while the Lie Algebra of L +is the space L of vector fields which admit zeros (L is an ideal in the Lie +Algebra A). +If X is not uniruled, then L = {1}. +We explain now how to derive from Roth’s definition a simpler one. +We have an action T × X → X, and we denote by Tx the orbit of x, image +of T × {x}. +Hence the orbits Tx of T give a variety V of dimension n − k in the Hilbert +scheme H (Douady space) of X, and the restriction of the universal family +to V, φ : U → V, yields U which maps isomorphically to X. +In fact, +dim(U) = n = dim(X), and the identity of X factors as: +X → U → V × X → X, +where x ∈ X �→ (Tx, x). +Hence we may write: φ : X → V, and since the action of T, +a : T × X → X +commutes with the projection over V, and is effective, then the general fibre +of φ is isomorphic to T. +The other fibres are instead of the form T/G′, where G′ is a finite subgroup +of T. Since the general fibres are isomorphic, over an open set V′ of V we + +10 +FABRIZIO CATANESE +have a holomorphic bundle (for instance, as a consequence of Kuranishi’s +theorem). +Because of the action of T on the fibres the monodromy of the bundle +centralizes the group of translations hence the monodromy transformations +consist of translations, and we have a principal bundle. +If X is projective, then the monodromy is finite, hence we get a finite group +G ⊂ T. +The fibration is isotrivial, hence there exists a finite Galois base change +f : Y → V with group G such that the fibre product is birational to a +product +Y ×V X ∼ Y × T. +At each point of V the local monodromy G′ is a subgroup of G, hence the +fibre product Y ×V X yields an unramified covering of X ( since G′ yields +an unramified covering of the corresponding fibre). +Hence the fibre product Y ×V X is smooth, is birational to Y × T, and all +the fibres are isomorphic to T: therefore, Y ×V X ∼= Y ×T, compatibly with +the projection onto Y . +This motivates an equivalent definition, and some related definitions: +Definition 3.2. (I) A complex manifold X of dimension n is said to be a k- +Pseudo-Torus (or Pseudo-Torus of order k) if there is a torus T of dimension +k, a manifold Y , and a finite Abelian group G acting on T faithfully via +translations, acting faithfully on Y , such that the quotient of the product +action is isomorphic to X: +X = (Y × T)/G. +(II) If moreover X is projective, we shall say that X is a Pseudo-Abelian +Variety of order k in the strong sense. +(III) A compact K¨ahler Manifold X is said to be Seifert fibred, cf. [Li78], +if there is a finite abelian unramified covering Z → X with Abelian Galois +group G (hence X = Z/G) such that Z is a principal bundle Z → Y , with +fibre a (positive dimensional) complex torus T, and where the action of G +commutes with the action of T on Z. +(IV) A compact complex manifold X is called ([AMN12]) a suspension over +a complex torus T = Ck/Λ if there is a compact complex manifold Y and a +homomorphism ρ : Λ → Aut(Y ) such that +X = (Ck × Y )/Λ, λ(z, y) := (z + λ, ρ(λ)(y)). +Remark 3.2. (a) The reason for definition (III) is that if X is only a cKM, +and it has an action of a complex torus T, with all the orbits tori of the +same dimension, then the global monodromy is not necessarily finite. But +the local monodromies around the points corresponding to multiple fibres +are finite subgroups of T, and since we have a finite number of them, the +local monodromies generate a finite subgroup G of T. Associated to this +subgroup there is an covering Y → V which is a principal T-bundle. + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT..11 +(b) In the case of definition (IV) of a suspension over a torus, Ck acts on +X via translations on the first summand, but in general there is no complex +torus acting on X. +Indeed the Ck-orbits are the projections π(Ck × {y}), which are isomorphic +to Ck/Staby, hence they do not need be compact. +The suspension over a torus is a Pseudo-torus if and only if the subgroup +Im(ρ) ⊂ Aut(Y ) is finite. Equivalently, if and only if all orbits are tori +(which amounts to Staby being a subgroup of finite index in Λ for all y ∈ Y ): +since then Ker(ρ) has finite index. +In particular the suspension is Pseudo-Abelian if and only if it is a Pseudo- +torus. +(c) Observe that for such a suspension we have a splitting of the tangent +bundle +ΘX ∼= Ok +X ⊕ F, +hence also of the cotangent bundle. +In [AMN12] Example 2.4, page 1005, it is observed that a suspension X as +above is K¨ahler if and only if Y is K¨ahler and a finite index subgroup Λ′ ⊂ Λ +maps to Aut0(Y ). +In this case, taking G := Λ/Λ′, there is a Galois covering Z of X with +group G. If moreover Aut0(Y ) is trivial, then Z = T ′ × Y and we have a +pseudo-torus. +If Aut0(Y ) is non-trivial, one cannot conclude that Z is a principal torus +bundle, since the action of Λ′ on Y need not be properly discontinuous. +(d) If X is Seifert fibred, then X = Z/G, and we have an exact sequence +corresponding to the principal bundle Z → Y : +0 → Ok +Z → ΘZ → FZ → 0. +Since the action of G commutes with the action of T, the exact sequence +descends to X, and we have +0 → Ok +X → ΘX → F → 0. +As we shall soon see, the above notions of Seifert-fibred manifold, or of +pseudo-torus, and of suspension over a torus are not general enough in order +to deal with manifolds with Chern classes trivial in real cohomology, hence +we give another definition (important for the case of projective varieties): +Definition 3.3. A compact complex manifold X of dimension n is said to +be +• (i) Isogenous to a k-Torus product +(MITP of order k ), if X +is isogenous to a product Y × T, where T is a torus of dimension +k > 0, Y is a compact complex manifold, and k is maximal with this +property; +• (ii) Strongly Isogenous to a k-Torus product if there is a torus +T of dimension k > 0, a compact complex manifold Y , and a finite + +12 +FABRIZIO CATANESE +group G acting on T, and Y such that G acts freely on the product +Y × T via the induced diagonal action (g(t, y) := (g(t), g(y)) and +moreover: +X = (Y × T)/G, +and k is maximal with this property. +If X is projective, we shall say that it is a Variety Isogenous to a k-Torus +product. +Proposition 3.3. If X is not uniruled, and X is a compact K¨ahler Manifold, +the two notions (i) and (ii) are equivalent. +Proof. Clearly (ii) implies (i), hence it suffices to show that (i) implies (ii), +and we shall do it by induction on the number h of finite unramified maps +which make X isogenous to such a product Y × T. +Step (a(1)) For h = 1, assume that there is a finite unramified map f : +X → Y × T. +Then, taking the Galois closure Z, and letting G be the Galois group, we +see that Z is associated to an epimorphism ψ : π1(Y ) × π1(T) → G, and we +denote G1 := ψ(π1(Y )), G2 := ψ(π1(T)). +Then G is a quotient of G1 × G2 by the normal subgroup G1 ∩ G2. +To the epimorphism φ : π1(Y ) × π1(T) → (G1 × G2) corresponds a product +Manifold Y1 × T1, and clearly Z = (Y1 × T1)/G1 ∩ G2, hence X is as desired, +since G1 × G2 acts on T1 diagonally (and on T1 via translations. Hence X +is also a quotient of (Y1 × T1) by another subgroup of G1 × G2. +Step (b(1)) For h = 1, assume that there is a finite unramified map f : +(Y × T) → X. If f is not Galois, take the Galois closure Z, and denote by +Γ the Galois group of Z → X, and by G the subgroup which is the Galois +group of Z → (Y × T). Arguing as in Step (a(1)), we find a Galois cover +(Y1 × T1) → (Y × T) with group G1 × G2. +We claim that the unramified cover (Y1 × T1) → X is Galois; to establish +this claim it suffices to show that the elements γ ∈ Γ admit lifts to (Y1 ×T1). +Now, Z has a split tangent bundle ΘZ = F⊕Ok +Z, and since Z is not uniruled, +it follows from the Theorem of Fujiki and Lieberman, and by the maximality +of k, that H0(ΘZ) = H0(Ok +Z). +Hence the splitting is uniquely determined, and the group T2 maps onto +Aut0(Z). The fibration Z → (Z/T2) is then isotrivial, and (Y1 × T1) is the +canonical product pull back under Y 1 → (Z/T2). +Therefore the lifting property holds true. +Step (a(h)) Assume that X2 = (Y2 × T2)/G2, as in (ii), and that there is +a finite unramified map f : X → X2. +Then, taking a component X′ of the fibre product X ×X2 (Y2 × T2), and ap- +plying (a(1)) we get that X′, hence also X, admits (Y1×T1) as an unramified +covering, hence we can conclude because of Step (b(1)). +Step (b(h)) Assume then that X2 = (Y2×T2)/G2, as in (ii), and that there +is a finite unramified map f : X2 → X. + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT..13 +Then taking the composition we get an unramified map (Y2 ×T2) → X, and +again we are done by Step (b(1)). +□ +The following are easy important properties of such Manifolds and Varieties +Isogenous to a Torus product. +Proposition 3.4. Let the complex Manifold X of dimension n be Isogenous +to a k-Torus product Y × T where dim(T) = k > 0. +Then the integral Chern classes ci(Y × T) ∈ H∗(Y × T, Z) vanish for i ≥ +n − k + 1. +And the rational Chern classes ci(X) ∈ H∗(X, Q) vanish for i ≥ n − k + 1. +Moreover all the Chern numbers of X vanish. +If moreover X is (respectively is isogenous to) a pseudo-torus, or more gen- +erally the suspension over a torus, or is Seifert fibred, then all the integral +Chern classes ci(X) ∈ H∗(X, Z) vanish for i ≥ n − k + 1 (respectively the +corresponding rational Chern classes vanish). +Proof. Since the tangent bundle of Y × T is the direct sum of the pull back +of the tangent bundle of Y with the pull back of the tangent bundle of T +which is trivial, the total Chern class of Y × T is the pull back of the total +Chern class of Y , and this proves the first assertion. +The second assertion follows since the Chern classes of X pull back to the +Chern classes of the product Y × T, and H∗(X, Q) = H∗(Y × T, Q)G. +The third assertion follows since any isobaric polynomial of weight n in the +Chern classes of X is a rational multiple of the same isobaric polynomial of +weight n in the Chern classes of Y , and we are assuming k > 0. +The last assertion follows since the tangent bundle of X either splits as +ΘX = Ok +X ⊕ F, or has a bundle exact sequence +0 → Ok +X → ΘX → F → 0. +□ +4. On the non K¨ahler case +In the realm of compact complex manifolds, the Manifolds X with a holo- +morphically trivial tangent bundle ΘX ∼= On +X are called the parallelizable +Manifolds. +By a Theorem of Wang [Wang54] every parallelizable Manifold is a quotient +X = G/Λ, where G is a complex Lie group and Λ is a cocompact discrete +subgroup. +Since every Manifold isogenous to a parallelizable Manifold has trivial real +Chern classes, it would be interesting to study this class in more detail (in +the K¨ahler case the parallelizable Manifolds are the complex tori, and the +Manifolds isogenous to a torus are the Hyperelliptic Manifolds). + +14 +FABRIZIO CATANESE +Proposition 4.1. Every Manifold Y isogenous to a parallelizable Mani- +fold X′ = G/Λ′ is obtained from another parallelizable Manifold X = G/Λ +dividing by the action of a finite group G acting freely on X. +G is a group of affine transformations of X, of the form f(x) = gF(x) where +g ∈ G and F has a fixed point x0 and has derivative DF determined by the +value of the derivative at x0. +The quotient Manifold Y := X/G will be called a twisted hyperelliptic +Manifold. +Proof. If Y is isogenous to X′, then Y has also G as universal cover, as it is +trivial to see. +Hence we can write Y = G/Γ, where Γ acts freely on G and with compact +quotient. +Observe that any isogeny amounts, up to isomorphism, to a finite index +containment between fundamental groups, viewed as discrete groups of au- +tomorphisms of G (since G is simply connected). +It is a general fact that the intersection H ∩K of two finite index subgroups +H, K of a group Γ′ is again of finite index in Γ′ (it is the stabilizer of the +orbit of H × K on Γ′/H × Γ′/K 3). +Hence, if Γ2 is of finite index in Γ1 and Γ3, and Γ4 is of finite index in Γ3 +and Γ5, then Γ2 ∩ Γ4 is of finite index in Γ3, hence also in Γ1 and Γ5. +Therefore, since we can always assume that the number of direct relations +yielding the isogeny is even (introducing an equality step), we can reduce +to the case where Γ and Λ′ contain a finite index subgroup in both of them, +call it Λ′′. By replacing X′′ = G/Λ′′ by the Galois closure of X′′ → Y , we +obtain Y = X/G as desired. +Now, if f ∈ G, there exists g ∈ G such that F := g−1f has a fixed point x0 := +Λ on X. The derivative of F is a holomorphic section of the trivial bundle +DF ∈ H0(End(On +X)) gotten by the action on the space of left invariant +vector fields: since X is compact, DF is constant, hence f(x) = gF(x) is +affine. +□ +Of course each such automorphism F lifts to an automorphism ˜F of G which +must have the property of fixing the identity and of sending Λ to Λ, since +we must have +˜F(gλ) = ˜F(g)π1(F)(λ) ⇒ ˜F(λ) = π1(F)(λ), +this is exactly as it happens for automorphisms of complex tori. +We give now a simple example of such Manifolds. +Proposition 4.1. Consider the Iwasawa Manifold X = G/Λ, where G is +the complex Lie group of unipotent 3 × 3 complex matrices, that is, upper +triangular matrices and with diagonal entries equal to 1. +3see for instance: stack Overflow + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT..15 +And let Λ be the subgroup of matrices with upper-diagonal entries belonging +to the ring Z[i] of Gaussian integers. +Then we define the following automorphism of X, such that +r[(z12, z21, z13)] := [(iz12, z21 + 1/4, iz13)]. +Then r has order 4, and generates a group G ∼= Z/4 acting freely on X. +The quotient is a twisted Hyperelliptic Manifold with H0(Ω1 +Y ) of dimension +1 and generated by a closed holomorphic 1-form.. +Proof. The automorphism is well defined, since the product zλ has coordi- +nates +(z12 + λ12, z21 + λ21, z13 + λ13 + z12λ21), +hence we infer [r(z)] = [r(zλ)] since +[r(zλ)] = [(iz12 + iλ12, z21 + λ21 + 1/4, iz13 + iλ13 + iz12λ21)] = += [(iz12, z21 + λ21 + 1/4, iz13 + iz12λ21)] = [r(z)λ′], +where λ′ = (0, λ21, 0). +That the action is free follows since for the second coordinate +z21 �→ z21 + 1/4. +It is known that H0(Ω1 +X) is generated by +ω1 := dz12, ω2 := dz21, ω3 := dz13 − z21dz12, +which are mapped by r to iω1, ω2, iω3 − (i/4)ω1. +Hence H0(Ω1 +X)G is generated by ω1. +□ +We leave aside here the general discussion of the structure of twisted Hy- +perelliptic Manifolds, but we raise a question. +Question 4.2. The twisted Hyperelliptic Manifolds, being isogenous to par- +allelizable Manifolds, are examples of compact complex manifolds with all +the real Chern classes ci,R(X) = 0, ∀i. +Are there other examples ? +We can also define another general class of Manifolds, which we call Sus- +pensions over a parallelizable Manifold. +Definition 4.3. (S-P-M) A compact complex manifold X is called a sus- +pension over a parallelizable Manifold Z = G/Λ if there is a compact +complex manifold Y and a homomorphism ρ : Λ → Aut(Y ) such that +X = (G × Y )/Λ, λ(z, y) := (zλ, yρ(λ)). +Proposition 4.2. If X is a suspension over a parallelizable Manifold Z, +then we have a splitting of the tangent bundle +ΘX = Ok +X ⊕ F, +where k = dim(Z). + +16 +FABRIZIO CATANESE +Proof. The splitting follows right away since Λ has a product action. +Moreover, the first summand is trivial because Z = G/Λ is parallelizable +and the first summand is just the pull-back of the tangent bundle ΘZ for +the projection X → Z, with fibres isomorphic to Y . +□ +Remark 4.4. In the case where ρ has finite image, we get that, Λ′ being +defined as Λ′ := ker(ρ), then X is isogenous to Z′ × Y , where Z′ := G/Λ′. +And we get then triviality of the Chern numbers of X. +In the next section we shall show how the occurrence of such a splitting +ΘX = Ok +X ⊕ F is understood under the K¨ahler assumption as the structure +of a suspension over a torus. +It is interesting to see whether a similar characterization of suspensions over +a parallelizable Manifold holds also for compact complex manifolds. +For this purpose one has to take k maximal, and the first key point would +be to see whether H0(Ok +X) ⊂ H0(ΘX) is a Lie subalgebra G: then we would +have an action on X of the simply connected Lie group G associated to G. +5. Partially framed and co-framed manifolds +We begin with a simple definition yielding complex manifolds with all the +last k integral Chern classes equal to zero. +Definition 5.1. A complex manifold X of dimension n is said to be k- +tangentially framed, or simply k-framed, if the holomorphic tangent bundle +ΘX admits a trivial subbundle ∼= Ok +X, and where k > 0 is maximal. +A complex manifold X of dimension n is said to be k-cotangentially framed, +or simply k-co-framed, if the holomorphic cotangent bundle Ω1 +X admits a +trivial subbundle ∼= Ok +X, and where k > 0 is maximal. +In the rest of the section we shall use the results of [Li78], [Fuj78], [AMN12], +often for simplicity we might refer to the exposition given in the last paper. +Proposition 5.2. Assume that X is a k-coframed compact K¨ahler man- +ifold and let W ⊂ h0(Ω1 +X) be the corresponding maximal vector subspace +consisting entirely of nowhere vanishing holomorphic 1-forms (of dimension +k ≥ 1). Then W determines an analytically integrable foliation with trivial +normal bundle. +i) If moreover the subspace W corresponds to a quotient torus T ′ of Alb(X) = +H0(Ω1 +X)∨/(H1(X, Z)/Tors), then the foliation is algebraically integrable, +consisting of the fibres of Ψ : X → T ′, which is a differentiable fibre bundle. +ii) If we have a splitting Ω1 +X ∼= Ok +X ⊕ F∨, then Ψ is a holomorphic fibre +bundle with fibre Y . +iii) If moreover Y has finite automorphism group, then X is a k-Pseudo- +Torus product X = (Y × T)/G with dim(T) = k > 0. +Proof. A subspace W which is maximal with the property that all forms +ω ∈ W \ {0} are nowhere vanishing, has a basis ω1, . . . , ωk such that the + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT..17 +forms ωj are linearly independent at each point, hence they generate a trivial +rank k subbundle of Ω1 +X. +The associated foliation is analitically integrable, because X is a cKM, and +holomorphic 1-forms are closed, hence the distribution induced by W is +integrable, and spans a trivial subbundle of the cotangent bundle. +The foliation on X is induced by a foliation on Alb(X), corresponding to the +annihilator of W. This foliation is algebraically integrable if, as we assume, +it corresponds to the projection onto a quotient torus T ′. +The composed map Ψ : X → T ′ has fibres of dimension k, which are there- +fore union of leaves. Since the fibres are smooth, if they are not connected, +we would get by the Stein factorization on unramified covering of T ′, which +is again a quotient of Alb(X) by the universal property of the Albanese +variety. +Hence we may assume that the fibres of Φ are connected, and we have a +differentiable fibre bundle. +The splitting Ω1 +X ∼= Ok +X ⊕ F∨ guarantees that the Kodaira-Spencer map +for the family is identically zero, hence by Kuranishi’s theorem we have a +holomorphic bundle. +If this is a holomorphic bundle, with fibre Y , and Aut(Y ) is finite, there is +an unramified covering T → T ′ with group G such that the pull back is a +product. Therefore X = (T × Y )/G, where G acts on T via translations. +□ +Theorem 5.1. Let X be a compact K¨ahler complex Manifold of dimension +n: then X is the suspension over a torus T with dim(T) = k > 0 if and only +if there is a k-framing yielding a partial tangential splitting +ΘX ∼= Ok +X ⊕ F. +A k-framing of X yields the structure of a Seifert fibration on X in the case +where h0(ΘX) = k. +Moreover, a k-framed projective manifold X is a suspension over an Abelian +Variety and is indeed a Pseudo-Abelian variety if KX is nef. +Proof. We have already seen that for a suspension over a torus we have such +a partial tangential splitting. +Conversely, recall that H0(ΘX) is the Lie algebra of the Lie group Aut(X). +The Lie Algebra H0(ΘX) =: AX contains the Lie ideal H1 +X of the vector +fields admitting zeros, and there is (see [AMN12], page 1002), a direct sum +AX = H1 +X ⊕ A, +where A is a maximal Abelian subalgebra generated by nowhere vanishing +vector fields. +By Theorem 3.14 of [Li78], H1 +X is precisely the subspace of HX yielding the +zero flow on Alb(X). +If H1 +X = 0 then the trivial subbundle yields k everywhere linearly inde- +pendent vector fields, which, see [Fuj78], [Li78], and also Theorem 1.2 of + +18 +FABRIZIO CATANESE +[AMN12], generate the action of a k-dimensional complex torus T with +smooth orbits Tx, quotients of T by a finite group Hx. +Hence X is Seifert fibred, as we have discussed earlier, cf. Theorem 4.9 by +Lieberman [Li78]. +If we have a tangential splitting ΘX ∼= Ok +X ⊕ F, we get a corresponding +cotangent splitting, +Ω1 +X ∼= Ok +X ⊕ F∨. +It suffices, by the preceding proposition, to show that the coframing defines +a subspace W ⊂ H0(Ω1 +X) corresponding to a quotient torus T ′ of Alb(X). +Since every automorphism of X yields an affine action on Alb(X), the action +of the torus T, which spans W ∨, yields a subtorus A of Alb(X). Then we +define T ′ := Alb(X)/A, and we get Ψ : X → T ′ which is is a holomorphic +bundle, with parallel transport given by the action of T: hence X is the +suspension over a torus. +Finally, if X is projective, T ′ is an Abelian variety. We defer the reader to +Theorem 0.3 of [AMN12] for the proof of the last assertions, see also [Li77]. +□ +Remark 5.2. Theorem 0.3 of [AMN12] proves the following very interesting +result: if X is a compact K¨ahler manifold which is k-framed, then X admits +a small deformation which is a suspension over a k-dimensional torus, and +which is a k-Pseudo-Torus if Kod(X) ≥ 0. +If X is projective and k-framed, as we saw, they show that X is Pseudo- +Abelian. +Theorem 5.3. (i) Assume that X is a k-coframed compact K¨ahler manifold, +and that q := h0(Ω1 +X) = k: then the Albanese map aX is a differentiable +fibre bundle. +(ii) If X is projective, k = q = n − 1, and KX is nef, then X is a pseudo- +Abelian variety. +Proof. The first assertion follows from i) of Proposition 5.2. +We prove now the second assertion (ii): by the assumption q := h0(Ω1 +X) = +n − 1 (i) applies, and the fibres of aX are smooth curves of genus g ≥ 1. +Since the fibration induces a holomorphic map to the Teichm¨uller space Tg +which is biholomorphic to a bounded domain, this map is constant and we +have a holomorphic fibre bundle. +Since X is projective, the monodromy is finite, so there exists a finite un- +ramified map A′ → A := Alb(X) such that the pull back is a product, hence +X is a Pseudo-Abelian variety. +□ +Remark 5.3. If k = n − 1, but q > n − 1, by the exact sequence +0 → On−1 +X +→ Ω1 +X → F → 0, +the quotient line bundle F ∼= OX(KX). + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT..19 +Since q > n − 1, there is a section σ of Ω1 +X inducing a non zero element of +H0(F), hence we have an effective divisor D which is linearly equivalent to +KX, hence D is nef. +If we knew that h0(OD) = 1, or that h1(OX(−KX)) = 0, we could conclude +that we have a splitting of the above exact sequence, since such a σ would +be given in local charts Uα by vectors σ1,α, σ2,α such that D = {σ2,α = 0}, +and +σ1,β = σ1,α + ψα,βσ2,β. +Then σ1,β ≡ σ1,α on D, hence it is a vector of constants, therefore we may +assume that σ1,α vanishes on D, therefore σ yields a splitting. Hence X is +a Pseudo-Abelian Variety. +Example 5.4. This example concerns k-coframed varieties. +(1) A smooth fibration onto an Abelian variety need not be isotrivial, and +not even a holomorphic bundle. +Let in fact A be an elliptic curve and assume that we have an embedding +j : A → S, where S is a surface of general type, say with KS ample. +Let Γ ⊂ A × S be the graph of j, and take X to be the blow up of A × S +with centre the smooth curve Γ. +Then the differentiable fibre bundle f : X → A is not a holomorphic bundle, +since Aut(S) is finite, and the pairs (S, p1) and (S, P2) are not isomorphic +for general P1, P2 ∈ j(A). +(2) In this case KX is not nef. Else, one may ask whether the fibration is +isotrivial. This has been shown by Kovacs in the case where the fibres have +ample canonical divisor [Kov97]. +We pose now the following general questions: +Question 5.5. (a) Assume that X is a k-coframed projective manifold. +Does there exist a coframing V (a subbundle V ∼= Ok +X of Ω1 +X) such that i) of +Proposition 5.2 holds, namely the subspace W = H0(V ) ⊂ H0(Ω1 +X) defines +a quotient Abelian variety A′? +(b) If X is projective with KX nef and it admits a fibration f : X → A′ +onto an Abelian variety A′ with all fibres smooth, is then f is a holomorphic +bundle and hence X is a pseudo-Abelian variety? +As already discussed, question (b) is motivated by the result of [Kov97], and +it fits into a pattern of conjectures of classification theory. +Remark 5.4. Question (a) asks, in the case where X is projective, whether +the subspace W ⊂ H0(Ω1 +X) corresponds to an Abelian subvariety of Alb(X), +equivalently, whether Λ2k(W ⊕ ¯W) is a point defined over Q in the Grass- +mann manifold Grass(2k, H1(X, C)) ⊂ P(Λ2k(H1(X, C)). +One may conjecture that this is true if V is geometrically defined, that +is, it is unique and invariant by all automorphisms in Aut(Q). Here it is +important that X is projective, and one may reduce to the case where X is +defined over an algebraic extension of Q. + +20 +FABRIZIO CATANESE +I initially thought that question (a) has a positive answer, trying to use (see +[Miya87]) the generic semipositivity of Ω1 +X for a non uniruled variety, and +its Harder-Narasimhan filtration to define a geometrically unique coframing +V . But Deligne spotted a trivial mistake in my reasoning. +6. Mathematical and Historical comments on Baldassarri’s +paper [Bald56] and the questions it suggests +Baldassarri [Bald56] was trying to characterize the smooth projective man- +ifolds X whose first h canonical systems K0(X), . . . , Kh−1(X) have degree +zero. +At that time, even if Baldassarri in his Ergebnisse book [Bald56book] (the +book was rather influential, it was for instance translated in Russian by +Manin) was exposing the new methods in the theory of Algebraic Varieties, +the concept of canonical systems of all dimension was rather based on more +geometric approaches. +The canonical systems of a manifold (see [Roth56]), defined in a geometric +way by Todd, Eger and later in a simpler way by Beniamino Segre [Seg52], +[Seg54] 4 after proposals made by Severi, were shown in 1955 by Nakano +[Nak55] to be the so called Chern classes of the cotangent bundle (see [At98] +for an historical account and [Ful84] as a general reference): more precisely, +up to sign, to the system Kh(X) corresponds the Chern class cn−h(X), where +n is the dimension of X (and we can consider the Chern classes either as +elements of the Chow ring of X, or as integral cohomology classes). +Hence we formulate the questions raised by the work of Baldassarri in terms +of Chern classes: Baldassarri dealt with the question of characterizing all the +projective varieties X such that the rational Chern classes ci(X) ∈ H∗(X, Q) +vanish for n − k + 1 ≤ i ≤ n, but not for i = n − k. +This question is still wide open, except for the case k = n, as we saw in +Theorem 1.2. +Already in the surface case the condition that all the Chern numbers are zero +(this means K2 +X = e(X) = 0) is not sufficient, as we have seen, to imply that +X is a complex torus or a hyperelliptic surface (in the old notation a torus +was also called a hyperelliptic surface, of grade, or rank, 1). But it implies +(see Theorem 2.6) that the surface is either a torus or it is birationally (but +not necessarily biregularly) covered by a 1-dimensional family of isomorphic +elliptic curves. +Baldassarri’s question also suggests (see question (IV) below) to classify, +if possible, the varieties (cKM) X whose integral Chern classes ci(X) ∈ +H∗(X, Z) vanish for n − k + 1 ≤ i ≤ n. +Again, the question is quite open, even for the case k = n, as we saw in Part +I [Cat23]. +4using the embedding covariants for the case of the diagonal ∆X ⊂ X × X: this +approach was also later followed by Grothendieck. + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT..21 +The paper [Bald56] by Baldassarri, suggesting that the Pseudo Abelian Va- +rieties of Roth might be the varieties with such vanishing top real Chern +classes, motivates indeed some more general questions. But, in view of what +we have seen in the case of surfaces, one has to include first of all a condition +of minimality of X in a strong sense, for instance that KX is nef. Or require +only a birational isomorphism with a product (because a P1-bundle over an +elliptic curve T is only birational to P1 ×T), or that X be only birational to +(Y × T)/G, where the action is only free in codimension 1 (as in [HS21a]). +Todd’s review of [Bald56] pointed out a wrong intermediate result, admitting +as counterexample the blow up X of P3 with centre a smooth curve of genus +3, which is a regular manifold X having c3 = 0. +Todd’s counterexample could somehow be quickly dismissed as only pointing +out the need to assume that X is a minimal manifold, say with KX nef, as +we already mentioned. +We observe here however that the flaw is not simply a technical problem, the +main claim by Baldassarri that the solutions are Roth’s Pseudo-Abelian va- +rieties is incorrect also for minimal manifolds. Because for instance the class +of Manifolds isogenous to a k-torus product is a larger class than the class +of Roth’s Pseudo-Abelian varieties, and varieties in this class are solutions +to Baldassarri’s problem. +On the other hand Todd saw that the crucial flaw in Baldassarri’s argument +was the attempt to show that manifolds with top Chern class equal to zero +have positive irregularity. +What is historically interesting is to observe that the ‘original sin’ of [Bald56] +was to try to extend to higher dimension some wrong results by Enriques, +Dantoni and Roth (indeed the error of Enriques is also reproduced in the +classification theorem of Castelnuovo and Enriques [CastEn15]). +In fact, the work by Baldassarri and Roth is inspired by a paper by Dan- +toni [Dant43], devoted to the minimal surfaces X with c2(X) = 0. Dantoni +uses surface classification, especially an article by Enriques of 1905 [Enr05b], +claiming that the non ruled surfaces with these properties are the hyperel- +liptic manifolds and the ‘elliptic’ surfaces. But ‘elliptic’ for Enriques here +does not have the same standard meaning introduced later by Kodaira and +others: Enriques requires the action of a fixed elliptic curve on X, with all +orbits of dimension 1. +Enriques and Dantoni in their classification omit to consider the case of +quotients X = (E × C)/G where the action of the finite group G is free, of +product type, but G does not act on the elliptic curve E via translations, +and moreover C is a curve of genus g ≥ 2, such that the quotient C/G is +an elliptic curve (see section 2, and for instance [CatLi19] or [Bea78] for the +special case pg = 0, and [CB] or [Cat22] for the general case). Indeed, in +this latter case the automorphism group of X has dimension zero. +Dantoni’s paper inspired Roth [Roth54] [Roth55] who defined the Pseudo- +Abelian varieties as the manifolds admitting the action of a complex torus + +22 +FABRIZIO CATANESE +of positive dimension = k having all orbits of dimension k, and such that k +is maximal with this property. +But for instance, in [Roth53], Roth does not realize about the existence of +Hyperelliptic threefolds with automorphism groups of dimension zero, and +believes that these are only Pseudo-Abelian varieties with k ≥ 1. +The first conclusion is simple: the ‘original sin’ was to consider only quo- +tients X = (T × Y )/G where T is a torus, G acts via a product action, +which is free and such that G acts on T via translations. Obviously under +these assumptions T acts on X and the orbits have all the same dimension +k = dim(T)! +Baldassarri’ s paper suggests the following other questions: +Question 6.1. (I) What can be said about a projective Manifold (respec- +tively, a compact K¨ahler manifold) X with KX nef and such that all its +Chern numbers are equal to zero? +(II) Is a projective Manifold (respectively, a compact K¨ahler manifold) X +with KX nef and such that its rational Chern classes ci(X) ∈ H∗(X, R) +vanish for k + 1 ≤ i ≤ n isogenous to a k-Torus Product (respectively, +isogenous to a k-framed or k-coframed manifold)? +(III) Same question for a projective Manifold (or compact K¨ahler manifold) +X with KX nef and with all the Chern numbers equal to zero. +(IV) What can we say about a projective Manifold (or cKM) X whose +integral Chern classes ci(X) ∈ H∗(X, Z) vanish for k + 1 ≤ i ≤ n? +We shall see in the next section that work of Chad Schoen [Schoen88] gives +a negative answer to Questions (II) and (III). +Remark 6.2. a) We have mentioned that Baldassarri’s assertion is wrong +already for surfaces. +b) The assertion is also wrong in dim = 3 since here, as shown in Prop. +1.5 of Part I, [Cat23], there are Hyperelliptic Threefolds whose group of +Automorphisms is discrete, hence they are not Pseudo-Abelian. Indeed, if +one looks at the paper by Roth [Roth53] on Hyperelliptic threefolds, one +sees that Roth does not consider the case of Hyperelliptic Threefolds for +which Aut0(X) consists only of the Identity. +c) In the same paper Roth calls, following Enriques, [Enr05a], [Enr05b], ‘the +elliptic surfaces’ the surfaces such that Aut0(X) is an elliptic curve. +Here the modern terminology, introduced by Kodaira, differs: an elliptic +surface is a surface admitting a fibration f : S → C with fibres elliptic +curves; in general it does not possess non-trivial automorphisms. +As we saw in Part I [Cat23], however, there are Hyperelliptic Threefolds and +Varieties (hence for them cn(X) = 0) which are regular (H1(OX) = 0). +Hence the crucial fact that Baldassarri wants to use, that for cn(X) = 0 we +have an irregular variety possessing a holomorphic 1-form without zeros has +to be taken as an assumption. + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT..23 +To give an idea of the difficulty of the type of questions considered by Bal- +dassarri, let us notice that for instance the investigation of varieties such +that there is ω ∈ H0(Ω1 +X) without zeros has been taken up (before our +present investigations) in the last years by Schreieder and Hao ([Schre21] +[HS21b]), and a classification has been achieved only in dimension 3. +What is more interesting is that, under this much stronger assumption of +the existence of such a form ω, the results confirms, in a special case, the +result claimed by Baldassarri (see also our Theorem 5.3). +Theorem 6.3. (Hao-Schreieder [HS21b]) +Let X be a smooth projective threefold satisfying property +(A) : admitting a holomorphic 1-form ω ∈ H0(Ω1 +X) without zeros. +Then the minimal model program for X yields a birational morphism σ : +X → Xmin blowing up smooth elliptic curves which are not contracted by +the Albanese map, and such that +(2) there is a smooth morphism π : Xmin → A to an Abelian variety of +positive dimension; +(3) If the Kodaira dimension is non negative, then a finite ´etale cover of +Xmin is a product X′ ∼= A′ × S′, where S′ is smooth projective, and the +composite map A′ ∼= A′ × {s′} → A is finite and ´etale. +(4) If the Kodaira dimension is equal to −∞, then either +(4a) Xmin has a smooth Del Pezzo fibration over an elliptic curve, or +(4b) Xmin has a conic bundle fibration f : Xmin → S over a smooth surface +S satisfying property (A). Moreover, either f is smooth, or A is smooth and +the degeneracy locus of f is a finite union of elliptic curves which are ´etale +over A. +Corollary 6.1. If X is a smooth projective threefold satisfying property +(A) of admitting a holomorphic 1-form ω ∈ H0(Ω1 +X) without zeros, and the +Kodaira dimension of X is non negative, then Xmin is a Pseudo-Abelian +variety in the sense of Roth. +Proof. We are in case (3), and obviously A′ acts on the product A′ × S′. +Since the map A′ → A is finite and unramified, it is a quotient map, with +Galois group G, a finite Abelian group of translations of A′. +We have A′ ×S′ → Xmin → A, and since A′ ×S′ → A′ is a smooth fibration +with fibre S′, hence also Xmin → A is a smooth fibration with fibre S′. +Since the pull back of Xmin → A to A′ is isomorphic to the product A′ ×S′, +we conclude that A′ acts on Xmin and Xmin = (A′ × S′)/G, where G acts +on A′ by translations. +□ +The historical conclusion that we can draw is that Baldassarri’s paper, even +if vitiated by the ‘original sin’ of trying to extend results which were not +correct already in small dimension n = 2, 3, poses some problems which, in +spite of the tremendous substantial and technical progress which took place +in the last 60 or more years, are still quite open and very difficult. + +24 +FABRIZIO CATANESE +The typical example is in our opinion the question of describing the k-co- +framed manifolds X ( see Proposition 5.2 and Example 5.4). +7. Manifolds with vanishing Chern numbers +The title of this section is on purpose ambiguous: one may ask about Man- +ifolds for which a certain Chern number vanishes, or all the way consider +Manifolds for which all the Chern numbers are equal to zero. +We have given the example of Manifolds X Isogenous to a k-Torus Product +as a prototype of manifolds with all the Chern numbers equal to zero. +We have also observed that, at least in dimension 2, all the surfaces with all +the Chern numbers c2 +1 = c2 = 0, or the minimal surfaces with c2 = 0, in view +of Theorem 2.6, are the manifolds of this type, if KS is nef, or birational to +one of this type if S is ruled. +Because if S is minimal and not elliptic ruled, there exists a Galois ´etale +covering S′ → S such that S′ ∼= T × Y , where T is a complex torus with +dim(T) > 0. +More generally, we have the class of manifolds X isogenous to a partially +cotangentially framed manifold X. +If we go up to dimension 3, there are three Chern numbers, c3 +1, c3, and +c1c2 = χ(OX). +A recent result by Hao and Schreieder goes in the direction of answering the +question, [HS21a] in the special case where the Kodaira dimension is n − 1: +Theorem 7.1. (Hao-Schreieder) Let X be a minimal model with dim(X) = +n and Kodaira dimension n − 1. +Then cn−2 +1 +c2(X) = 0 if and only if X is birational to a quotient Z = (E × +Y )/G, where +(1) Z has canonical singularities; +(2) E is an elliptic curve and Y is a normal projective variety with KY +ample; +(3) G acts diagonally, faithfully on each factor, and freely in codimension +two on E × Y . +Now, the case where X is a threefold of general type (in this case it can +be c1c2 = 0, as shown by Ein and Lazarsfeld, [EL97]), is excluded if X is +minimal, since then c3 +1 > 0. +Hence the missing cases are the cases of Kodaira dimension 0 and 1. For +Kodaira dimension 0, c1(X) = 0 ∈ H2(X, Q), hence remains to see what +happens for c3 = 0. Some examples of simply connected Calabi-Yau three- +folds with c3 = 0 have been constructed by Chad Schoen [Schoen88] and +other examples were later found by Volker Braun [Bra12] as hypersurfaces +in a toric fourfold. +We want to discuss now the former examples by Schoen, and show some +partial results which seem to indicate that they should not be birational to +a quotient of a torus product. + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT..25 +These examples are constructed as small resolutions of fibre products X = +S1 ×P1 S2 where fi : Si → P1 is a rational elliptic surface with a section. Let +f : X → P1 be the fibre product of f1 and f2. +We are interested in the special case where the critical values of f1, f2 are +different; then the fibre product X is smooth, and all the fibres F of f are +either a product of two elliptic curves, or the product of an elliptic curve +with a degenerate fibre. Hence all the fibres F have Euler number e(F) = 0, +and c3(X) = e(X) = 0. +The canonical divisor of X is trivial, for instance we can take the fi = Fi +Gi +to be given by a pencil of plane cubics with simple base points: then X is +the small resolution of a hypersurface of bidegree (3, 3) X′ ⊂ P2 × P2, +X′ = {(x, y)|F1(x)G2(y) = F2(y)G1(x)} +hence X′ and X have trivial canonical divisor (see [Schoen88] page 181). +Moreover, essentially by the hyperplane theorem of Lefschetz, X is simply +connected (see at any rate (2.1) of [Schoen88], page 181). +Thus X is a Calabi-Yau threefold with c3(X) = 0. +Proposition 7.1. The Schoen threefolds X cannot be birationally covered +by a 1-dimensional family of subvarieties which are isomorphic to a fixed +Abelian surface T. +Proof. Assume that X is birationally covered by a family T × C (where by +the way C = P1 since q(X) = 0). +For a general b ∈ C, Tb := T ×{b} cannot map to a fibre of f : X → P1, since +f is the fibre product of f1 and f2 and we may assume that the fibrations +fi do not have constant moduli. +Hence Tb, which is a subvariety of X, dominates P1 through the morphism +f. +But a linear system of dimension one on an Abelian surface yields a mor- +phism to P1 only if the system is not ample, that is, the fibre is a union +of translates of an elliptic curve E ⊂ T. Varying b, the elliptic curve E is +fixed (since it corresponds to a subgroup of the first homology group pf T), +therefore all the fibres of f contain an elliptic curve isomorphic to E. This +is a contradiction, as the general fibres of f1 and f2 are not even isogenous +to a fixed elliptic curve E. +□ +Proposition 7.2. The Schoen threefolds X do not admit a fibration ψ : +X → S onto a surface S such that the general fibre is isomorphic to a fixed +elliptic curve E. +Proof. Since X is simply connected, and the general fibre of ψ is connected, +it follows that S is also simply connected. +Let D be the divisorial part of the set of critical values of ψ, and let D∗ be +its smooth locus: define +S∗ := (S \ Sing(D)), X∗ = ψ−1(S∗). + +26 +FABRIZIO CATANESE +Then we have an exact sequence +π1(E) → π1(X∗) → π1(S∗) → 1, +and we observe that also X∗, S∗ are simply connected, since we have removed +a subvariety of real codimension 4. +The image of π1(E) → π1(X∗) is the quotient of π1(E) by the local mon- +odromies around the irreducible components Dj of D∗. +To understand these local monodromies, take a general curve section C of +S. The inverse image of C is an elliptic fibration Σ → C over C such that +all the smooth fibres are isomorphic to E. +The fibration is isotrivial, hence Σ = (C′ × E)/G, where C′ → C is Galois +with group G. +G acts on C′ × E via a product action. If there are fixpoints for the action +of G on C′, then, since the quotient is smooth, it follows that the isotropy +subgroups act freely on E, hence by translations. So the local holomorphic +monodromies are translations by torsion points, and the monodromy acts +trivially on the first homology of E. +The conclusion is that the image of π1(E) → π1(X∗) is an infinite group, +and this is a contradiction since π1(X∗) is trivial. +□ +The construction of Chad Schoen [Schoen88] applied to other elliptic fibra- +tions leads to threefolds X with Kodaira dimension 1, and c3(X) = 0. We +have not yet investigated whether we can achieve with this construction +trivial Chern number c1(X)c2(X) = 0. +Acknowledgements: I would like to thank Francesco Baldassarri for bring- +ing the paper [Bald56] by Mario Baldassarri to our attention, thus raising +my interest in these questions, Ciro Ciliberto and Flaminio Flamini for pro- +viding me with the text of [Dant43], Thomas Peternell for mentioning the +article by Braun. +Many thanks to Matthias Sch¨utt for bringing the examples of [Schoen88] to +my attention and explaining their key features. +Thanks to Pierre Deligne for answering an email query (see Remark 5.4). +Thanks to Adriano Tomassini for pointing out Wang’s reference [Wang54] +to me. +References +[AMN12] Jaume Amor´os, M`onica Manjar´ın, Marcel Nicolau: +Deformations of +K¨ahler manifolds with nonvanishing holomorphic vector fields. J. Eur. Math. +Soc. 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Complex and symplectic geometry, 39-49, Springer INdAM +Ser., 21, Springer, Cham (2017). +[CatLi19] Fabrizio Catanese, Binru Li: Enriques’ classification in characteristic p > 0: +the P12 -theorem. Nagoya Math. J. 235, 201–226 (2019). +[CatLiu21] Fabrizio Catanese, Wenfei Liu: On topologically trivial automorphisms +of compact K¨ahler manifolds and algebraic surfaces. Atti Accad. Naz. Lincei, +Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 32, No. 2, 181–211 +(2021). +[Dant43] +Giovanni Dantoni: Determinazione delle superficie con serie di Severi di +ordine nullo o negativo. Atti Accad. Italia, Mem., Cl. Sci. Fis. Mat. Nat. 14, +39–49 (1943). +[EL97] +Lawrence Ein, Robert Lazarsfeld: Singularities of theta divisors and the +birational geometry of irregular varieties. J. Am. Math. Soc. 10, No. 1, 243–258 +(1997). +[Enr05a] Federigo Enriques Sulle superficie algebriche che ammettono un gruppo con- +tinuo di trasformazioni birazionali in se stesse. 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Lieberman: Holomorphic vector fields on projective varieties. Several +complex Variables, Proc. Symp. Pure Math. 30, Part 1, Williamstown 1975, 273– +276 (1977). +[Lu-Ta18] +Steven Lu, Behrouz Taji: A characterization of finite quotients of abelian +varieties. Int. Math. Res. Not. 2018, No. 1, 292–319 (2018). +[Mar71] +Masaki Maruyama: On automorphism groups of ruled surfaces. J. Math. Ky- +oto Univ. 11 (1971), 89–112. +[Miya87] Yoichi Miyaoka: The Chern classes and Kodaira dimension of a minimal vari- +ety. Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. +10, 449–476 (1987). +[Mum70] David Mumford: Abelian varieties. Studies in Mathematics. Tata Institute of +Fundamental Research 5. London: Oxford University Press. VIII, 242 p. (1970), +with notes by C. P. Ramanujam. +[Nak55] +Shigeo Nakano Tangential vector bundle and Todd canonical systems of an +algebraic variety. Mem. Coll. Sci. Univ. Kyoto, Ser. A 29, 145–149 (1955). +[Roth53] +Leonard Roth: Hyperelliptic threefolds. Proc. Camb. Philos. Soc. 49, 397–409 +(1953). +[Roth54] +Leonard Roth: Pseudo-Abelian varieties. Proc. Camb. Philos. Soc. 50, 360– +371 (1954). +[Roth55] +Leonard Roth: Some properties of pseudo-Abelian varieties. Ann. Mat. Pura +Appl., IV. Ser. 38, 281–302 (1955). +[Roth56] +Leonard Roth: Sistemi canonici ed anticanonici. Univ. Genova, Pubbl. Ist. +Mat. 21, 118 p. (1956). +[Schre21] +Stefan Schreieder: Zeros of holomorphic one-forms and topology of K¨ahler +manifolds. Int. Math. Res. Not. IMRN 2021, no. 8, 6169–6183. +[Schoen88] +Chad Schoen: On fiber products of rational elliptic surfaces with section. +Math. Z. 197, No. 2, 177—199 (1988). + +VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT..29 +[Seg52] +Beniamino Segre: Vari´et´es covariantes d ’ immersion et vari´et´es canoniques +sur une vari´et´e alg´ebrique ou topologique. C. R. Acad. Sci., Paris 234, 1731–1733 +(1952). +[Seg54] +Beniamino Segre: Dilatazioni e variet`a canoniche sulle variet`a algebriche. +Ann. Mat. Pura Appl., IV. Ser. 37, 139–155 (1954). +[Sev51] +Francesco Severi Fondamenti per la geometria sulle variet´a algebriche. II. +Ann. Mat. Pura Appl., IV. Ser. 32, 1–81 (1951). +[Som74] +Andrew John Sommese: Holomorphic vector-fields on compact K¨ahler man- +ifolds. Math. Ann. 210, 75–82 (1974). +[Yau78] +Shing-Tung Yau: On the Ricci curvature of a compact K¨ahler manifold and +the complex Monge-Amp`ere equation. I. Commun. Pure Appl. Math. 31, 339– +411 (1978). +[Wang54] Hsien-Chung Wang: Complex parallelisable manifolds. Proc. Am. Math. Soc. +5, 771–776 (1954). +Mathematisches Institut der Universit¨at Bayreuth, NW II, Universit¨atsstr. +30, 95447 Bayreuth +Email address: fabrizio.catanese@uni-bayreuth.de +Korea Institute for Advanced Study, Hoegiro 87, Seoul, 133–722. + diff --git a/zNFKT4oBgHgl3EQfMi2t/content/tmp_files/load_file.txt b/zNFKT4oBgHgl3EQfMi2t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5254a7953fdc667d7059e15e99d0eca9a2a4e23c --- /dev/null +++ b/zNFKT4oBgHgl3EQfMi2t/content/tmp_files/load_file.txt @@ -0,0 +1,881 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf,len=880 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='11751v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='AG] 27 Jan 2023 MANIFOLDS WITH TRIVIAL CHERN CLASSES II: MANIFOLDS ISOGENOUS TO A TORUS PRODUCT, COFRAMED MANIFOLDS AND A QUESTION BY BALDASSARRI FABRIZIO CATANESE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Motivated by a general question addressed by Mario Bal- dassarri in 1956, we discuss the Pseudo-Abelian Varieties introduced by Roth, and we introduce a first new notion, of Manifolds Isogenous to a k-Torus Product: the latter have the last k Chern classes trivial in rational cohomology and vanishing Chern numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We show that in dimension 2 the latter class is the correct substitute for some incorrect assertions by Enriques, Dantoni, Roth and Baldas- sarri: these are the surfaces with KX nef and c2(X) = 0 ∈ H4(X, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We observe in the last section, using a construction by Chad Schoen, that a similar picture does not hold in such a simple way in higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We discuss then, as a class of solutions to Baldassarri’s question, (manifolds isogenous to) projective (respectively: K¨ahler) manifolds whose tangent bundle (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' cotangent bundle) has a trivial subbun- dle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The former class of ‘partially framed’ projective manifolds, that is, whose tangent bundle has a trivial subbundle, consists, in the case where KX is nef, of the Pseudo-Abelian varieties of Roth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' while the latter class of ‘partially co-framed’ projective manifolds is not yet understood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' we can however state some new results and formulate open questions and conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the course of the paper we address also the general case of com- pact complex Manifolds, introducing the new notions of suspensions over parallelizable Manifolds, and of twisted hyperelliptic Manifolds, and de- scribe the known results under the K¨ahler assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In memory of Mario Baldassarri (1920-1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Introduction and history of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Surfaces with second Chern class equal to zero, and a Chern class characterization of Hyperelliptic and Abelian surfaces 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Pseudo Abelian Varieties and Varieties Isogenous to a k-Torus- Product 9 Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' AMS Classification: 14F, 14K, 14C25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 1 2 FABRIZIO CATANESE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' On the non K¨ahler case 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Partially framed and co-framed manifolds 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Mathematical and Historical comments on Baldassarri’s paper [Bald56] and the questions it suggests 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Manifolds with vanishing Chern numbers 24 References 26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Introduction and history of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The initial purpose of this article was to discuss and reformulate a question coming from Mario Baldassarri ’s work [Bald56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In order to do this we first have to explain how the question arose, relying on generalizations of some incorrect assertions concerning the classification of algebraic surfaces (for which we present here the correct results);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' then, we explain the state of the art, prove some new results, define new classes of compact complex Manifolds, and pose some new questions, related to these older questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Baldassarri’s paper [Bald56] aims at determining the manifolds whose last Chern classes are zero in rational cohomology, namely ci(X) = 0 ∈ H2i(X, Q) for i ≥ k + 1: but he made the false claim that these manifolds are exactly the Pseudo-Abelian varieties introduced by Roth [Roth54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A first basic principle that Baldassarri and his precedessors missed to con- sider (as the use of topology was not sufficiently established at the time) was the following: Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (Isogeny principle): If we have a finite unramified map f : Z → X, then ci(Z) = 0 ∈ H2i(Z, Q) if and only if ci(X) = 0 ∈ H2i(X, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Defining isogeny between manifolds as the equivalence relation generated by the existence of such finite unramified maps, we see that the set of man- ifolds which are solutions to Baldassarri’s question consists of a union of isogeny classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Baldassarri’s problem has been solved in the extremal case where k = 0: the characterization of the compact K¨ahler Manifolds with all Chern classes zero in rational (or real) cohomology was solved in 1978, thanks to Yau’s celebrated theorem [Yau78] on the existence of K¨ahler-Einstein metrics on manifolds with c1(X) = 0 ∈ H2(X, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' It follows that these Manifolds are the ones isogenous to complex tori (they are the so-called Hyperelliptic Manifolds, quotients X = T/G of a complex torus by a finite group G acting freely on T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' More precisely, we have: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (Yau) A compact K¨ahler manifold X such that c1(X) = 0, c2(X) = 0, in H∗(X, R), is a Hyperelliptic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='. 3 What happens for k ≥ 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The first significant case is the case where n := dim(X) = 2, k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Already to describe this case we need some definition: a compact K¨ahler Manifold X is said to be a torus product iff X ∼= T × Y , where T is a complex torus of dimension ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Whereas one says that X is strongly isogenous to a torus product if X = (T × Y )/G, where G acts on T, Y and freely by its diagonal action on T × Y (we shall see later that, if X is not uniruled, and it is a compact K¨ahler Manifold, the above two notions are equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The case of Roth’s Pseudo Abelian Varieties is the case where X = (T × Y )/G, X is projective, and moreover G acts on T freely (hence, as a group of translations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the case n = 2, k = 1 we have a complete answer (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='6 for more details): Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Let X be a compact K¨ahler surface with c2(X) = 0 ∈ H4(X, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then: (1) if KX is nef, then X is strongly isogenous to a torus product, namely X = (T ×Y )/G, where G acts freely via a diagonal action on T ×Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (2) if X is a minimal surface, but KX is not nef, then X is a P1-bundle over an elliptic curve E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (3) if X is non minimal, then X is a blow up of a P1-bundle over a curve C of genus g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the first two cases c1(X)2 = c2(X) = 0, while in the last case χ(X) = 1 − g, hence c1(X)2 = K2 X < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If c1(X) = 0 ∈ H2(X, R), then S is either a complex torus, or a hyperelliptic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If c1(X) = 0 ∈ Pic(X), then X is a complex torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' There are three features in our above theorem for n = 2: (I): in case (1) where KX is nef, X need not be a Pseudo-Abelian variety;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (II): since χ(OX) ≤ 0, then X always possesses a non zero holomor- phic 1-form ω ∈ H0(Ω1 X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (III): if moreover X is minimal, then ω is everywhere non vanishing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (IV): if X is minimal, then X is birationally covered by a a trivial family of complex tori, that is, there exists a dominant rational map ψ : T × Y ��� X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The first feature (I) contradicts results of Dantoni [Dant43] and Enriques [Enr05b], (indeed the error of Enriques is also reproduced in the classifica- tion theorem of Castelnuovo and Enriques [CastEn15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' These papers put Baldassarri and Roth on the wrong track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The second feature (II) is the one which, as we shall see, fails to hold true in dimension n ≥ 3 as a consequence of vanishing of the top Chern class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 4 FABRIZIO CATANESE while, if (II) holds true, since (−1)ncn(X) is the expected number of zeros of a holomorphic 1-form, then (III) looks plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If one takes (III) as an assumption, that there exists a holomorphic 1-form without zeros, then Baldassarri’s claim holds at least in dimension 3, as shown by [HS21b] (see also a similar result in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The failure of (I) for n ≥ 3 is due to work of Chad Schoen [Schoen88], which shows that in dimension 3 there are simply connected manifolds, actually Calabi Yau manifolds (KX trivial in Pic0(X)) with c3(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Concerning the failure of feature (IV) for n ≥ 3, our result here is that the Schoen threefolds are not birationally covered by a family of isomorphic Abelian surfaces, see Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1, and that they do not admit a fibration onto a surface with general fibres isomorphic to a fixed elliptic curve E, see Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence that they are also birationally far away from being isogenous to a torus product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' What’s left is then to proceed essentially assuming feature (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' More precisely, an easy class of solutions to Baldassarri’s question is provided by the manifolds X which are isogenous to a partially framed or co-framed manifold Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Our definition here is that Z is partially (tangentially) framed if the tangent bundle ΘZ admits a maximal trivial subbundle Ok X, with k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Whereas Z is said to be partially co-framed (cotangentially framed) if the cotangent bundle Ω1 Z admits such a maximal trivial subbundle Ok X, with k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The first definition brings back into play Roth’s Pseudo-Abelian Varieties, see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1, which contains also more general results: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A tangentially k-framed projective Manifold with KX nef is a pseudo-Abelian variety, in particular ΘX ∼= Ok X ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The previous classical Theorem was essentially proven by Roth and, thanks to the work of Fujiki [Fuj78] and Lieberman [Li78], the class of partially (tangentially) framed projective manifolds is understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We also observe, using results of [Li78], [AMN12] (which also contains an excellent survey of the theory of framed K¨ahler manifolds), that the picture becomes more com- plicated if we enlarge our consideration to the wider realm of cKM (compact K¨ahler Manifolds), where more complicated constructions such as Seifert fi- brations, principal torus bundles and torus suspensions enter the picture (see section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We also consider in Section 3 the non K¨ahler case, where we consider a more general notion of suspension over a parallelizable Manifold for which a splitting of the type ΘX ∼= Ok X ⊕ F holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' And we consider all the Manifolds isogenous to a parallelizable Manifold, we call them twisted Hyperelliptic Manifolds since they also enjoy the property of having all the real Chern classes equal to zero (but a converse result is missing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='. 5 Passing to the consideration of the second class of partially co-framed man- ifolds, we see how this interesting class is more mysterious and presents some intriguing questions (see the discussion following Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A new result which we prove here is the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (i) Assume that X is a k-coframed compact K¨ahler manifold, and that q := h0(Ω1 X) = k: then the Albanese map aX is a differentiable fibre bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (ii) If X is projective, k = q = n − 1, and KX is nef, then X is a pseudo- Abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the projective case with KX nef we do not have yet more examples of k-coframed Manifolds than the class of the Pseudo-Abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In fact, we ask two general questions such that a positive answer to both of them would imply that we have nothing more than just the Pseudo-Abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Observe that for k- framed Manifolds and for k-coframed Manifolds the last k Chern classes are zero (in the Chow ring if the Manifolds are projective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' More generally, for the Manifolds isogenous to a k-Torus product it follows from Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1 not only that the last k Chern classes are zero in rational cohomology, but also that all the Chern numbers of X are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the last section we briefly discuss Manifolds with KX nef and with van- ishing Chern numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' For n = 2, as we already illustrated, there are only manifolds isogenous to a torus product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' but, in dimension n = 3, in spite of a quite recent result by Hao-Schreieder [HS21a], describing threefolds with c1c2 = 0, and Kodaira dimension 2, as being birationally isogenous to a torus product, we have also the Schoen threefolds which do not exhibit this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Surfaces with second Chern class equal to zero, and a Chern class characterization of Hyperelliptic and Abelian surfaces The main goal of this section is to establish Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3 mentioned in the Introduction: this is done rerunning the proof of the classification of surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The classification theorem by Castelnuovo and Enriques [CastEn15], was ex- tended by Kodaira [Kod68], and a crucial result is the following (see [Bea78], Chapter 6, Theorem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4 of [CatLi19], and especially the cru- cial Theorem of [Cat22]): Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Let S be a compact smooth complex surface, minimal in the strong sense that KS is nef, and such that χ(S) = 0 (which implies that K2 S = 0 and the topological Euler number e(S) = c2(S) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Equivalently, assume that KS is nef, and that e(S) = c2(S) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 6 FABRIZIO CATANESE Then X is strongly isogenous to a torus product, namely X = (T × Y )/G, where T is a torus of positive dimension (1 or 2) and G acts freely via a diagonal action on T × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' More precisely, either 1) pg(S) = 1, q(S) = 2, and S is a complex torus A (a hyperelliptic surface of grade 1), or 2) pg(S) = p, q(S) = p + 1 and S is isogenous to an elliptic product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' S is the quotient (C1 × C2)/G of a product of curves of genera g1 := g(C1) = 1, g2 := g(C2) ≥ 1, by a free action of a finite group of product type (that is, G acts faithfully on C1, C2 and we take the diagonal action g(x, y) := (gx, gy)), such that, if we denote by g′ j = g(Cj/G), then g′ 1 + g′ 2 = p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Case 2) bifurcates into two subcases: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1,p) g′ 1 = 1 (hence G acts on C1 by translations), C2/G has genus p, and we assume 1, for p = 1, that g2 ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='0,p) g′ 1 = 0 (hence C1/G ∼= P1), C2/G has genus p + 1 = q(S), and we assume 2, for p = 0, that g2 ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' here the image of Albanese map α : S → Alb(S) equals C2/G ⊂ Alb(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1,0) with g2 = 1 is the case where S is a properly hyperelliptic (bielliptic) surface (a hyperelliptic surface of grade ≥ 2): S = (E1 ×C2)/G, where E1, C2 are elliptic curves, and G acts via an action of product type, such that G acts on E1 via translations, and faithfully on C2 with C2/G ∼= P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In this case all the fibres of the Albanese map are isomorphic to C2, P12(S) = 1, and S admits also an elliptic fibration ψ : S → C2/G ∼= P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the other cases (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='0,p), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1,p), for p ≥ 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1,0) with g2 ≥ 2, S is isogenous to a higher genus elliptic product, this means that C2 has genus g2 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Here S is properly elliptic and P12(S) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The cases are distinguished mainly by the geometric genus pg(S) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the case of the torus and of the hyperelliptic surfaces KS is numerically equivalent to zero, whereas in the other cases KS is not numerically equiva- lent to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Moreover, the three cases are also distinguished (notice that c1(S) is the class of the divisor −KS) by KS = 0 ∈ Pic(S) for the case 1) of a complex torus, c1(S) = 0 ∈ H2(S, Z) but KS ̸= 0 ∈ Pic(S) in the case of properly hyperelliptic surfaces, 1to exclude that we are in case 1) 2to exclude that we are in case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1,0) VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='. 7 c1(S) ̸= 0 ∈ H2(S, Q) in the other cases where S is isogenous to a higher genus elliptic product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The part which is not contained in the cited sources concerns how to distinguish the several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Everything is straightforward, except the assertion that for hyperelliptic surfaces (case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1,0) with g1 = 1)) the first integral Chern class is zero: but this is a special case of our result on Bagnera de Franchis manifolds, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1 of [Cat23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ The following can be readily checked: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In case 1) A acts transitively and freely on A, so Aut0(S) has dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1,p), Aut0(S) has dimension 1: C1 = E1 acts on S, transitively on the orbit closures, but with stabilizer H ⊂ G ⊂ E1 for the classes of points (x, y) ∈ E1 × C2 such that Hy = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Finally, in case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='0,p ), Aut0(S) has dimension 0 : there is no action of C1 on S, but the general fibres of the Albanese map α : S → C2/G are isomorphic to C1 (a finite number shall only be isogenous to C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' With a weaker notion of minimality we have a counterexample to the pre- vious Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1, as we shall now see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If the smooth surface satisfies c2 1 = c2 = 0, that is K2 S = e(S) = 0, then S is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' By the Noether’s formula, from K2 S = 0 and e(S) = c2(S) = 0 follows χ(OS) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' And χ(OS) is a birational invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If S is not minimal, then S is the blow-up of a minimal surface S′ with e(S′) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' By Castelnuovo’s theorem S′ is a ruled surface with q(S′) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' But then 0 > χ(OS′) = χ(OS) = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Consider the minimal surfaces S with Chern numbers c2 1 = c2 = 0, that is with K2 S = e(S) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If KS is nef, S is isogenous to a product Y ×A, with a torus A of dimension ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If KS is not nef, then S is a P1-bundle over an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1 there remains only to consider the case where KS is not nef, hence S is ruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Therefore, since the case of S = P2 yields e(S) = 3, we must have a P1-bundle over a curve C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In this case, since 0 = e(S) = 4(1 − g(C)), we get that C has genus g(C) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If we take an elliptic curve B, and a vector bundle V of rank 2 on B, say V = L ⊕ M, where deg(L) = deg(M), then P(V ) is minimal, 8 FABRIZIO CATANESE but the group of automorphisms of P(V ) consist of C∗ for general choice of L, M (see [Mar71], also Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3 of [CatLiu21] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence the action is not transitive on the orbit closures, because the two sections of P(V ) are left invariant by the automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' For completeness we show that: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Surfaces in the class (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='0,0) do exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Let G := (Z/2)3, and make it first act on an elliptic curve C1 as the group of transformations z �→ ±z + η, 2η = 0, so that G has generators η1, η2, ǫ, where ǫ(z) = −z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' To get a second action on C2 such that C2/G =: E2 is an elliptic curve, we take E2 to be an elliptic curve, B = {x1, x2} a branch set, so that π1(E2 \\ B) = ⟨α, β, γ1, γ2|γ1γ2 = [α, β]⟩, hence H1(E2 \\ B) = Zα ⊕ Zβ ⊕ Zγ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Define µ : H1(E2 \\ B) → G by: µ(α) = ǫ, µ(β) = η1, µ(γ1) = η2 ⇒ µ(γ2) = η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We want to prove that the product action of G on C1 × C2 is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' To this purpose we observe that η2 has eight fixed points on C2, and, since C2 has genus 5, E′ 2 := C2/⟨η2⟩ is an elliptic curve, so that E′ 2 → E2 is ´etale, that is, G/⟨η2⟩ acts freely on E′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The conclusion is that the only element acting on C2 with fixed points is η2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' since η2 atcs freely on C1, the product action is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ We end this section observing that if S is a minimal surface with e(S) = 0, then either S is a P1 bundle over an elliptic curve, or KS is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the latter case it must be K2 S = 0, since for K2 S < 0 S is ruled, hence KS is not nef, and for K2 S > 0 S is of general type, and then e(S) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The following is the correction of the theorem of Dantoni [Dant43];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' it show that if S is not a torus then the surface is ‘elliptic’ only in the weak sense that it has a rational map with fibres elliptic curves, and not in the strong sense that a torus of dimension at least 1 should act on S, as we saw in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2 and in the previous remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4 (it was also stated as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3 in the Introduction with only slightly different wording).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A surface S with e(S) = 0 is either the blow up of a P1- bundle over a curve of genus at least 2, or it is minimal, and then it is either a complex torus, or a hyperelliptic surface, or it is isogenous to a higher genus elliptic product, or it is a P1-bundle over an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the last three cases S contains a 1-dimensional family of isomorphic el- liptic curves whose union is dense in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If the canonical divisor is numerically trivial, then S is either a complex torus, or a hyperelliptic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='. 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Pseudo Abelian Varieties and Varieties Isogenous to a k-Torus-Product The first aim of this section is to show how to derive from Roth’s original definition of the Pseudo Abelian Varieties the modern definition which we adopt in this article;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' and how Roth’s definition leads, in the case of compact K¨ahler Manifolds, to the notion of Seifert fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Later on we dwell on other notions, for instance we give a new definition, of suspension over a parallelizable Manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Leonard Roth [Roth54], [Roth55] defined the Psudo-Abelian varieties as follows: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (Roth) A projective variety X of dimension n is said to be Pseudo Abelian of order k if i) Aut0(X) contains a complex torus T, of maximal dimension = k, with the property that its orbits are all of dimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Recall in fact the following theorem by Fujiki and Lieberman [Fuj78], [Li78] Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If X is a compact K¨ahler manifold (or more generally in the Fujiki class) then there is an exact sequence of groups 1 → L → Aut0(X) → TX → 1 where L is a linear algebraic group, and TX is a complex torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The Lie Algebra of Aut0(X) is A := H0(ΘX), while the Lie Algebra of L is the space L of vector fields which admit zeros (L is an ideal in the Lie Algebra A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If X is not uniruled, then L = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We explain now how to derive from Roth’s definition a simpler one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We have an action T × X → X, and we denote by Tx the orbit of x, image of T × {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence the orbits Tx of T give a variety V of dimension n − k in the Hilbert scheme H (Douady space) of X, and the restriction of the universal family to V, φ : U → V, yields U which maps isomorphically to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In fact, dim(U) = n = dim(X), and the identity of X factors as: X → U → V × X → X, where x ∈ X �→ (Tx, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence we may write: φ : X → V, and since the action of T, a : T × X → X commutes with the projection over V, and is effective, then the general fibre of φ is isomorphic to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The other fibres are instead of the form T/G′, where G′ is a finite subgroup of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since the general fibres are isomorphic, over an open set V′ of V we 10 FABRIZIO CATANESE have a holomorphic bundle (for instance, as a consequence of Kuranishi’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Because of the action of T on the fibres the monodromy of the bundle centralizes the group of translations hence the monodromy transformations consist of translations, and we have a principal bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If X is projective, then the monodromy is finite, hence we get a finite group G ⊂ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The fibration is isotrivial, hence there exists a finite Galois base change f : Y → V with group G such that the fibre product is birational to a product Y ×V X ∼ Y × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' At each point of V the local monodromy G′ is a subgroup of G, hence the fibre product Y ×V X yields an unramified covering of X ( since G′ yields an unramified covering of the corresponding fibre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence the fibre product Y ×V X is smooth, is birational to Y × T, and all the fibres are isomorphic to T: therefore, Y ×V X ∼= Y ×T, compatibly with the projection onto Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' This motivates an equivalent definition, and some related definitions: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (I) A complex manifold X of dimension n is said to be a k- Pseudo-Torus (or Pseudo-Torus of order k) if there is a torus T of dimension k, a manifold Y , and a finite Abelian group G acting on T faithfully via translations, acting faithfully on Y , such that the quotient of the product action is isomorphic to X: X = (Y × T)/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (II) If moreover X is projective, we shall say that X is a Pseudo-Abelian Variety of order k in the strong sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (III) A compact K¨ahler Manifold X is said to be Seifert fibred, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' [Li78], if there is a finite abelian unramified covering Z → X with Abelian Galois group G (hence X = Z/G) such that Z is a principal bundle Z → Y , with fibre a (positive dimensional) complex torus T, and where the action of G commutes with the action of T on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (IV) A compact complex manifold X is called ([AMN12]) a suspension over a complex torus T = Ck/Λ if there is a compact complex manifold Y and a homomorphism ρ : Λ → Aut(Y ) such that X = (Ck × Y )/Λ, λ(z, y) := (z + λ, ρ(λ)(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (a) The reason for definition (III) is that if X is only a cKM, and it has an action of a complex torus T, with all the orbits tori of the same dimension, then the global monodromy is not necessarily finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' But the local monodromies around the points corresponding to multiple fibres are finite subgroups of T, and since we have a finite number of them, the local monodromies generate a finite subgroup G of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Associated to this subgroup there is an covering Y → V which is a principal T-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='.11 (b) In the case of definition (IV) of a suspension over a torus, Ck acts on X via translations on the first summand, but in general there is no complex torus acting on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Indeed the Ck-orbits are the projections π(Ck × {y}), which are isomorphic to Ck/Staby, hence they do not need be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The suspension over a torus is a Pseudo-torus if and only if the subgroup Im(ρ) ⊂ Aut(Y ) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Equivalently, if and only if all orbits are tori (which amounts to Staby being a subgroup of finite index in Λ for all y ∈ Y ): since then Ker(ρ) has finite index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In particular the suspension is Pseudo-Abelian if and only if it is a Pseudo- torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (c) Observe that for such a suspension we have a splitting of the tangent bundle ΘX ∼= Ok X ⊕ F, hence also of the cotangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In [AMN12] Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4, page 1005, it is observed that a suspension X as above is K¨ahler if and only if Y is K¨ahler and a finite index subgroup Λ′ ⊂ Λ maps to Aut0(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In this case, taking G := Λ/Λ′, there is a Galois covering Z of X with group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If moreover Aut0(Y ) is trivial, then Z = T ′ × Y and we have a pseudo-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If Aut0(Y ) is non-trivial, one cannot conclude that Z is a principal torus bundle, since the action of Λ′ on Y need not be properly discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (d) If X is Seifert fibred, then X = Z/G, and we have an exact sequence corresponding to the principal bundle Z → Y : 0 → Ok Z → ΘZ → FZ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since the action of G commutes with the action of T, the exact sequence descends to X, and we have 0 → Ok X → ΘX → F → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' As we shall soon see, the above notions of Seifert-fibred manifold, or of pseudo-torus, and of suspension over a torus are not general enough in order to deal with manifolds with Chern classes trivial in real cohomology, hence we give another definition (important for the case of projective varieties): Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A compact complex manifold X of dimension n is said to be (i) Isogenous to a k-Torus product (MITP of order k ), if X is isogenous to a product Y × T, where T is a torus of dimension k > 0, Y is a compact complex manifold, and k is maximal with this property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (ii) Strongly Isogenous to a k-Torus product if there is a torus T of dimension k > 0, a compact complex manifold Y , and a finite 12 FABRIZIO CATANESE group G acting on T, and Y such that G acts freely on the product Y × T via the induced diagonal action (g(t, y) := (g(t), g(y)) and moreover: X = (Y × T)/G, and k is maximal with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If X is projective, we shall say that it is a Variety Isogenous to a k-Torus product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If X is not uniruled, and X is a compact K¨ahler Manifold, the two notions (i) and (ii) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Clearly (ii) implies (i), hence it suffices to show that (i) implies (ii), and we shall do it by induction on the number h of finite unramified maps which make X isogenous to such a product Y × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Step (a(1)) For h = 1, assume that there is a finite unramified map f : X → Y × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then, taking the Galois closure Z, and letting G be the Galois group, we see that Z is associated to an epimorphism ψ : π1(Y ) × π1(T) → G, and we denote G1 := ψ(π1(Y )), G2 := ψ(π1(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then G is a quotient of G1 × G2 by the normal subgroup G1 ∩ G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' To the epimorphism φ : π1(Y ) × π1(T) → (G1 × G2) corresponds a product Manifold Y1 × T1, and clearly Z = (Y1 × T1)/G1 ∩ G2, hence X is as desired, since G1 × G2 acts on T1 diagonally (and on T1 via translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence X is also a quotient of (Y1 × T1) by another subgroup of G1 × G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Step (b(1)) For h = 1, assume that there is a finite unramified map f : (Y × T) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If f is not Galois, take the Galois closure Z, and denote by Γ the Galois group of Z → X, and by G the subgroup which is the Galois group of Z → (Y × T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Arguing as in Step (a(1)), we find a Galois cover (Y1 × T1) → (Y × T) with group G1 × G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We claim that the unramified cover (Y1 × T1) → X is Galois;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' to establish this claim it suffices to show that the elements γ ∈ Γ admit lifts to (Y1 ×T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Now, Z has a split tangent bundle ΘZ = F⊕Ok Z, and since Z is not uniruled, it follows from the Theorem of Fujiki and Lieberman, and by the maximality of k, that H0(ΘZ) = H0(Ok Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence the splitting is uniquely determined, and the group T2 maps onto Aut0(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The fibration Z → (Z/T2) is then isotrivial, and (Y1 × T1) is the canonical product pull back under Y 1 → (Z/T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Therefore the lifting property holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Step (a(h)) Assume that X2 = (Y2 × T2)/G2, as in (ii), and that there is a finite unramified map f : X → X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then, taking a component X′ of the fibre product X ×X2 (Y2 × T2), and ap- plying (a(1)) we get that X′, hence also X, admits (Y1×T1) as an unramified covering, hence we can conclude because of Step (b(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Step (b(h)) Assume then that X2 = (Y2×T2)/G2, as in (ii), and that there is a finite unramified map f : X2 → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='.13 Then taking the composition we get an unramified map (Y2 ×T2) → X, and again we are done by Step (b(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ The following are easy important properties of such Manifolds and Varieties Isogenous to a Torus product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Let the complex Manifold X of dimension n be Isogenous to a k-Torus product Y × T where dim(T) = k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then the integral Chern classes ci(Y × T) ∈ H∗(Y × T, Z) vanish for i ≥ n − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' And the rational Chern classes ci(X) ∈ H∗(X, Q) vanish for i ≥ n − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Moreover all the Chern numbers of X vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If moreover X is (respectively is isogenous to) a pseudo-torus, or more gen- erally the suspension over a torus, or is Seifert fibred, then all the integral Chern classes ci(X) ∈ H∗(X, Z) vanish for i ≥ n − k + 1 (respectively the corresponding rational Chern classes vanish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since the tangent bundle of Y × T is the direct sum of the pull back of the tangent bundle of Y with the pull back of the tangent bundle of T which is trivial, the total Chern class of Y × T is the pull back of the total Chern class of Y , and this proves the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The second assertion follows since the Chern classes of X pull back to the Chern classes of the product Y × T, and H∗(X, Q) = H∗(Y × T, Q)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The third assertion follows since any isobaric polynomial of weight n in the Chern classes of X is a rational multiple of the same isobaric polynomial of weight n in the Chern classes of Y , and we are assuming k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The last assertion follows since the tangent bundle of X either splits as ΘX = Ok X ⊕ F, or has a bundle exact sequence 0 → Ok X → ΘX → F → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' On the non K¨ahler case In the realm of compact complex manifolds, the Manifolds X with a holo- morphically trivial tangent bundle ΘX ∼= On X are called the parallelizable Manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' By a Theorem of Wang [Wang54] every parallelizable Manifold is a quotient X = G/Λ, where G is a complex Lie group and Λ is a cocompact discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since every Manifold isogenous to a parallelizable Manifold has trivial real Chern classes, it would be interesting to study this class in more detail (in the K¨ahler case the parallelizable Manifolds are the complex tori, and the Manifolds isogenous to a torus are the Hyperelliptic Manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 14 FABRIZIO CATANESE Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Every Manifold Y isogenous to a parallelizable Mani- fold X′ = G/Λ′ is obtained from another parallelizable Manifold X = G/Λ dividing by the action of a finite group G acting freely on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' G is a group of affine transformations of X, of the form f(x) = gF(x) where g ∈ G and F has a fixed point x0 and has derivative DF determined by the value of the derivative at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The quotient Manifold Y := X/G will be called a twisted hyperelliptic Manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If Y is isogenous to X′, then Y has also G as universal cover, as it is trivial to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence we can write Y = G/Γ, where Γ acts freely on G and with compact quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Observe that any isogeny amounts, up to isomorphism, to a finite index containment between fundamental groups, viewed as discrete groups of au- tomorphisms of G (since G is simply connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' It is a general fact that the intersection H ∩K of two finite index subgroups H, K of a group Γ′ is again of finite index in Γ′ (it is the stabilizer of the orbit of H × K on Γ′/H × Γ′/K 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence, if Γ2 is of finite index in Γ1 and Γ3, and Γ4 is of finite index in Γ3 and Γ5, then Γ2 ∩ Γ4 is of finite index in Γ3, hence also in Γ1 and Γ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Therefore, since we can always assume that the number of direct relations yielding the isogeny is even (introducing an equality step), we can reduce to the case where Γ and Λ′ contain a finite index subgroup in both of them, call it Λ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' By replacing X′′ = G/Λ′′ by the Galois closure of X′′ → Y , we obtain Y = X/G as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Now, if f ∈ G, there exists g ∈ G such that F := g−1f has a fixed point x0 := Λ on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The derivative of F is a holomorphic section of the trivial bundle DF ∈ H0(End(On X)) gotten by the action on the space of left invariant vector fields: since X is compact, DF is constant, hence f(x) = gF(x) is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ Of course each such automorphism F lifts to an automorphism ˜F of G which must have the property of fixing the identity and of sending Λ to Λ, since we must have ˜F(gλ) = ˜F(g)π1(F)(λ) ⇒ ˜F(λ) = π1(F)(λ), this is exactly as it happens for automorphisms of complex tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We give now a simple example of such Manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Consider the Iwasawa Manifold X = G/Λ, where G is the complex Lie group of unipotent 3 × 3 complex matrices, that is, upper triangular matrices and with diagonal entries equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 3see for instance: stack Overflow VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='.15 And let Λ be the subgroup of matrices with upper-diagonal entries belonging to the ring Z[i] of Gaussian integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then we define the following automorphism of X, such that r[(z12, z21, z13)] := [(iz12, z21 + 1/4, iz13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then r has order 4, and generates a group G ∼= Z/4 acting freely on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The quotient is a twisted Hyperelliptic Manifold with H0(Ω1 Y ) of dimension 1 and generated by a closed holomorphic 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='. Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The automorphism is well defined, since the product zλ has coordi- nates (z12 + λ12, z21 + λ21, z13 + λ13 + z12λ21), hence we infer [r(z)] = [r(zλ)] since [r(zλ)] = [(iz12 + iλ12, z21 + λ21 + 1/4, iz13 + iλ13 + iz12λ21)] = = [(iz12, z21 + λ21 + 1/4, iz13 + iz12λ21)] = [r(z)λ′], where λ′ = (0, λ21, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' That the action is free follows since for the second coordinate z21 �→ z21 + 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' It is known that H0(Ω1 X) is generated by ω1 := dz12, ω2 := dz21, ω3 := dz13 − z21dz12, which are mapped by r to iω1, ω2, iω3 − (i/4)ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence H0(Ω1 X)G is generated by ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ We leave aside here the general discussion of the structure of twisted Hy- perelliptic Manifolds, but we raise a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The twisted Hyperelliptic Manifolds, being isogenous to par- allelizable Manifolds, are examples of compact complex manifolds with all the real Chern classes ci,R(X) = 0, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Are there other examples ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We can also define another general class of Manifolds, which we call Sus- pensions over a parallelizable Manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (S-P-M) A compact complex manifold X is called a sus- pension over a parallelizable Manifold Z = G/Λ if there is a compact complex manifold Y and a homomorphism ρ : Λ → Aut(Y ) such that X = (G × Y )/Λ, λ(z, y) := (zλ, yρ(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If X is a suspension over a parallelizable Manifold Z, then we have a splitting of the tangent bundle ΘX = Ok X ⊕ F, where k = dim(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 16 FABRIZIO CATANESE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The splitting follows right away since Λ has a product action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Moreover, the first summand is trivial because Z = G/Λ is parallelizable and the first summand is just the pull-back of the tangent bundle ΘZ for the projection X → Z, with fibres isomorphic to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the case where ρ has finite image, we get that, Λ′ being defined as Λ′ := ker(ρ), then X is isogenous to Z′ × Y , where Z′ := G/Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' And we get then triviality of the Chern numbers of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the next section we shall show how the occurrence of such a splitting ΘX = Ok X ⊕ F is understood under the K¨ahler assumption as the structure of a suspension over a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' It is interesting to see whether a similar characterization of suspensions over a parallelizable Manifold holds also for compact complex manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' For this purpose one has to take k maximal, and the first key point would be to see whether H0(Ok X) ⊂ H0(ΘX) is a Lie subalgebra G: then we would have an action on X of the simply connected Lie group G associated to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Partially framed and co-framed manifolds We begin with a simple definition yielding complex manifolds with all the last k integral Chern classes equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A complex manifold X of dimension n is said to be k- tangentially framed, or simply k-framed, if the holomorphic tangent bundle ΘX admits a trivial subbundle ∼= Ok X, and where k > 0 is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A complex manifold X of dimension n is said to be k-cotangentially framed, or simply k-co-framed, if the holomorphic cotangent bundle Ω1 X admits a trivial subbundle ∼= Ok X, and where k > 0 is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In the rest of the section we shall use the results of [Li78], [Fuj78], [AMN12], often for simplicity we might refer to the exposition given in the last paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Assume that X is a k-coframed compact K¨ahler man- ifold and let W ⊂ h0(Ω1 X) be the corresponding maximal vector subspace consisting entirely of nowhere vanishing holomorphic 1-forms (of dimension k ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then W determines an analytically integrable foliation with trivial normal bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' i) If moreover the subspace W corresponds to a quotient torus T ′ of Alb(X) = H0(Ω1 X)∨/(H1(X, Z)/Tors), then the foliation is algebraically integrable, consisting of the fibres of Ψ : X → T ′, which is a differentiable fibre bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' ii) If we have a splitting Ω1 X ∼= Ok X ⊕ F∨, then Ψ is a holomorphic fibre bundle with fibre Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' iii) If moreover Y has finite automorphism group, then X is a k-Pseudo- Torus product X = (Y × T)/G with dim(T) = k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A subspace W which is maximal with the property that all forms ω ∈ W \\ {0} are nowhere vanishing, has a basis ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' , ωk such that the VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='.17 forms ωj are linearly independent at each point, hence they generate a trivial rank k subbundle of Ω1 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The associated foliation is analitically integrable, because X is a cKM, and holomorphic 1-forms are closed, hence the distribution induced by W is integrable, and spans a trivial subbundle of the cotangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The foliation on X is induced by a foliation on Alb(X), corresponding to the annihilator of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' This foliation is algebraically integrable if, as we assume, it corresponds to the projection onto a quotient torus T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The composed map Ψ : X → T ′ has fibres of dimension k, which are there- fore union of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since the fibres are smooth, if they are not connected, we would get by the Stein factorization on unramified covering of T ′, which is again a quotient of Alb(X) by the universal property of the Albanese variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence we may assume that the fibres of Φ are connected, and we have a differentiable fibre bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The splitting Ω1 X ∼= Ok X ⊕ F∨ guarantees that the Kodaira-Spencer map for the family is identically zero, hence by Kuranishi’s theorem we have a holomorphic bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If this is a holomorphic bundle, with fibre Y , and Aut(Y ) is finite, there is an unramified covering T → T ′ with group G such that the pull back is a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Therefore X = (T × Y )/G, where G acts on T via translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Let X be a compact K¨ahler complex Manifold of dimension n: then X is the suspension over a torus T with dim(T) = k > 0 if and only if there is a k-framing yielding a partial tangential splitting ΘX ∼= Ok X ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A k-framing of X yields the structure of a Seifert fibration on X in the case where h0(ΘX) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Moreover, a k-framed projective manifold X is a suspension over an Abelian Variety and is indeed a Pseudo-Abelian variety if KX is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We have already seen that for a suspension over a torus we have such a partial tangential splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Conversely, recall that H0(ΘX) is the Lie algebra of the Lie group Aut(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The Lie Algebra H0(ΘX) =: AX contains the Lie ideal H1 X of the vector fields admitting zeros, and there is (see [AMN12], page 1002), a direct sum AX = H1 X ⊕ A, where A is a maximal Abelian subalgebra generated by nowhere vanishing vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='14 of [Li78], H1 X is precisely the subspace of HX yielding the zero flow on Alb(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If H1 X = 0 then the trivial subbundle yields k everywhere linearly inde- pendent vector fields, which, see [Fuj78], [Li78], and also Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2 of 18 FABRIZIO CATANESE [AMN12], generate the action of a k-dimensional complex torus T with smooth orbits Tx, quotients of T by a finite group Hx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence X is Seifert fibred, as we have discussed earlier, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='9 by Lieberman [Li78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If we have a tangential splitting ΘX ∼= Ok X ⊕ F, we get a corresponding cotangent splitting, Ω1 X ∼= Ok X ⊕ F∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' It suffices, by the preceding proposition, to show that the coframing defines a subspace W ⊂ H0(Ω1 X) corresponding to a quotient torus T ′ of Alb(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since every automorphism of X yields an affine action on Alb(X), the action of the torus T, which spans W ∨, yields a subtorus A of Alb(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then we define T ′ := Alb(X)/A, and we get Ψ : X → T ′ which is is a holomorphic bundle, with parallel transport given by the action of T: hence X is the suspension over a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Finally, if X is projective, T ′ is an Abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We defer the reader to Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3 of [AMN12] for the proof of the last assertions, see also [Li77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3 of [AMN12] proves the following very interesting result: if X is a compact K¨ahler manifold which is k-framed, then X admits a small deformation which is a suspension over a k-dimensional torus, and which is a k-Pseudo-Torus if Kod(X) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If X is projective and k-framed, as we saw, they show that X is Pseudo- Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (i) Assume that X is a k-coframed compact K¨ahler manifold, and that q := h0(Ω1 X) = k: then the Albanese map aX is a differentiable fibre bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (ii) If X is projective, k = q = n − 1, and KX is nef, then X is a pseudo- Abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The first assertion follows from i) of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We prove now the second assertion (ii): by the assumption q := h0(Ω1 X) = n − 1 (i) applies, and the fibres of aX are smooth curves of genus g ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since the fibration induces a holomorphic map to the Teichm¨uller space Tg which is biholomorphic to a bounded domain, this map is constant and we have a holomorphic fibre bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since X is projective, the monodromy is finite, so there exists a finite un- ramified map A′ → A := Alb(X) such that the pull back is a product, hence X is a Pseudo-Abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If k = n − 1, but q > n − 1, by the exact sequence 0 → On−1 X → Ω1 X → F → 0, the quotient line bundle F ∼= OX(KX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='.19 Since q > n − 1, there is a section σ of Ω1 X inducing a non zero element of H0(F), hence we have an effective divisor D which is linearly equivalent to KX, hence D is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If we knew that h0(OD) = 1, or that h1(OX(−KX)) = 0, we could conclude that we have a splitting of the above exact sequence, since such a σ would be given in local charts Uα by vectors σ1,α, σ2,α such that D = {σ2,α = 0}, and σ1,β = σ1,α + ψα,βσ2,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then σ1,β ≡ σ1,α on D, hence it is a vector of constants, therefore we may assume that σ1,α vanishes on D, therefore σ yields a splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence X is a Pseudo-Abelian Variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' This example concerns k-coframed varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (1) A smooth fibration onto an Abelian variety need not be isotrivial, and not even a holomorphic bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Let in fact A be an elliptic curve and assume that we have an embedding j : A → S, where S is a surface of general type, say with KS ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Let Γ ⊂ A × S be the graph of j, and take X to be the blow up of A × S with centre the smooth curve Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then the differentiable fibre bundle f : X → A is not a holomorphic bundle, since Aut(S) is finite, and the pairs (S, p1) and (S, P2) are not isomorphic for general P1, P2 ∈ j(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (2) In this case KX is not nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Else, one may ask whether the fibration is isotrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' This has been shown by Kovacs in the case where the fibres have ample canonical divisor [Kov97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We pose now the following general questions: Question 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (a) Assume that X is a k-coframed projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Does there exist a coframing V (a subbundle V ∼= Ok X of Ω1 X) such that i) of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2 holds, namely the subspace W = H0(V ) ⊂ H0(Ω1 X) defines a quotient Abelian variety A′?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (b) If X is projective with KX nef and it admits a fibration f : X → A′ onto an Abelian variety A′ with all fibres smooth, is then f is a holomorphic bundle and hence X is a pseudo-Abelian variety?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' As already discussed, question (b) is motivated by the result of [Kov97], and it fits into a pattern of conjectures of classification theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Question (a) asks, in the case where X is projective, whether the subspace W ⊂ H0(Ω1 X) corresponds to an Abelian subvariety of Alb(X), equivalently, whether Λ2k(W ⊕ ¯W) is a point defined over Q in the Grass- mann manifold Grass(2k, H1(X, C)) ⊂ P(Λ2k(H1(X, C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' One may conjecture that this is true if V is geometrically defined, that is, it is unique and invariant by all automorphisms in Aut(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Here it is important that X is projective, and one may reduce to the case where X is defined over an algebraic extension of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 20 FABRIZIO CATANESE I initially thought that question (a) has a positive answer, trying to use (see [Miya87]) the generic semipositivity of Ω1 X for a non uniruled variety, and its Harder-Narasimhan filtration to define a geometrically unique coframing V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' But Deligne spotted a trivial mistake in my reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Mathematical and Historical comments on Baldassarri’s paper [Bald56] and the questions it suggests Baldassarri [Bald56] was trying to characterize the smooth projective man- ifolds X whose first h canonical systems K0(X), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' , Kh−1(X) have degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' At that time, even if Baldassarri in his Ergebnisse book [Bald56book] (the book was rather influential, it was for instance translated in Russian by Manin) was exposing the new methods in the theory of Algebraic Varieties, the concept of canonical systems of all dimension was rather based on more geometric approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The canonical systems of a manifold (see [Roth56]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' defined in a geometric way by Todd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Eger and later in a simpler way by Beniamino Segre [Seg52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' [Seg54] 4 after proposals made by Severi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' were shown in 1955 by Nakano [Nak55] to be the so called Chern classes of the cotangent bundle (see [At98] for an historical account and [Ful84] as a general reference): more precisely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' up to sign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' to the system Kh(X) corresponds the Chern class cn−h(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' where n is the dimension of X (and we can consider the Chern classes either as elements of the Chow ring of X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' or as integral cohomology classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence we formulate the questions raised by the work of Baldassarri in terms of Chern classes: Baldassarri dealt with the question of characterizing all the projective varieties X such that the rational Chern classes ci(X) ∈ H∗(X, Q) vanish for n − k + 1 ≤ i ≤ n, but not for i = n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' This question is still wide open, except for the case k = n, as we saw in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Already in the surface case the condition that all the Chern numbers are zero (this means K2 X = e(X) = 0) is not sufficient, as we have seen, to imply that X is a complex torus or a hyperelliptic surface (in the old notation a torus was also called a hyperelliptic surface, of grade, or rank, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' But it implies (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='6) that the surface is either a torus or it is birationally (but not necessarily biregularly) covered by a 1-dimensional family of isomorphic elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Baldassarri’s question also suggests (see question (IV) below) to classify, if possible, the varieties (cKM) X whose integral Chern classes ci(X) ∈ H∗(X, Z) vanish for n − k + 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Again, the question is quite open, even for the case k = n, as we saw in Part I [Cat23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 4using the embedding covariants for the case of the diagonal ∆X ⊂ X × X: this approach was also later followed by Grothendieck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='.21 The paper [Bald56] by Baldassarri, suggesting that the Pseudo Abelian Va- rieties of Roth might be the varieties with such vanishing top real Chern classes, motivates indeed some more general questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' But, in view of what we have seen in the case of surfaces, one has to include first of all a condition of minimality of X in a strong sense, for instance that KX is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Or require only a birational isomorphism with a product (because a P1-bundle over an elliptic curve T is only birational to P1 ×T), or that X be only birational to (Y × T)/G, where the action is only free in codimension 1 (as in [HS21a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Todd’s review of [Bald56] pointed out a wrong intermediate result, admitting as counterexample the blow up X of P3 with centre a smooth curve of genus 3, which is a regular manifold X having c3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Todd’s counterexample could somehow be quickly dismissed as only pointing out the need to assume that X is a minimal manifold, say with KX nef, as we already mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We observe here however that the flaw is not simply a technical problem, the main claim by Baldassarri that the solutions are Roth’s Pseudo-Abelian va- rieties is incorrect also for minimal manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Because for instance the class of Manifolds isogenous to a k-torus product is a larger class than the class of Roth’s Pseudo-Abelian varieties, and varieties in this class are solutions to Baldassarri’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' On the other hand Todd saw that the crucial flaw in Baldassarri’s argument was the attempt to show that manifolds with top Chern class equal to zero have positive irregularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' What is historically interesting is to observe that the ‘original sin’ of [Bald56] was to try to extend to higher dimension some wrong results by Enriques, Dantoni and Roth (indeed the error of Enriques is also reproduced in the classification theorem of Castelnuovo and Enriques [CastEn15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' In fact, the work by Baldassarri and Roth is inspired by a paper by Dan- toni [Dant43], devoted to the minimal surfaces X with c2(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Dantoni uses surface classification, especially an article by Enriques of 1905 [Enr05b], claiming that the non ruled surfaces with these properties are the hyperel- liptic manifolds and the ‘elliptic’ surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' But ‘elliptic’ for Enriques here does not have the same standard meaning introduced later by Kodaira and others: Enriques requires the action of a fixed elliptic curve on X, with all orbits of dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Enriques and Dantoni in their classification omit to consider the case of quotients X = (E × C)/G where the action of the finite group G is free, of product type, but G does not act on the elliptic curve E via translations, and moreover C is a curve of genus g ≥ 2, such that the quotient C/G is an elliptic curve (see section 2, and for instance [CatLi19] or [Bea78] for the special case pg = 0, and [CB] or [Cat22] for the general case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Indeed, in this latter case the automorphism group of X has dimension zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Dantoni’s paper inspired Roth [Roth54] [Roth55] who defined the Pseudo- Abelian varieties as the manifolds admitting the action of a complex torus 22 FABRIZIO CATANESE of positive dimension = k having all orbits of dimension k, and such that k is maximal with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' But for instance, in [Roth53], Roth does not realize about the existence of Hyperelliptic threefolds with automorphism groups of dimension zero, and believes that these are only Pseudo-Abelian varieties with k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The first conclusion is simple: the ‘original sin’ was to consider only quo- tients X = (T × Y )/G where T is a torus, G acts via a product action, which is free and such that G acts on T via translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Obviously under these assumptions T acts on X and the orbits have all the same dimension k = dim(T)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Baldassarri’ s paper suggests the following other questions: Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (I) What can be said about a projective Manifold (respec- tively, a compact K¨ahler manifold) X with KX nef and such that all its Chern numbers are equal to zero?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (II) Is a projective Manifold (respectively, a compact K¨ahler manifold) X with KX nef and such that its rational Chern classes ci(X) ∈ H∗(X, R) vanish for k + 1 ≤ i ≤ n isogenous to a k-Torus Product (respectively, isogenous to a k-framed or k-coframed manifold)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (III) Same question for a projective Manifold (or compact K¨ahler manifold) X with KX nef and with all the Chern numbers equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (IV) What can we say about a projective Manifold (or cKM) X whose integral Chern classes ci(X) ∈ H∗(X, Z) vanish for k + 1 ≤ i ≤ n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We shall see in the next section that work of Chad Schoen [Schoen88] gives a negative answer to Questions (II) and (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' a) We have mentioned that Baldassarri’s assertion is wrong already for surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' b) The assertion is also wrong in dim = 3 since here, as shown in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='5 of Part I, [Cat23], there are Hyperelliptic Threefolds whose group of Automorphisms is discrete, hence they are not Pseudo-Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Indeed, if one looks at the paper by Roth [Roth53] on Hyperelliptic threefolds, one sees that Roth does not consider the case of Hyperelliptic Threefolds for which Aut0(X) consists only of the Identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' c) In the same paper Roth calls, following Enriques, [Enr05a], [Enr05b], ‘the elliptic surfaces’ the surfaces such that Aut0(X) is an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Here the modern terminology, introduced by Kodaira, differs: an elliptic surface is a surface admitting a fibration f : S → C with fibres elliptic curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' in general it does not possess non-trivial automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' As we saw in Part I [Cat23], however, there are Hyperelliptic Threefolds and Varieties (hence for them cn(X) = 0) which are regular (H1(OX) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence the crucial fact that Baldassarri wants to use, that for cn(X) = 0 we have an irregular variety possessing a holomorphic 1-form without zeros has to be taken as an assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='.23 To give an idea of the difficulty of the type of questions considered by Bal- dassarri, let us notice that for instance the investigation of varieties such that there is ω ∈ H0(Ω1 X) without zeros has been taken up (before our present investigations) in the last years by Schreieder and Hao ([Schre21] [HS21b]), and a classification has been achieved only in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' What is more interesting is that, under this much stronger assumption of the existence of such a form ω, the results confirms, in a special case, the result claimed by Baldassarri (see also our Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (Hao-Schreieder [HS21b]) Let X be a smooth projective threefold satisfying property (A) : admitting a holomorphic 1-form ω ∈ H0(Ω1 X) without zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then the minimal model program for X yields a birational morphism σ : X → Xmin blowing up smooth elliptic curves which are not contracted by the Albanese map, and such that (2) there is a smooth morphism π : Xmin → A to an Abelian variety of positive dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (3) If the Kodaira dimension is non negative, then a finite ´etale cover of Xmin is a product X′ ∼= A′ × S′, where S′ is smooth projective, and the composite map A′ ∼= A′ × {s′} → A is finite and ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (4) If the Kodaira dimension is equal to −∞, then either (4a) Xmin has a smooth Del Pezzo fibration over an elliptic curve, or (4b) Xmin has a conic bundle fibration f : Xmin → S over a smooth surface S satisfying property (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Moreover, either f is smooth, or A is smooth and the degeneracy locus of f is a finite union of elliptic curves which are ´etale over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If X is a smooth projective threefold satisfying property (A) of admitting a holomorphic 1-form ω ∈ H0(Ω1 X) without zeros, and the Kodaira dimension of X is non negative, then Xmin is a Pseudo-Abelian variety in the sense of Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We are in case (3), and obviously A′ acts on the product A′ × S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since the map A′ → A is finite and unramified, it is a quotient map, with Galois group G, a finite Abelian group of translations of A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We have A′ ×S′ → Xmin → A, and since A′ ×S′ → A′ is a smooth fibration with fibre S′, hence also Xmin → A is a smooth fibration with fibre S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since the pull back of Xmin → A to A′ is isomorphic to the product A′ ×S′, we conclude that A′ acts on Xmin and Xmin = (A′ × S′)/G, where G acts on A′ by translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ The historical conclusion that we can draw is that Baldassarri’s paper, even if vitiated by the ‘original sin’ of trying to extend results which were not correct already in small dimension n = 2, 3, poses some problems which, in spite of the tremendous substantial and technical progress which took place in the last 60 or more years, are still quite open and very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 24 FABRIZIO CATANESE The typical example is in our opinion the question of describing the k-co- framed manifolds X ( see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2 and Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Manifolds with vanishing Chern numbers The title of this section is on purpose ambiguous: one may ask about Man- ifolds for which a certain Chern number vanishes, or all the way consider Manifolds for which all the Chern numbers are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We have given the example of Manifolds X Isogenous to a k-Torus Product as a prototype of manifolds with all the Chern numbers equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We have also observed that, at least in dimension 2, all the surfaces with all the Chern numbers c2 1 = c2 = 0, or the minimal surfaces with c2 = 0, in view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='6, are the manifolds of this type, if KS is nef, or birational to one of this type if S is ruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Because if S is minimal and not elliptic ruled, there exists a Galois ´etale covering S′ → S such that S′ ∼= T × Y , where T is a complex torus with dim(T) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' More generally, we have the class of manifolds X isogenous to a partially cotangentially framed manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If we go up to dimension 3, there are three Chern numbers, c3 1, c3, and c1c2 = χ(OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' A recent result by Hao and Schreieder goes in the direction of answering the question, [HS21a] in the special case where the Kodaira dimension is n − 1: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (Hao-Schreieder) Let X be a minimal model with dim(X) = n and Kodaira dimension n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Then cn−2 1 c2(X) = 0 if and only if X is birational to a quotient Z = (E × Y )/G, where (1) Z has canonical singularities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (2) E is an elliptic curve and Y is a normal projective variety with KY ample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (3) G acts diagonally, faithfully on each factor, and freely in codimension two on E × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Now, the case where X is a threefold of general type (in this case it can be c1c2 = 0, as shown by Ein and Lazarsfeld, [EL97]), is excluded if X is minimal, since then c3 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence the missing cases are the cases of Kodaira dimension 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' For Kodaira dimension 0, c1(X) = 0 ∈ H2(X, Q), hence remains to see what happens for c3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Some examples of simply connected Calabi-Yau three- folds with c3 = 0 have been constructed by Chad Schoen [Schoen88] and other examples were later found by Volker Braun [Bra12] as hypersurfaces in a toric fourfold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We want to discuss now the former examples by Schoen, and show some partial results which seem to indicate that they should not be birational to a quotient of a torus product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='.25 These examples are constructed as small resolutions of fibre products X = S1 ×P1 S2 where fi : Si → P1 is a rational elliptic surface with a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Let f : X → P1 be the fibre product of f1 and f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We are interested in the special case where the critical values of f1, f2 are different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' then the fibre product X is smooth, and all the fibres F of f are either a product of two elliptic curves, or the product of an elliptic curve with a degenerate fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence all the fibres F have Euler number e(F) = 0, and c3(X) = e(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The canonical divisor of X is trivial, for instance we can take the fi = Fi Gi to be given by a pencil of plane cubics with simple base points: then X is the small resolution of a hypersurface of bidegree (3, 3) X′ ⊂ P2 × P2, X′ = {(x, y)|F1(x)G2(y) = F2(y)G1(x)} hence X′ and X have trivial canonical divisor (see [Schoen88] page 181).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Moreover, essentially by the hyperplane theorem of Lefschetz, X is simply connected (see at any rate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1) of [Schoen88], page 181).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Thus X is a Calabi-Yau threefold with c3(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The Schoen threefolds X cannot be birationally covered by a 1-dimensional family of subvarieties which are isomorphic to a fixed Abelian surface T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Assume that X is birationally covered by a family T × C (where by the way C = P1 since q(X) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' For a general b ∈ C, Tb := T ×{b} cannot map to a fibre of f : X → P1, since f is the fibre product of f1 and f2 and we may assume that the fibrations fi do not have constant moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Hence Tb, which is a subvariety of X, dominates P1 through the morphism f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' But a linear system of dimension one on an Abelian surface yields a mor- phism to P1 only if the system is not ample, that is, the fibre is a union of translates of an elliptic curve E ⊂ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Varying b, the elliptic curve E is fixed (since it corresponds to a subgroup of the first homology group pf T), therefore all the fibres of f contain an elliptic curve isomorphic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' This is a contradiction, as the general fibres of f1 and f2 are not even isogenous to a fixed elliptic curve E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The Schoen threefolds X do not admit a fibration ψ : X → S onto a surface S such that the general fibre is isomorphic to a fixed elliptic curve E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Since X is simply connected, and the general fibre of ψ is connected, it follows that S is also simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Let D be the divisorial part of the set of critical values of ψ, and let D∗ be its smooth locus: define S∗ := (S \\ Sing(D)), X∗ = ψ−1(S∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 26 FABRIZIO CATANESE Then we have an exact sequence π1(E) → π1(X∗) → π1(S∗) → 1, and we observe that also X∗, S∗ are simply connected, since we have removed a subvariety of real codimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The image of π1(E) → π1(X∗) is the quotient of π1(E) by the local mon- odromies around the irreducible components Dj of D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' To understand these local monodromies, take a general curve section C of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The inverse image of C is an elliptic fibration Σ → C over C such that all the smooth fibres are isomorphic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The fibration is isotrivial, hence Σ = (C′ × E)/G, where C′ → C is Galois with group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' G acts on C′ × E via a product action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' If there are fixpoints for the action of G on C′, then, since the quotient is smooth, it follows that the isotropy subgroups act freely on E, hence by translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' So the local holomorphic monodromies are translations by torsion points, and the monodromy acts trivially on the first homology of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' The conclusion is that the image of π1(E) → π1(X∗) is an infinite group, and this is a contradiction since π1(X∗) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' □ The construction of Chad Schoen [Schoen88] applied to other elliptic fibra- tions leads to threefolds X with Kodaira dimension 1, and c3(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' We have not yet investigated whether we can achieve with this construction trivial Chern number c1(X)c2(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Acknowledgements: I would like to thank Francesco Baldassarri for bring- ing the paper [Bald56] by Mario Baldassarri to our attention, thus raising my interest in these questions, Ciro Ciliberto and Flaminio Flamini for pro- viding me with the text of [Dant43], Thomas Peternell for mentioning the article by Braun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Many thanks to Matthias Sch¨utt for bringing the examples of [Schoen88] to my attention and explaining their key features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Thanks to Pierre Deligne for answering an email query (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Thanks to Adriano Tomassini for pointing out Wang’s reference [Wang54] to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' References [AMN12] Jaume Amor´os, M`onica Manjar´ın, Marcel Nicolau: Deformations of K¨ahler manifolds with nonvanishing holomorphic vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} 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+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 30, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 3, 305–316 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' VANISHING CHERN CLASSES, MANIFOLDS ISOGENOUS TO A TORUS PRODUCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='.27 [BdF08] Giuseppe Bagnera, Michele de Franchis: Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti, Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' di Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' e di Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (3) 15, 253–343 (1908).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' [Bald56] Mario Baldassarri: Una caratterizzazione delle variet`a abeliane e pseudo- abeliane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Pura Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' (4) 42 (1956), 227–252.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' [CastEn15] Guido Castelnuovo, federigo Enriques: Die algebraischen Fl¨achen vom Gesichtspunkte der birationalen Transformation aus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Enz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' III C 6 b, 674–768 (Band III 2, Heft 6) (1915) (Memorie Scelte 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Stefan Schreieder: Equality in the Bogomolov-Miyaoka-Yau inequality in the non-general type case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 775 (2021), 87– 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' [HS21b] Feng Hao, Stefan Schreieder: Holomorphic one-forms without zeros on threefolds, Geom.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 5, 771–776 (1954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' Mathematisches Institut der Universit¨at Bayreuth, NW II, Universit¨atsstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content=' 30, 95447 Bayreuth Email address: fabrizio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='catanese@uni-bayreuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} +page_content='de Korea Institute for Advanced Study, Hoegiro 87, Seoul, 133–722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFKT4oBgHgl3EQfMi2t/content/2301.11751v1.pdf'} diff --git a/zdE4T4oBgHgl3EQfAQuo/content/tmp_files/2301.04842v1.pdf.txt b/zdE4T4oBgHgl3EQfAQuo/content/tmp_files/2301.04842v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f265091523ae93b2f2d26e6c59d4c74479fdd76 --- /dev/null +++ b/zdE4T4oBgHgl3EQfAQuo/content/tmp_files/2301.04842v1.pdf.txt @@ -0,0 +1,903 @@ +Towards High Performance One-Stage Human Pose Estimation +Ling Li1,2, Lin Zhao1,2†, Linhao Xu1, Jie Xu1 +1 PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education, +and Jiangsu Key Lab of Image and Video Understanding for Social Security, Nanjing University of Science and +Technology +2 State Key Laboratory of Integrated Services Networks (Xidian University) +{lingl,linzhao,linh,jiexu}@njust.edu.cn +Abstract +Making top-down human pose estimation method present both +good performance and high efficiency is appealing. Mask RCNN +can largely improve the efficiency by conducting person detection +and pose estimation in a single framework, as the features provided +by the backbone are able to be shared by the two tasks. However, +the performance is not as good as traditional two-stage methods. +In this paper, we aim to largely advance the human pose estimation +results of Mask-RCNN and still keep the efficiency. Specifically, +we make improvements on the whole process of pose estimation, +which contains feature extraction and keypoint detection. The part +of feature extraction is ensured to get enough and valuable infor- +mation of pose. Then, we introduce a Global Context Module into +the keypoints detection branch to enlarge the receptive field, as it is +crucial to successful human pose estimation. On the COCO val2017 +set, our model using the ResNet-50 backbone achieves an AP of 68.1, +which is 2.6 higher than Mask RCNN (AP of 65.5). Compared to +the classic two-stage top-down method SimpleBaseline, our model +largely narrows the performance gap (68.1 𝐴𝑃𝑘𝑝 vs. 68.9 𝐴𝑃𝑘𝑝) +with a much faster inference speed (77 ms vs. 168 ms), demonstrat- +ing the effectiveness of the proposed method. Code is available at: +https://github.com/lingl_space/maskrcnn_keypoint_refined. +CCS Concepts +• Computing methodologies → Activity recognition and un- +derstanding. +Keywords +human pose estimation, one-stage, efficiency +ACM Reference Format: +Ling Li1,2, Lin Zhao1,2†, Linhao Xu1, Jie Xu1. 2022. Towards High Per- +formance One-Stage Human Pose Estimation. In ACM Multimedia Asia +(MMAsia ’22), December 13–16, 2022, Tokyo, Japan. ACM, New York, NY, +USA, 5 pages. https://doi.org/10.1145/3551626.3564968 +† Corresponding authors. Permission to make digital or hard copies of all or part of +this work for personal or classroom use is granted without fee provided that copies are +not made or distributed for profit or commercial advantage and that copies bear this +notice and the full citation on the first page. Copyrights for components of this work +owned by others than ACM must be honored. Abstracting with credit is permitted. To +copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior +specific permission and/or a fee. Request permissions from permissions@acm.org. +MMAsia ’22, December 13–16, 2022, Tokyo, Japan +© 2022 Association for Computing Machinery. +ACM ISBN 978-1-4503-9478-9/22/12...$15.00 +https://doi.org/10.1145/3551626.3564968 +Figure 1: The trade-off between speed and accuracy. Infer- +ence time is measured on a single TITAN V GPU. +1 +Introduction +Multi-person pose estimation has always been a fundamental and +challenging task in computer vision. It has many complex down- +stream applications, such as human parsing [5, 15, 28, 39], action +recognition [1, 19, 24], human-computer interaction [3, 21], video +surveillance [14, 33, 34, 36], and animation generation [11, 12]. +With the continuous development of deep convolutional neural +networks (CNN), the task of human pose estimation has achieved +remarkable progress. The main ideas are roughly divided into two +groups: top-down methods [6, 27, 32, 40] and bottom-up methods +[2, 18, 22, 29]. Although bottom-up methods have better real-time +performance, it is difficult to achieve the same high performance as +top-down methods, due to the diversity of human instances scales. +Generally, top-down methods obtain more accurate keypoint local- +ization based on the detected bounding boxes. But because through- +out the process features cannot be shared by two independent steps, +the inference time can be much longer. +Mask RCNN [8] proposes the technique of RoIAlign to replace +the traditional RoIPooling, which solves the problem of misalign- +ment between CNN features and the input image. With this tech- +nique, Mask RCNN successfully integrates keypoint detection into +the framework of person instance detection. Thus, the efficiency +can be largely improved, since the keypoint detection branch can di- +rectly use the features from the backbone. However, Mask RCNN’s +performance is far behind traditional two-stage top-down methods +like SimpleBaseline [40]. We make a thorough investigation on the +keypoint detection process of Mask RCNN, and aim to largely refine +the results but still keep the efficiency (as shown in Fig. 1). +First of all, due to the inaccuracy of the candidate boxes, the +contextual information around some keypoints can easily exceed +the box and be discarded, it may affect the keypoint detection. More- +over, the Feature Pyramid Network (FPN) [16] can make the object +detection results robust to the various size. Yet pose estimation is a +pixel-level task, which requires finer feature granularity, and high +resolution is important to get accurate results, as demonstrated in +arXiv:2301.04842v1 [cs.CV] 12 Jan 2023 + +72 +MasKRCNN +SimpleBaselin +70 +PiiPal +HigherHRNet +68 +66 +64 +62 +50 +100 +150 +200 +250 +300 +infer.time(msMMAsia ’22, December 13–16, 2022, Tokyo, Japan +Ling Li, et al. +Figure 2: The refinements based on Mask RCNN. (a) de- +scribes the complete process. (b) explains different strate- +gies for FPN layer selection. The strategy of enlarging box +is shown in (c). +HRNet [32]. Thus, it may not be proper to use the same feature se- +lection strategy as object detection. Third, the perception of human +body structure is very crucial to pose estimation. It is necessary to +expand the receptive field to provide more contextual information. +However, the original design in Mask RCNN focuses more on local +information mining since the network depth is limited. +In this paper, we demonstrate that a comprehensive solution (as +shown in Fig. 2) of all the above aspects is able to improve the pose +estimation performance of Mask RCNN greatly, and still keep the +superior efficiency over traditional two-stage top-down methods. +2 +Related Work +2.1 +Traditional Two-Stage Methods +Top-down Methods. The top-down strategy is composed of +two phases: detecting all human instances for image cropping +and then acquiring pose by single person keypoint detector. Well- +known methods include Hourglass [23], Simplebaseline [40], HRNet +[32], PPNet [42] and so on. Since top-down methods normalize the +cropped image size, it is more robust to human instances of different +scales. And it is able to obtain relatively high-resolution features, +which is helpful to keypoint location. But the inference speed is +slow due to the inability to share computation and features with +the object detector. This main drawback makes the inference time +increase linearly with the number of human instances in one image. +Bottom-up Methods. Unlike top-down methods, bottom-up +methods predict all possible keypoints, and assign them to cor- +responding human instances [37]. There are various association +algorithms, such as dynamic programming [2], tag matching [22], +greedy decoding algorithm [26]. Their computational complexity is +not related to the number of human instances and can compute all +features once. However, it is difficult to obtain accurate localization +due to the sensitivity to instance scale changes, and the matching +during inference is an NP-hard problem with high complexity. +2.2 +One-Stage Methods +Two-stage methods, especially those using top-down strategies, +obtain good performances in accuracy, but have low inference effi- +ciency. So one-stage methods are proposed, which combine human +detection with pose estimation. A one-stage solution for pose esti- +mation, the SPM model [25], is presented by Nie et al. It uses a root +joint to represent the human body position and then uses offsets +to represent keypoints. While ensuring the speed advantage, this +scheme achieves performance on par with the two-stage bottom-up +(a) Keypoints miss detection situation +(b) The information needed for keypoints +Figure 3: (a) shows keypoint missed detections resulting +from training with either FPN layer. And (b) shows using +tight bounding boxes tends to discard important features. +methods. Recently, Geng et al. [7] present a competitive one-stage +method, DEKR, that employs a novel pose-specific neural network +to solve keypoint regression. Based on the foundation of anchor-free +object detector FCOS [35], Mao et al. propose a fully convolutional +multi-person pose estimation network, FCPose [20], which uses +dynamic instance-aware convolutions for pose estimation. +2.3 +Our approach +Mask RCNN [8] can also be viewed as a one-stage method. But +different from one-stage methods mentioned above, it also uses +top-down strategy as the bounding boxes are needed for separating +different persons. In this paper, we aim to thoroughly refine the +process of pose estimation based on the Mask RCNN framework, +achieving the performance close to two-stage top-down methods, +and keeping the efficiency of one-stage methods. +3 +Method +3.1 +Feature extraction during RoIAlign +3.1.1 +The selection of feature layers. The selection of output +feature layers in FPN [16] has a certain influence on the subsequent +tasks. We can regard keypoints as small-scale objects, thus requiring +strong spatial information for detection. Typically, RoIAlign uses +box regions to select in layers P2 to P5 of FPN to extract features. +However, the low resolution of high-level features can easily lead +to missed detection of keypoints. As shown in Fig. 3, we analyzed +the miss situation when taking features on each layer, and found +that small-scale persons have the most serious misses when taking +features on 𝑃4 and 𝑃5. Here, we propose to use only the 𝑃2 feature +layer for keypoint detection, which not only has high resolution and +more precise spatial information, but also retains strong semantic +information due to the top-down feature fusion process in FPN. +3.1.2 +The strategy for bounding box enlarging. The size of +bounding box determines the amount of pose-related information +the keypoint detection branch can obtain. We need to choose the +best-fit box size to provide sufficient and effective features. +We use heatmaps to represent sensitive feature regions during +keypoint detection, as shown in Fig. 3. The heat values represent +the importance of the surrounding contextual features. We map the +area back to the original image and use red boxes to represent the +box detection result. As can be seen, due to the given proposal box +is not always very accurate and may surround the person instance +tightly, RoIAlign is very probable to discard the features important + +features +for box +features +for keypoint +(b)layer selection +(c)enlarge +feature +Box +7x7 +Detection +post- +RolAlign +feature +Keypoint process Final +14x14 +Detection +Results +(a)structure0.25 +P2 +P3 +P4 +0.20 +P5 +0.15 +Gaps +AP +0.10 +0.05 +0.00 +xI +all +xxI +mTowards High Performance One-Stage Human Pose Estimation +MMAsia ’22, December 13–16, 2022, Tokyo, Japan +(a) The refined keypoint detection branch +(b) Fusion structure +Figure 4: Our keypoint branch and two fusion structures. +for keypoint detection, especially for the keypoints adjacent to +the edge of detection boxes. It can be seen from the visualization +results in Fig. 3 that shoulders, elbows, wrists, and ankles are more +prone to be out-of-bounds. The loss of keypoint-sensitive features is +irreparable, which will have a great impact on the results. To solve +this problem, we experimentally determine that a magnification of +1.3× is appropriate. Meanwhile, we fill the area beyond image with +black backgrounds to prevent human body center deviation. +3.2 +The Global Context Module +A successful pose estimation model needs to incorporate contextual +information in a sufficiently large receptive field to learn the com- +plex relationship of body parts. Mask RCNN uses 8 convolutions +for detection with a receptive field of 17, which is not enough. +3.2.1 +The Structure of Module. Blindly stacking convolutions +to enlarge the receptive field will greatly increase the model size and +reduce the inference speed. Inspired by the successful applications +of transformer models like ViT [41], we utilize the multi-head self- +attention (MHSA) to solve the problem. +We refer to the added part as Global Context Module (GCM). As +shown in the Fig. 4, it is composed of two parts. One is a multi-head +self-attention module referring to the BoTNet [31], which is used to +extract global information. The other is a 3×3 convolution kernel to +perform preliminary integration of global features while aligning +the channels. In addition, the residual structure is added. By using +this module, we can provide the model with global information, +and the performance of keypoint detection is able to be improved. +3.2.2 +Fusion Structure. After acquiring global features, we fuse +them with local features in a series connection. As shown in the +Fig. 4 (a), first, we use the GCM to extract global context, and then +use 4 convolutions to further process the features. This process is +repeated twice to obtain the information for final pose estimation. +4 +Experiment results +4.1 +Dataset +The COCO dataset [17] contains over 250K person instances, each +of which contains 17 annotated keypoints. We train our model on +the COCO train2017, which contains 57K images and 150K human +instances. The model is then validated on COCO val2017 and COCO +test-dev2017, with 5K images and 20K images respectively. +(a) Enlarging boxes of different sizes +(b) Using different layers of FPN +Figure 5: Ablation study of Different Feature Extraction +Strategies in RoIAlign. The 1.0x magnification and multi- +select experiments refer to the strategy in the Mask RCNN. +4.2 +Implementation details +We use Detectron2 [38] for the model implementation. The model +is trained using a single TITAN V GPU with Stochastic Gradient +Descent. We initialize with a backbone model pretrained on the +ImageNet classification task [9]. The base learning rate is initially +set to 0.0025, and is reduced to a tenth of the original at 480K and +640K iterations in the 1× training schedule (i.e., 720K iterations). In +terms of data augmentation, we adjust the longer side of images to +be less than or equal to 1333 and the shorter side to be between 640 +and 800. Random flipping is used to enhance the learning ability. +All inference processes are measured on a single TITAN V GPU. +The shapes of the images are adjusted to have a length of 800 on the +short side, or no more than 1333 on the long side. Most importantly +considering the high efficiency maintenance, we do not use flip +operation and multi-scale testing strategy to produce better results. +4.3 +Ablation Experiments +In this section, we investigate the performance of our improved +model in 1x training schedule with ResNet-50 as the backbone. +4.3.1 +Comparison of Different Feature Extraction Strategies +in RoIAlign. We set the magnification of bounding boxes at inter- +vals of 0.05 in the range of 1 to 1.5 for experiments. Fig. 5 (a) shows +the experimental results. The AP increases almost linearly from 1.0 +to 1.3 due to the increased information around keypoints. However, +AP decreases slowly when the magnification exceeds 1.3 due to the +introduction of a lot of useless background features. Thus, we select +to enlarge bounding box by 1.3x to obtain the best performance. +As for the selection of feature levels, we also conduct separate +experiments for each level. The results are shown in Fig. 5 (b). +Mask RCNN uses the sizes of bounding boxes to select different +feature layers from FPN, which only obtain 64 AP. Using the 𝑃2 +layer alone to acquire features can achieve the best performance. +The experiments using the 𝑃4 and 𝑃5 layer alone show severe AP +drops, by 1.3 and 4.4 AP respectively. Referring to the results of +error analysis [30] in Fig. 3, we conclude that keypoint missed +detection is particularly serious when taking features at the 𝑃4 or +𝑃5 layer. +Based on the above experiments, we enlarge bounding boxes by +1.3x and select the 𝑃2 layer to get pose-related features. The AP is +improved by 2.1 from 64.0 to 66.1 using the ResNet-50 backbone. +4.3.2 +The Design of the Keypoint Branch. We present two +fusion methods in Fig. 4. The results given in Table 2 show that + +Conv. +in +Rn(H x 1 x d) +Up. +BN +content-position +MHSA +Rw(l x W xd) +BN +14x14 +x256 +Cony. +GCM +out +14x14 +x512 +Conv. x 4 +14x14 +x512 +Up +56x56 +x17 +Keypoint +DetectionGM +Conv. x4 +Conv. x4 +GM +Conv. x4 +GM +Conv. x4 +Conv. x4 +GM +Conv. x465.5 +65.3 +65.3 +65.1 +65.1 +64.9 +64.764 +64.7 +64 +64.5 +64.3 +64.1 +63.9 +63.7 +63.5 +1.05 +11 +1.15 +1.2 +1.25 +1.3 +1.35 +1.4 +1.45 +1.565 +64.8 +64 +64 +63 +60 +59 +multi-select +P2 +P3 +P4 +P5MMAsia ’22, December 13–16, 2022, Tokyo, Japan +Ling Li, et al. +Table 1: Comparison of experimental results. Mask RCNN uses the implementation on Detectron2, and − represents testing +with the boxes we measured, and turning off the flip test. The rescoring network is not used when DEKR tests. +COCO val2017 +COCO test-dev 2017 +Method +backbone +infer. +(ms) +𝐴𝑃𝑘𝑝 +𝐴𝑃𝑘𝑝 +50 +𝐴𝑃𝑘𝑝 +75 +𝐴𝑃𝑘𝑝 +𝑀 +𝐴𝑃𝑘𝑝 +𝐿 +𝐴𝑃𝑘𝑝 +𝐴𝑃𝑘𝑝 +50 +𝐴𝑃𝑘𝑝 +75 +𝐴𝑃𝑘𝑝 +𝑀 +𝐴𝑃𝑘𝑝 +𝐿 +Top-down +G-RMI [27] +ResNet-101 +- +- +- +- +- +- +64.9 +85.5 +71.3 +62.3 +70.0 +SimpleBaseline- [40] +ResNet-50 +168 +68.9 +88.2 +76.5 +65.5 +75.2 +67.7 +89.0 +75.2 +64.4 +73.6 +SimpleBaseline- [40] +ResNet-101 +192 +70.3 +89.0 +78.0 +67.0 +76.4 +69.0 +89.9 +77.1 +65.9 +74.7 +HRNet- [32] +HRNet-W32 +- +73.4 +90.0 +80.6 +69.7 +79.9 +71.5 +90.5 +79.6 +68.1 +77.4 +Bottom-up +SimplePose [13] +IMHN +- +66.1 +85.9 +71.6 +59.8 +76.2 +68.5 +86.7 +74.9 +66.4 +71.9 +PersonLab [26] +ResNet-152 +- +66.5 +86.2 +71.9 +62.3 +73.2 +66.5 +88.0 +72.6 +62.4 +72.3 +PifPaf [10] +ResNet-152 +263 +67.4 +- +- +- +- +66.7 +87.8 +73.6 +62.4 +72.9 +HigherHRNet [4] +HRNet-W32 +128 +67.1 +86.2 +73.0 +61.5 +76.1 +64.7 +86.9 +71.0 +60.2 +71.2 +One-Stage +CenterNet [43] +Hourglass +147 +64.0 +85.6 +70.2 +59.4 +72.1 +63.0 +86.8 +69.6 +58.9 +70.4 +Mask RCNN [8] +ResNet-50 +72 +65.5 +87.2 +71.1 +61.3 +73.4 +63.1 +87.3 +68.7 +57.8 +71.4 +Mask RCNN [8] +ResNet-101 +83 +66.1 +87.4 +72.0 +61.5 +74.4 +- +- +- +- +- +FCPose [20] +ResNet-50 +68 +- +- +- +- +- +64.3 +87.3 +71.0 +61.6 +70.5 +FCPose [20] +ResNet-101 +93 +- +- +- +- +- +65.6 +87.9 +72.6 +62.1 +72.3 +DEKR [7] +HRNet-W32 +63 +67.2 +86.3 +73.8 +61.7 +77.1 +66.6 +87.6 +73.5 +61.2 +75.6 +SPM [25] +Hourglass +- +- +- +- +- +- +66.9 +88.5 +72.9 +62.6 +73.1 +Our method +ResNet-50 +77 +68.1 +88.0 +74.5 +63.7 +76.2 +66.4 +88.4 +73.1 +62.2 +73.9 +Our method +ResNet-101 +90 +68.3 +88.0 +74.8 +63.8 +76.6 +67.1 +89.0 +74.0 +63.0 +74.7 +Table 2: Comparative experiments on network structures. +Method +𝐴𝑃𝑘𝑝 +𝐴𝑃𝑘𝑝 +50 +𝐴𝑃𝑘𝑝 +75 +𝐴𝑃𝑘𝑝 +𝑀 +𝐴𝑃𝑘𝑝 +𝐿 +Mask RCNN +64.0 +86.0 +69.7 +59.6 +72.1 +𝐺𝐶𝑀𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 +64.6 +86.2 +70.2 +60.0 +72.9 +𝐺𝐶𝑀𝑠𝑒𝑟𝑖𝑒𝑠 +65.2 +86.8 +71.1 +60.7 +73.4 +the series fusion method leads to better performance. One possible +explanation is that obtaining some global context information is +more beneficial to subsequent local feature extraction. +4.3.3 +Combination of All Techniques. Finally, we combine +all the refinement strategies used above. The final AP can be im- +proved to 66.8, outperforming original Mask RCNN by 2.8 AP, which +demonstrates the effectiveness of the proposed method. +4.4 +Comparison with the State-of-the-art Methods +In this section, we compare our model with other methods. Here +our model and Mask RCNN are trained using 3× training schedule. +The results of other methods are obtained using their public trained +models or from the corresponding papers. We report the results of +our model and other state-of-the-art methods in Table 1. +4.4.1 +Comparison with Two-Stage Methods. We mainly com- +pare our method with SimpleBaseline [40]. Because SimpleBaseline +is the most classic and its structure is as simple as ours. For a fair +comparison, we turn off the flip test, and the person instance de- +tection boxes from our models (Person AP of 55.4) are used. The +performance of our method still lags behind 0.8 AP and 2.0 AP +under ResNet-50 and ResNet-101 backbone respectively. However +considering the raw performance gap between Mask RCNN and +SimpleBaseline, this has been narrowed largely. Moreover, using the +ResNet-50 backbone, our method averagely takes 77 ms to detect +keypoints from one image, which is much faster than SimpleBase- +line’s 168 ms. Under the ResNet-101 backbone, our method is still +more than 2x faster. Because high efficiency is as important as per- +formance in real applications, our method may be a better solution. +Compared to other two-stage top-down methods like the famous +HRNet [32], the performance of our method is not competitive, +however, their inference time is much larger than ours. +Furthermore, we compare our model with bottom-up methods. +Our model using the ResNet-50 backbone can achieve higher detec- +tion performance than HigherHRNet using the HRNet-W32 back- +bone. Compared with most bottom-up methods, we have advan- +tages both in inference time and detection performance. +4.4.2 +Comparison with One-Stage Methods. Compared with +the benchmark model Mask RCNN, our model can be closer to +the state-of-the-art top-down methods while maintaining real-time +performance. Meanwhile, compared with other one-stage methods, +our model also obtains better results. Our model with the ResNet-50 +backbone outperforms CenterNet [43] 4.1 AP with about half the in- +ference time. And when we turn off the rescoring network in DEKR +[7] for a fair comparison, our model achieves better performance +with a small backbone ( 68.1 AP vs. 67.2 AP). +4.4.3 +Results on COCO test-dev Our model is able to achieve +better or comparable AP performance on the COCO test-dev2017 +compared to most bottom-up and one-stage methods. Our model +using the ResNet-101 backbone achieves 67.1 AP. Compared to +the most recent FCPose [20] with the same backbone, our model +receives 2.1 AP (ResNet-50) and 1.5 AP (ResNet-101) improvements, +respectively. +5 +Conclusion +In this paper, we make thorough improvements to the Mask RCNN +based human pose estimation. The experimental results demon- +strate that our refinement largely narrows the performance gap +between one-stage and two-stage top-down methods, and we suc- +cessfully keep the superior inference efficiency. This success mainly +stems from a more suitable feature extraction strategy in RoIAlign +and a better network structure for keypoint detection: (1) select the +P2 layer of FPN to ensure high-resolution spatial information; (2) +enlarge the person instance bounding box to prevent the loss of use- +ful features of human pose; and (3) add global feature information +through a Global Context Module to increase the receptive field for +human pose estimation. Our improved model, using ResNet-50 as +the backbone, achieved 66.8 AP with 1x training schedule, which is +2.8 AP higher than Mask RCNN. We hope our research and find- +ings can inspire further study on one-stage human pose estimation +methods, which can be more popular in real applications. + +Towards High Performance One-Stage Human Pose Estimation +MMAsia ’22, December 13–16, 2022, Tokyo, Japan +Acknowledgments +This work was supported by NSF of China (No. 62172222), China +Postdoctoral Science Foundation (No. 2020M681609). +References +[1] Fabien Baradel, Christian Wolf, Julien Mille, and Graham W Taylor. 2018. Glimpse +clouds: Human activity recognition from unstructured feature points. In Proceed- +ings of the IEEE Conference on Computer Vision and Pattern Recognition. 469–478. +[2] Zhe Cao, Tomas Simon, Shih-En Wei, and Yaser Sheikh. 2017. Realtime multi- +person 2d pose estimation using part affinity fields. In Proceedings of the IEEE +Conference on Computer Vision and Pattern Recognition. 7291–7299. +[3] Zhe Chen, Jing Zhang, and Dacheng Tao. 2021. Recursive context routing for +object detection. 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Objects as points. +arXiv preprint arXiv:1904.07850 (2019). + diff --git a/zdE4T4oBgHgl3EQfAQuo/content/tmp_files/load_file.txt b/zdE4T4oBgHgl3EQfAQuo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f645d4122aa1c3710183c2650f7a533828f7bd0 --- /dev/null +++ b/zdE4T4oBgHgl3EQfAQuo/content/tmp_files/load_file.txt @@ -0,0 +1,683 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf,len=682 +page_content='Towards High Performance One-Stage Human Pose Estimation Ling Li1,2, Lin Zhao1,2†, Linhao Xu1, Jie Xu1 1 PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education, and Jiangsu Key Lab of Image and Video Understanding for Social Security, Nanjing University of Science and Technology 2 State Key Laboratory of Integrated Services Networks (Xidian University) {lingl,linzhao,linh,jiexu}@njust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='cn Abstract Making top-down human pose estimation method present both good performance and high efficiency is appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Mask RCNN can largely improve the efficiency by conducting person detection and pose estimation in a single framework, as the features provided by the backbone are able to be shared by the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' However, the performance is not as good as traditional two-stage methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' In this paper, we aim to largely advance the human pose estimation results of Mask-RCNN and still keep the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Specifically, we make improvements on the whole process of pose estimation, which contains feature extraction and keypoint detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The part of feature extraction is ensured to get enough and valuable infor- mation of pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Then, we introduce a Global Context Module into the keypoints detection branch to enlarge the receptive field, as it is crucial to successful human pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' On the COCO val2017 set, our model using the ResNet-50 backbone achieves an AP of 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1, which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 higher than Mask RCNN (AP of 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Compared to the classic two-stage top-down method SimpleBaseline, our model largely narrows the performance gap (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 𝐴𝑃𝑘𝑝 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 𝐴𝑃𝑘𝑝) with a much faster inference speed (77 ms vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 168 ms), demonstrat- ing the effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Code is available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='com/lingl_space/maskrcnn_keypoint_refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' CCS Concepts Computing methodologies → Activity recognition and un- derstanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Keywords human pose estimation, one-stage, efficiency ACM Reference Format: Ling Li1,2, Lin Zhao1,2†, Linhao Xu1, Jie Xu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Towards High Per- formance One-Stage Human Pose Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' In ACM Multimedia Asia (MMAsia ’22), December 13–16, 2022, Tokyo, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' ACM, New York, NY, USA, 5 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1145/3551626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3564968 † Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' MMAsia ’22, December 13–16, 2022, Tokyo, Japan © 2022 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' ACM ISBN 978-1-4503-9478-9/22/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1145/3551626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3564968 Figure 1: The trade-off between speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Infer- ence time is measured on a single TITAN V GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 1 Introduction Multi-person pose estimation has always been a fundamental and challenging task in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' It has many complex down- stream applications, such as human parsing [5, 15, 28, 39], action recognition [1, 19, 24], human-computer interaction [3, 21], video surveillance [14, 33, 34, 36], and animation generation [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' With the continuous development of deep convolutional neural networks (CNN), the task of human pose estimation has achieved remarkable progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The main ideas are roughly divided into two groups: top-down methods [6, 27, 32, 40] and bottom-up methods [2, 18, 22, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Although bottom-up methods have better real-time performance, it is difficult to achieve the same high performance as top-down methods, due to the diversity of human instances scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Generally, top-down methods obtain more accurate keypoint local- ization based on the detected bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' But because through- out the process features cannot be shared by two independent steps, the inference time can be much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Mask RCNN [8] proposes the technique of RoIAlign to replace the traditional RoIPooling, which solves the problem of misalign- ment between CNN features and the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' With this tech- nique, Mask RCNN successfully integrates keypoint detection into the framework of person instance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Thus, the efficiency can be largely improved, since the keypoint detection branch can di- rectly use the features from the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' However, Mask RCNN’s performance is far behind traditional two-stage top-down methods like SimpleBaseline [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We make a thorough investigation on the keypoint detection process of Mask RCNN, and aim to largely refine the results but still keep the efficiency (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' First of all, due to the inaccuracy of the candidate boxes, the contextual information around some keypoints can easily exceed the box and be discarded, it may affect the keypoint detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' More- over, the Feature Pyramid Network (FPN) [16] can make the object detection results robust to the various size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Yet pose estimation is a pixel-level task, which requires finer feature granularity, and high resolution is important to get accurate results, as demonstrated in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='04842v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='CV] 12 Jan 2023 72 MasKRCNN SimpleBaselin 70 PiiPal HigherHRNet 68 66 64 62 50 100 150 200 250 300 infer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='time(msMMAsia ’22, December 13–16, 2022, Tokyo, Japan Ling Li, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Figure 2: The refinements based on Mask RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' (a) de- scribes the complete process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' (b) explains different strate- gies for FPN layer selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The strategy of enlarging box is shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' HRNet [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Thus, it may not be proper to use the same feature se- lection strategy as object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Third, the perception of human body structure is very crucial to pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' It is necessary to expand the receptive field to provide more contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' However, the original design in Mask RCNN focuses more on local information mining since the network depth is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' In this paper, we demonstrate that a comprehensive solution (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 2) of all the above aspects is able to improve the pose estimation performance of Mask RCNN greatly, and still keep the superior efficiency over traditional two-stage top-down methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 Traditional Two-Stage Methods Top-down Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The top-down strategy is composed of two phases: detecting all human instances for image cropping and then acquiring pose by single person keypoint detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Well- known methods include Hourglass [23], Simplebaseline [40], HRNet [32], PPNet [42] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Since top-down methods normalize the cropped image size, it is more robust to human instances of different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' And it is able to obtain relatively high-resolution features, which is helpful to keypoint location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' But the inference speed is slow due to the inability to share computation and features with the object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' This main drawback makes the inference time increase linearly with the number of human instances in one image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Bottom-up Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Unlike top-down methods, bottom-up methods predict all possible keypoints, and assign them to cor- responding human instances [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' There are various association algorithms, such as dynamic programming [2], tag matching [22], greedy decoding algorithm [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Their computational complexity is not related to the number of human instances and can compute all features once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' However, it is difficult to obtain accurate localization due to the sensitivity to instance scale changes, and the matching during inference is an NP-hard problem with high complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 One-Stage Methods Two-stage methods, especially those using top-down strategies, obtain good performances in accuracy, but have low inference effi- ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' So one-stage methods are proposed, which combine human detection with pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' A one-stage solution for pose esti- mation, the SPM model [25], is presented by Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' It uses a root joint to represent the human body position and then uses offsets to represent keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' While ensuring the speed advantage, this scheme achieves performance on par with the two-stage bottom-up (a) Keypoints miss detection situation (b) The information needed for keypoints Figure 3: (a) shows keypoint missed detections resulting from training with either FPN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' And (b) shows using tight bounding boxes tends to discard important features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Recently, Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' [7] present a competitive one-stage method, DEKR, that employs a novel pose-specific neural network to solve keypoint regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Based on the foundation of anchor-free object detector FCOS [35], Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' propose a fully convolutional multi-person pose estimation network, FCPose [20], which uses dynamic instance-aware convolutions for pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 Our approach Mask RCNN [8] can also be viewed as a one-stage method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' But different from one-stage methods mentioned above, it also uses top-down strategy as the bounding boxes are needed for separating different persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' In this paper, we aim to thoroughly refine the process of pose estimation based on the Mask RCNN framework, achieving the performance close to two-stage top-down methods, and keeping the efficiency of one-stage methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 3 Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 Feature extraction during RoIAlign 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 The selection of feature layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The selection of output feature layers in FPN [16] has a certain influence on the subsequent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We can regard keypoints as small-scale objects, thus requiring strong spatial information for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Typically, RoIAlign uses box regions to select in layers P2 to P5 of FPN to extract features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' However, the low resolution of high-level features can easily lead to missed detection of keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 3, we analyzed the miss situation when taking features on each layer, and found that small-scale persons have the most serious misses when taking features on 𝑃4 and 𝑃5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Here, we propose to use only the 𝑃2 feature layer for keypoint detection, which not only has high resolution and more precise spatial information, but also retains strong semantic information due to the top-down feature fusion process in FPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 The strategy for bounding box enlarging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The size of bounding box determines the amount of pose-related information the keypoint detection branch can obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We need to choose the best-fit box size to provide sufficient and effective features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We use heatmaps to represent sensitive feature regions during keypoint detection, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The heat values represent the importance of the surrounding contextual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We map the area back to the original image and use red boxes to represent the box detection result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' As can be seen, due to the given proposal box is not always very accurate and may surround the person instance tightly, RoIAlign is very probable to discard the features important features for box features for keypoint (b)layer selection (c)enlarge feature Box 7x7 Detection post- RolAlign feature Keypoint process Final 14x14 Detection Results (a)structure0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='25 P2 P3 P4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='20 P5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='15 Gaps AP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='00 xI all xxI mTowards High Performance One-Stage Human Pose Estimation MMAsia ’22, December 13–16, 2022, Tokyo, Japan (a) The refined keypoint detection branch (b) Fusion structure Figure 4: Our keypoint branch and two fusion structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' for keypoint detection, especially for the keypoints adjacent to the edge of detection boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' It can be seen from the visualization results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 3 that shoulders, elbows, wrists, and ankles are more prone to be out-of-bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The loss of keypoint-sensitive features is irreparable, which will have a great impact on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' To solve this problem, we experimentally determine that a magnification of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3× is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Meanwhile, we fill the area beyond image with black backgrounds to prevent human body center deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 The Global Context Module A successful pose estimation model needs to incorporate contextual information in a sufficiently large receptive field to learn the com- plex relationship of body parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Mask RCNN uses 8 convolutions for detection with a receptive field of 17, which is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 The Structure of Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Blindly stacking convolutions to enlarge the receptive field will greatly increase the model size and reduce the inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Inspired by the successful applications of transformer models like ViT [41], we utilize the multi-head self- attention (MHSA) to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We refer to the added part as Global Context Module (GCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' As shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4, it is composed of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' One is a multi-head self-attention module referring to the BoTNet [31], which is used to extract global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The other is a 3×3 convolution kernel to perform preliminary integration of global features while aligning the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' In addition, the residual structure is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' By using this module, we can provide the model with global information, and the performance of keypoint detection is able to be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 Fusion Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' After acquiring global features, we fuse them with local features in a series connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' As shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4 (a), first, we use the GCM to extract global context, and then use 4 convolutions to further process the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' This process is repeated twice to obtain the information for final pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4 Experiment results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 Dataset The COCO dataset [17] contains over 250K person instances, each of which contains 17 annotated keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We train our model on the COCO train2017, which contains 57K images and 150K human instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The model is then validated on COCO val2017 and COCO test-dev2017, with 5K images and 20K images respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' (a) Enlarging boxes of different sizes (b) Using different layers of FPN Figure 5: Ablation study of Different Feature Extraction Strategies in RoIAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0x magnification and multi- select experiments refer to the strategy in the Mask RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 Implementation details We use Detectron2 [38] for the model implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The model is trained using a single TITAN V GPU with Stochastic Gradient Descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We initialize with a backbone model pretrained on the ImageNet classification task [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The base learning rate is initially set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0025, and is reduced to a tenth of the original at 480K and 640K iterations in the 1× training schedule (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=', 720K iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' In terms of data augmentation, we adjust the longer side of images to be less than or equal to 1333 and the shorter side to be between 640 and 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Random flipping is used to enhance the learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' All inference processes are measured on a single TITAN V GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The shapes of the images are adjusted to have a length of 800 on the short side, or no more than 1333 on the long side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Most importantly considering the high efficiency maintenance, we do not use flip operation and multi-scale testing strategy to produce better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 Ablation Experiments In this section, we investigate the performance of our improved model in 1x training schedule with ResNet-50 as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 Comparison of Different Feature Extraction Strategies in RoIAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We set the magnification of bounding boxes at inter- vals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='05 in the range of 1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 5 (a) shows the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The AP increases almost linearly from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 due to the increased information around keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' However, AP decreases slowly when the magnification exceeds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 due to the introduction of a lot of useless background features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Thus, we select to enlarge bounding box by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3x to obtain the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' As for the selection of feature levels, we also conduct separate experiments for each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Mask RCNN uses the sizes of bounding boxes to select different feature layers from FPN, which only obtain 64 AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Using the 𝑃2 layer alone to acquire features can achieve the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The experiments using the 𝑃4 and 𝑃5 layer alone show severe AP drops, by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 AP respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Referring to the results of error analysis [30] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 3, we conclude that keypoint missed detection is particularly serious when taking features at the 𝑃4 or 𝑃5 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Based on the above experiments, we enlarge bounding boxes by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3x and select the 𝑃2 layer to get pose-related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The AP is improved by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 from 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 to 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 using the ResNet-50 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 The Design of the Keypoint Branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We present two fusion methods in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The results given in Table 2 show that Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' in Rn(H x 1 x d) Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' BN content-position MHSA Rw(l x W xd) BN 14x14 x256 Cony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' GCM out 14x14 x512 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' x 4 14x14 x512 Up 56x56 x17 Keypoint DetectionGM Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' x4 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' x4 GM Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' x4 GM Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' x4 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' x4 GM Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' x465.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='565 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 64 64 63 60 59 multi-select P2 P3 P4 P5MMAsia ’22, December 13–16, 2022, Tokyo, Japan Ling Li, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Table 1: Comparison of experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Mask RCNN uses the implementation on Detectron2, and − represents testing with the boxes we measured, and turning off the flip test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The rescoring network is not used when DEKR tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' COCO val2017 COCO test-dev 2017 Method backbone infer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' (ms) 𝐴𝑃𝑘𝑝 𝐴𝑃𝑘𝑝 50 𝐴𝑃𝑘𝑝 75 𝐴𝑃𝑘𝑝 𝑀 𝐴𝑃𝑘𝑝 𝐿 𝐴𝑃𝑘𝑝 𝐴𝑃𝑘𝑝 50 𝐴𝑃𝑘𝑝 75 𝐴𝑃𝑘𝑝 𝑀 𝐴𝑃𝑘𝑝 𝐿 Top-down G-RMI [27] ResNet-101 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 SimpleBaseline- [40] ResNet-50 168 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 SimpleBaseline- [40] ResNet-101 192 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 HRNet- [32] HRNet-W32 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 71.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 71.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 PifPaf [10] ResNet-152 263 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 HigherHRNet [4] HRNet-W32 128 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 One-Stage CenterNet [43] Hourglass 147 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 Mask RCNN [8] ResNet-50 72 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 Mask RCNN [8] ResNet-101 83 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 FCPose [20] ResNet-50 68 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 FCPose [20] ResNet-101 93 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 DEKR [7] HRNet-W32 63 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 SPM [25] Hourglass 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 Our method ResNet-50 77 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 Our method ResNet-101 90 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 Table 2: Comparative experiments on network structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Method 𝐴𝑃𝑘𝑝 𝐴𝑃𝑘𝑝 50 𝐴𝑃𝑘𝑝 75 𝐴𝑃𝑘𝑝 𝑀 𝐴𝑃𝑘𝑝 𝐿 Mask RCNN 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 𝐺𝐶𝑀𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='9 𝐺𝐶𝑀𝑠𝑒𝑟𝑖𝑒𝑠 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 the series fusion method leads to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' One possible explanation is that obtaining some global context information is more beneficial to subsequent local feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 Combination of All Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Finally, we combine all the refinement strategies used above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The final AP can be im- proved to 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8, outperforming original Mask RCNN by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 AP, which demonstrates the effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4 Comparison with the State-of-the-art Methods In this section, we compare our model with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Here our model and Mask RCNN are trained using 3× training schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The results of other methods are obtained using their public trained models or from the corresponding papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We report the results of our model and other state-of-the-art methods in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 Comparison with Two-Stage Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We mainly com- pare our method with SimpleBaseline [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Because SimpleBaseline is the most classic and its structure is as simple as ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' For a fair comparison, we turn off the flip test, and the person instance de- tection boxes from our models (Person AP of 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The performance of our method still lags behind 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 AP and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='0 AP under ResNet-50 and ResNet-101 backbone respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' However considering the raw performance gap between Mask RCNN and SimpleBaseline, this has been narrowed largely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Moreover, using the ResNet-50 backbone, our method averagely takes 77 ms to detect keypoints from one image, which is much faster than SimpleBase- line’s 168 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Under the ResNet-101 backbone, our method is still more than 2x faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Because high efficiency is as important as per- formance in real applications, our method may be a better solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Compared to other two-stage top-down methods like the famous HRNet [32], the performance of our method is not competitive, however, their inference time is much larger than ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Furthermore, we compare our model with bottom-up methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Our model using the ResNet-50 backbone can achieve higher detec- tion performance than HigherHRNet using the HRNet-W32 back- bone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Compared with most bottom-up methods, we have advan- tages both in inference time and detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 Comparison with One-Stage Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Compared with the benchmark model Mask RCNN, our model can be closer to the state-of-the-art top-down methods while maintaining real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Meanwhile, compared with other one-stage methods, our model also obtains better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Our model with the ResNet-50 backbone outperforms CenterNet [43] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 AP with about half the in- ference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' And when we turn off the rescoring network in DEKR [7] for a fair comparison, our model achieves better performance with a small backbone ( 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 AP vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='2 AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='3 Results on COCO test-dev Our model is able to achieve better or comparable AP performance on the COCO test-dev2017 compared to most bottom-up and one-stage methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Our model using the ResNet-101 backbone achieves 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Compared to the most recent FCPose [20] with the same backbone, our model receives 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='1 AP (ResNet-50) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='5 AP (ResNet-101) improvements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' 5 Conclusion In this paper, we make thorough improvements to the Mask RCNN based human pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' The experimental results demon- strate that our refinement largely narrows the performance gap between one-stage and two-stage top-down methods, and we suc- cessfully keep the superior inference efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' This success mainly stems from a more suitable feature extraction strategy in RoIAlign and a better network structure for keypoint detection: (1) select the P2 layer of FPN to ensure high-resolution spatial information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' (2) enlarge the person instance bounding box to prevent the loss of use- ful features of human pose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' and (3) add global feature information through a Global Context Module to increase the receptive field for human pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Our improved model, using ResNet-50 as the backbone, achieved 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 AP with 1x training schedule, which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content='8 AP higher than Mask RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' We hope our research and find- ings can inspire further study on one-stage human pose estimation methods, which can be more popular in real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfAQuo/content/2301.04842v1.pdf'} +page_content=' Towards High Performance One-Stage Human Pose Estimation MMAsia ’22, December 13–16, 2022, Tokyo, Japan Acknowledgments This work was supported by NSF of China (No.' metadata={'source': 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